Advances in Microlocal and Time-Frequency Analysis (Applied and Numerical Harmonic Analysis) [1 ed.] 3030361373, 9783030361372

Table of contents :
ANHA Series Preface
Preface
Contents
Anisotropic Gevrey-Hörmander Pseudo-Differential Operators on Modulation Spaces
1 Introduction
2 Preliminaries
2.1 Weight Functions
2.2 Gelfand-Shilov Spaces
2.3 Short Time Fourier Transforms and Gelfand-Shilov Spaces
2.4 A Broad Family of Modulation Spaces
2.5 Pseudo-Differential Operators
2.6 Symbol Classes
3 Continuity on Modulation Spaces for Pseudo-Differential Operators with Symbols of Infinite Order
References
Hardy Spaces on Weighted Homogeneous Trees
1 Introduction
2 Weighted Homogeneous Trees
2.1 Doubling and Local Doubling Properties
2.2 Admissible Trapezoids and Calderón–Zygmund Sets
3 The Maximal Function
4 Hardy Spaces
4.1 Equivalence of Spaces H1,p(μ) for p(1,∞]
4.2 Real Interpolation Properties of H1(μ)
4.3 Boundedness of Singular Integrals on H1(μ)
References
The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces
1 Introduction and Main Result
2 Preliminaries
3 Well-Posedness for a Scalar Schrödinger-Type Equation
4 Well-Posedness for First Order Systems
5 Proof of Theorem 1
References
Cone-Adapted Shearlets and Radon Transforms
1 Introduction
2 Preliminaries
2.1 Notation
2.2 The Wavelet Transform
2.3 The Shearlet Transform
2.4 The Radon Transform
2.5 The Radon Transform Intertwines Wavelets and Shearlets
3 Cone-Adapted Shearlets and Radon Transforms
4 Generalizations
References
Linear Perturbations of the Wigner Transform and the Weyl Quantization
1 Introduction
2 Preliminaries
2.1 Function Spaces
2.2 Basic Properties of M1
2.3 Bilinear Coordinate Transformations
2.4 Partial Fourier Transforms
3 Matrix-Wigner Distributions
3.1 Connection to the Short-Time Fourier Transform
3.2 Main Properties of the Transformation BA
3.3 Cohen Class Members as Perturbations of the Wigner Transform
3.3.1 Main Properties of the Cohen Class
4 Pseudodifferential Operators
4.1 Boundedness Results
4.1.1 Operators on Lebesgue Spaces
4.1.2 Operators on Modulation Spaces
4.2 Symbols in the Sjöstrand Class
References
About the Nuclearity of S(Mp) and Sω
1 Introduction and Preliminaries
2 Results for the Space S(Mp)
3 Results for the Space Sω and Examples
References
Spaces of Ultradifferentiable Functions of Multi-anisotropic Type
1 Introduction
2 The Space EM( Ω)
3 The Space Es,Γ(Ω)
4 The Space of Multi-Anisotropic Gevrey Vectors
5 The Characterization of the Space Es,Γ(Ω)
6 The Space EM,Γ( Ω)
References
Comparison Principle for Non-cooperative Elliptic Systems and Applications
1 Introduction
2 Comparison Principle for Non-Cooperative Elliptic Systems
3 Existence of Classical Solution for Linear Non-Cooperative Elliptic System
References
On the Simple Layer Potential Ansatz for the n-Dimensional Helmholtz Equation
1 Introduction
2 Definitions
3 Reduction of a Certain Integral Equation
4 Representation Theorem
References
Decay Estimates and Gevrey Smoothing for a Strongly Damped Plate Equation
1 Introduction
2 Proof of Theorem 1
References
Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation
1 Introduction
2 Notation
3 Fundamental Solution and Decay Estimates
Appendix
References
On Density Operators with Gaussian Weyl Symbols
1 Introduction
2 Density Operators
2.1 Basic Definitions
2.2 Reduced Density Operators
2.3 Gaussian Symbols
2.4 A Lemma on Gaussians
2.5 Partial Traces of Gaussian Density Operators
3 Separability of Gaussian Density Operators
3.1 The Notion of Separability
3.2 The Gaussian Case: Necessary and Sufficient Conditions
References
On the Solvability of a Class of Second Order Degenerate Operators
1 Introduction
2 The Mixed-Type Case
2.1 Example
2.2 Example
2.3 Example
2.4 Example: A Mildly Complex Case
3 The Schrödinger-Type Case
3.1 Example
3.2 Example
3.3 Example
4 The Mixed-Schrödinger-Type Case
4.1 Example
4.2 Example
4.3 Example
5 Concluding Remarks
References
Small Data Solutions for Semilinear Waves with Time-Dependent Damping and Mass Terms
1 Introduction
2 The Linear Estimates
2.1 Examples
3 Proof of Theorem 1
References
Integrating Gauge Fields in the ζ-Formulation of Feynman's Path Integral
1 Introduction
2 The Free Real Scalar Quantum Field
2.1 The ζ-Regularized Vacuum Expectation Values of Hn and Hζ
2.2 The N→∞ Particle Limit
3 Free Complex Scalar Quantum Fields
4 The Dirac Field
5 Coupling a Fermion of Mass m to Light in 1+1 Dimensions
6 Conclusion
References
A Class of Well-Posed Parabolic Final Value Problems
1 Introduction
1.1 Background: Phenomena of Instability
1.2 Main Tool: Injectivity
1.3 The Abstract Final Value Problem
2 Proof of Theorem 1
3 The Heat Problem with Final Time Data
3.1 The Boundary Homogeneous Case
3.2 The Inhomogeneous Case
References
Localization of a Class of Muckenhoupt Weights by Using Mellin Pseudo-Differential Operators
1 Introduction
2 The C*-Algebras SO and QC
2.1 The C*-Algebra SO of Slowly Oscillating Functions
2.2 The C*-Algebra QC of Quasicontinuous Functions
3 The Banach Algebras Zp,w and Zp,wπ
4 Muckenhoupt Weights
4.1 Submultiplicative Functions and Their Indices
4.2 Functions in VMO0(Gamma) and SO0(Gamma)
4.3 A Subclass of Muckenhoupt Weights
4.4 Weights Locally Equivalent to Slowly Oscillating Muckenhoupt Weights
5 Mellin Pseudo-Differential Operators and Their Applications
5.1 Boundedness and Compactness of Mellin Pseudo-Differential Operators
5.2 Symbols of Mellin Pseudo-Differential Operators
5.3 Applications of Mellin Pseudo-Differential Operators
6 Localization of Muckenhoupt Weights Satisfying Condition (A)
References
Carleman Regularization and Hyperfunctions
1 Introduction
2 Notations and Review of Hyperfunctions
3 Carleman Regularization: Preparations
4 The Regularization
5 The Representation Theorems
6 Representation of Real Analytic Functions
References
Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space
1 Introduction
2 Statement of Results
3 Examples and Remarks
4 Definitions and Tools
4.1 Pseudodifferential Operators with Limited Smoothness
4.1.1 Mapping Properties
4.1.2 Composition, Adjoint and Sharp Gårding's Inequality
5 Proof
5.1 Regularization
5.2 Symbol Space
5.3 Transformation to a First-Order System
5.4 Diagonalization
5.4.1 First Step of Diagonalization
5.4.2 Second Step of Diagonalization
5.5 Conjugation
5.6 Conclusion
6 Concluding Remarks
References
Quantization and Coorbit Spaces for Nilpotent Groups
1 Introduction
2 Framework
3 Weyl Systems, the Fourier-Wigner Transform
4 Pseudo-Differential Operators
5 Phase-Space Shifts
6 Coorbit Spaces—A Short Overview
References
On the Measurability of Stochastic Fourier Integral Operators
1 Introduction
2 Classical Theory of Oscillatory Integrals and FIOs
2.1 Oscillatory Integrals
2.2 Classical Theory of FIOs
3 Stochastic Fourier Integral Operators
4 Applications
References
Convolution and Anti-Wick Quantisation on Ultradistribution Spaces
1 Introduction
2 Preliminaries
3 Existence of S*-Convolution and D*-Convolution
4 Convolution with the Gaussian Kernel
4.1 The Fourier-Laplace Transform on Ultradistributions
4.2 Convolution with the Gaussian Kernel
5 Anti-Wick Quantisation
6 An Extension of Convolution
References
Exact Formulas to the Solutions of Several Generalizations of the Nonlinear Schrödinger Equation
1 Introduction
2 Proof of Theorem 1
3 Proof of Theorem 2
References
Dirichlet-to-Neumann Operator and Zaremba Problem
1 Introduction
2 Reduction of Neumann Conditions to the Boundary
3 Reduction of Mixed Problems to the Boundary
4 The Relationship to the Edge Calculus
References
Extended Gevrey Regularity via the Short-Time Fourier Transform
1 Introduction
1.1 Basic Notation
2 Preliminaries
2.1 Extended Gevrey Regularity
2.2 The Associated Function to the Sequence Mτ,σp=pτp σ
2.3 Modulation Spaces
3 Decay Properties of the STFT
4 Wave Front Sets WFτ,σ and STFT
References
Wiener Estimates on Modulation Spaces
1 Introduction
2 Preliminaries
2.1 Gelfand-Shilov Spaces and Gevrey Classes
2.2 Ordered, Dual and Phase Split Bases
2.3 Invariant Quasi-Banach Spaces and Spaces of Mixed Quasi-Normed Spaces of Lebesgue Types
2.4 Modulation and Wiener Spaces
2.5 Classes of Periodic Elements
3 Estimates on Wiener Spaces and Periodic Elements in Modulation Spaces
3.1 Estimates of Wiener Spaces
3.2 Wiener Estimates on Short-Time Fourier Transforms, and Modulation Spaces
3.3 Periodic Elements in Modulation Spaces
References
The Gabor Wave Front Set of Compactly Supported Distributions
1 Introduction
2 Preliminaries
3 The Gabor and the Classical Wave Front Sets
4 Propagation of Singularities for Schrödinger Equations
References
Applied and Numerical Harmonic Analysis (99 volumes)

Citation preview

Applied and Numerical Harmonic Analysis

Paolo Boggiatto, Marco Cappiello, Elena Cordero, Sandro Coriasco, Gianluca Garello, Alessandro Oliaro, Jörg Seiler Editors

Advances in Microlocal and Time-Frequency Analysis

Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA

Advisory Editors Akram Aldroubi Vanderbilt University Nashville, TN, USA

Gitta Kutyniok Technical University of Berlin Berlin, Germany

Douglas Cochran Arizona State University Phoenix, AZ, USA

Mauro Maggioni Johns Hopkins University Baltimore, MD, USA

Hans G. Feichtinger University of Vienna Vienna, Austria

Zuowei Shen National University of Singapore Singapore, Singapore

Christopher Heil Georgia Institute of Technology Atlanta, GA, USA

Thomas Strohmer University of California Davis, CA, USA

Stéphane Jaffard University of Paris XII Paris, France

Yang Wang Hong Kong University of Science & Technology Kowloon, Hong Kong

Jelena Kovaˇcevi´c Carnegie Mellon University Pittsburgh, PA, USA

More information about this series at http://www.springer.com/series/4968

Paolo Boggiatto • Marco Cappiello • Elena Cordero • Sandro Coriasco • Gianluca Garello • Alessandro Oliaro • J¨org Seiler Editors

Advances in Microlocal and Time-Frequency Analysis

Editors Paolo Boggiatto Dipartimento di Matematica “G. Peano” Universit`a degli Studi di Torino Torino, Italy

Marco Cappiello Dipartimento di Matematica “G. Peano” Universit`a degli Studi di Torino Torino, Italy

Elena Cordero Dipartimento di Matematica “G. Peano” Universit`a degli Studi di Torino Torino, Italy

Sandro Coriasco Dipartimento di Matematica “G. Peano” Universit`a degli Studi di Torino Torino, Italy

Gianluca Garello Dipartimento di Matematica “G. Peano” Universit`a degli Studi di Torino Torino, Italy

Alessandro Oliaro Dipartimento di Matematica “G. Peano” Universit`a degli Studi di Torino Torino, Italy

J¨org Seiler Dipartimento di Matematica “G. Peano” Universit`a degli Studi di Torino Torino, Italy

ISSN 2296-5009 ISSN 2296-5017 (electronic) Applied and Numerical Harmonic Analysis ISBN 978-3-030-36137-2 ISBN 978-3-030-36138-9 (eBook) https://doi.org/10.1007/978-3-030-36138-9 Mathematics Subject Classification (2010): 35Lxx, 35Pxx, 35Qxx, 35Sxx, 42Axx, 42Bxx, 43Axx, 47G30, 47Nxx, 58Jxx, 58J40 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state-of-theart ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theory depends not only on classical Fourier analysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influence of wavelet theory in the aforementioned topics illustrates the justification for providing a means for centralizing and disseminating information from the broader, but still focused, area of harmonic analysis. This will be a key role of ANHA. We intend to publish with the scope and interaction that such a host of issues demands. v

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ANHA Series Preface

Along with our commitment to publish mathematically significant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish major advances in the following applicable topics in which harmonic analysis plays a substantial role: Antenna theory Prediction theory Biomedical signal processing Radar applications Digital signal processing Sampling theory Fast algorithms Spectral estimation Gabor theory and applications Speech processing Image processing Time-frequency and Numerical partial differential equations time-scaleanalysis Wavelet theory The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields. In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, e.g., the concept of “function.” Since the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, for example, by the mechanical synthesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineering, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers, but also provides the proper notion of spectrum for phenomena such as white light; this latter process leads to the Fourier analysis associated with correlation functions in filtering and prediction problems, and these problems, in turn, deal naturally with Hardy spaces in the theory of complex variables. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or the

ANHA Series Preface

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adaptive modeling inherent in time-frequency-scale methods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’être of the ANHA series! University of Maryland College Park, MD, USA

John J. Benedetto Series Editor

Preface

This volume contains contributions to the Conference Microlocal and TimeFrequency Analysis 2018 (MLTFA18) which took place at Torino University from 2nd to 6th July 2018. The event was organized in honor of Professor Luigi Rodino in the occasion of his 70th birthday. The choice of the conference title and the contents of the talks reflect the research interests of Luigi on his long and extremely prolific career at Torino University, enlightening as well the connections between the two above-mentioned main broad areas of modern mathematics, namely Microlocal and Time-Frequency Analysis. The starting ideas which laid the basis for Microlocal Analysis, broadly including pseudo-differential operators, wave front sets, propagation of singularities, hypoellipticity, Gevrey classes, etc., date back approximately to the 1960s with the pioneering works of J. J. Kohn and L. Nirenberg, which were promptly systematized by L. Hörmander. Luigi, after having spent a couple of years in the early 1970s in Sweden visiting L. Hörmander, was the first who brought these ideas to Torino University, which by that time had no experts in this new promising field. Soon he engaged not only in an extremely fruitful research activity but, as a dedicated teacher, he also started to mentor a number of students which constantly grew with time. Torino became in this way a renowned center for Microlocal Analysis that had many contacts with other research groups in Italy and abroad, among others the Universities of Bologna, Cagliari, Ferrara, and Padova, the University of Paris-Sud, the Max-Plank-Arbeitsgruppe in Potsdam, and the Bulgarian Academy of Sciences. The beginning of the new millennium brought new interests in addition to the old ones. Following contacts with the NuHAG group in Vienna and the York University in Toronto, the group of Luigi started to extend its research activities also to the area of Time-Frequency Analysis, which in the meantime had revealed deep connections with many aspects of Microlocal Analysis, especially for what concerns various types of pseudo-differential calculi. Since then Luigi has been actively working in both areas as researcher, editor, and mentor for an impressive number of students.

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Besides the contacts in Toronto and Vienna, numerous collaborations were born by that time which are still active today, among them those with the Universities of Hannover, Novi Sad, Valencia, and Växjö. This volume is necessarily restricted to contributions addressing only some of the topics related to the vast research activity of Luigi. In a certain sense, the volume is also incomplete because it focuses exclusively on the aspect of mathematical research, whereas the positive contributions of Luigi during all these years by far have not only been confined to this. His rare ability to present in his lectures the deepest concepts in a natural and simple way, always pointing directly and precisely at the core of their meaning, makes him a wonderful teacher highly appreciated among students at all levels. Not less worth mentioning is Luigi’s gentle character. All his students and colleagues know his patience and understanding of human nature in every circumstance of life. His calm and serene way of facing and handling mathematical and everyday problems has always been an example of constant positive inspiration for our group, which reaches far beyond the mere achievement of new mathematical results. We hope that his encouraging and inspiring guidance will accompany our group for many years to come. Happy Birthday Luigi! Torino, Italy Torino, Italy Torino, Italy Torino, Italy Torino, Italy Torino, Italy Torino, Italy

Paolo Boggiatto Marco Cappiello Elena Cordero Sandro Coriasco Gianluca Garello Alessandro Oliaro Jörg Seiler

Contents

Anisotropic Gevrey-Hörmander Pseudo-Differential Operators on Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmed Abdeljawad and Joachim Toft 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Continuity on Modulation Spaces for Pseudo-Differential Operators with Symbols of Infinite Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hardy Spaces on Weighted Homogeneous Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laura Arditti, Anita Tabacco, and Maria Vallarino 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Weighted Homogeneous Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Maximal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alessia Ascanelli 1 Introduction and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Well-Posedness for a Scalar Schrödinger-Type Equation . . . . . . . . . . . . . . . . . . . 4 Well-Posedness for First Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cone-Adapted Shearlets and Radon Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Francesca Bartolucci, Filippo De Mari, and Ernesto De Vito 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Cone-Adapted Shearlets and Radon Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 74 77

Linear Perturbations of the Wigner Transform and the Weyl Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Dominik Bayer, Elena Cordero, Karlheinz Gröchenig, and S. Ivan Trapasso 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3 Matrix-Wigner Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 About the Nuclearity of S(Mp ) and Sω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiara Boiti, David Jornet, and Alessandro Oliaro 1 Introduction and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Results for the Space S(Mp ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results for the Space Sω and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spaces of Ultradifferentiable Functions of Multi-anisotropic Type . . . . . . . . Chikh Bouzar 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Space E M (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Space E s,Γ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Space of Multi-Anisotropic Gevrey Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Characterization of the Space E s,Γ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Space E M,Γ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison Principle for Non-cooperative Elliptic Systems and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Georgi Boyadzhiev and Nikolay Kutev 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Comparison Principle for Non-Cooperative Elliptic Systems. . . . . . . . . . . . . . . 3 Existence of Classical Solution for Linear Non-Cooperative Elliptic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Simple Layer Potential Ansatz for the n-Dimensional Helmholtz Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alberto Cialdea, Vita Leonessa, and Angelica Malaspina 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Reduction of a Certain Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 123 125 129 131 131 132 134 137 139 141 143 145 145 147 150 155 157 157 158 159 161 168

Contents

Decay Estimates and Gevrey Smoothing for a Strongly Damped Plate Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marcello D’Abbicco 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marcello D’Abbicco and Alessandra De Luca 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fundamental Solution and Decay Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

169 169 174 179 181 181 183 184 186 189

On Density Operators with Gaussian Weyl Symbols . . . . . . . . . . . . . . . . . . . . . . . . Maurice A. de Gosson 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Separability of Gaussian Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191

On the Solvability of a Class of Second Order Degenerate Operators . . . . . Serena Federico and Alberto Parmeggiani 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Mixed-Type Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Schrödinger-Type Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Mixed-Schrödinger-Type Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207

Small Data Solutions for Semilinear Waves with Time-Dependent Damping and Mass Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giovanni Girardi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Linear Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrating Gauge Fields in the ζ -Formulation of Feynman’s Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tobias Hartung and Karl Jansen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Free Real Scalar Quantum Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Free Complex Scalar Quantum Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 192 201 205

207 211 218 221 224 225 227 227 233 236 239 241 241 244 250 251

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5 Coupling a Fermion of Mass m to Light in 1 + 1 Dimensions . . . . . . . . . . . . . 253 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 A Class of Well-Posed Parabolic Final Value Problems. . . . . . . . . . . . . . . . . . . . . . Jon Johnsen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Heat Problem with Final Time Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Localization of a Class of Muckenhoupt Weights by Using Mellin Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yu. I. Karlovich 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The C ∗ -Algebras SO  and QC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Banach Algebras Zp,w and Zπp,w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Muckenhoupt Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Mellin Pseudo-Differential Operators and Their Applications . . . . . . . . . . . . . . 6 Localization of Muckenhoupt Weights Satisfying Condition (A). . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carleman Regularization and Hyperfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Otto Liess 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notations and Review of Hyperfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Carleman Regularization: Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Representation of Real Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Lorenz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Definitions and Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 259 267 271 279 281 281 285 287 289 295 305 309 311 311 313 314 318 319 323 326 327 327 332 334 337 342 357 358

Quantization and Coorbit Spaces for Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . 361 M. M˘antoiu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

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xv

3 Weyl Systems, the Fourier-Wigner Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Phase-Space Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Coorbit Spaces—A Short Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

366 369 374 375 380

On the Measurability of Stochastic Fourier Integral Operators . . . . . . . . . . . . Michael Oberguggenberger and Martin Schwarz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Classical Theory of Oscillatory Integrals and FIOs . . . . . . . . . . . . . . . . . . . . . . . . . 3 Stochastic Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383

Convolution and Anti-Wick Quantisation on Ultradistribution Spaces . . . Stevan Pilipovi´c and Bojan Prangoski 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Existence of S†∗ -Convolution and D∗ -Convolution. . . . . . . . . . . . . . . . . . . . . . . . . 4 Convolution with the Gaussian Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Anti-Wick Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 An Extension of Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403

Exact Formulas to the Solutions of Several Generalizations of the Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Petar Popivanov and Angela Slavova 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirichlet-to-Neumann Operator and Zaremba Problem . . . . . . . . . . . . . . . . . . . . B.-W. Schulze 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Reduction of Neumann Conditions to the Boundary . . . . . . . . . . . . . . . . . . . . . . . . 3 Reduction of Mixed Problems to the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Relationship to the Edge Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended Gevrey Regularity via the Short-Time Fourier Transform . . . . . . Nenad Teofanov and Filip Tomi´c 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Decay Properties of the STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Wave Front Sets WFτ,σ and STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383 384 386 397 401

403 405 407 410 413 415 417 419 419 421 426 428 431 431 432 434 438 452 455 455 457 466 471 473

xvi

Contents

Wiener Estimates on Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joachim Toft 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Estimates on Wiener Spaces and Periodic Elements in Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

475

The Gabor Wave Front Set of Compactly Supported Distributions. . . . . . . . Patrik Wahlberg 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Gabor and the Classical Wave Front Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Propagation of Singularities for Schrödinger Equations . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

507

475 479 492 503

507 508 510 516 519

Anisotropic Gevrey-Hörmander Pseudo-Differential Operators on Modulation Spaces Ahmed Abdeljawad and Joachim Toft

Abstract We show continuity properties for the pseudo-differential operator Op(a) 0 (ω, ω ∈ P ), from M(ω0 ω, B) to M(ω, B), for fixed s, σ ≥ 1, ω, ω0 ∈ Ps,σ 0 s,σ σ,s;0 σ,s a ∈ (ω0 ) (a ∈ (ω0 ) ), and B is an invariant Banach function space. Keywords Pseudo-differential operators · Modulation spaces · Banach function spaces · Gelfand-Shilov spaces · Gevrey regularity

1 Introduction In this paper we consider pseudo-differential operators, whose symbols are of infinite order and possess suitable Gevrey regularities and which are allowed to grow sub-exponentially together with all their derivatives. It is well-known that such operators are continuous on suitable Gelfand-Shilov spaces and their distribution spaces (cf. [2, 4, 9]). In the isotropic case, it is also proved in [42] that such operators are continuous between suitable modulation spaces. Here we extend the results from [42] to the anisotropic case. More precisely, the symbols of the pseudo-differential operators depend on the phase space variables (x, ξ ), and in contrast to [42], we here allow the Gevrey parameters with respect to x and ξ to be different. As remarked in [1], it is well-motivated to consider such extensions, because in general, there are no links between such parameters. (See also [37] for analogous investigations within the framework of the ordinary distribution theory.)

A. Abdeljawad Department of Mathematics, University of Turin, Turin, Italy e-mail: [email protected] J. Toft () Department of Mathematics, Linnæus University, Växjö, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_1

1

2

A. Abdeljawad and J. Toft

More specifically, we consider pseudo-differential operators with symbols satisfying conditions of the form |∂xα ∂ξ a(x, ξ )|  h|α+β| α!σ β!s ω0 (x, ξ ), β

(1)

where ω0 should be a moderate weight on R2d and satisfy boundedness conditions like ω0 (x, ξ )  er(|x|

1 1 s +|ξ | σ

)

.

(2)

For such symbols a we prove that corresponding pseudo-differential operators Op(a) are continuous from the modulation space M(ω0 ω, B) to M(ω, B). Here ω, ω0 are moderate weights and B is a translation invariant BF-space. (See Sect. 2 for notations.) In particular, B can be any mixed normed space of Lebesgue type or an Orlicz space. Especially we note that the family of such modulation spaces p,q include the classical modulation spaces, M(ω) (Rd ), introduced by Feichtinger in [14]. Similar investigations were performed in [42] in the case s = σ (i. e. the isotropic case). Therefore, the results in the current paper are more general in the sense of the anisotropicity of the considered symbol classes. In our investigations, we shall use similar technique as in [42], but modify the arguments slightly to show that the arguments from [37] can also be used. We notice that in [37], the conditions (1) and (2) are replaced by β

|∂xα ∂ξ a(x, ξ )|  ω0 (x, ξ ) and ω0 (x, ξ )  (1 + |x| + |ξ |)N , respectively, for some N ≥ 0. That is, in [37], less regularity is required compared to [42] (as well as in Sect. 3). On the other hand, in [37], the symbols are not allowed to grow faster than polynomials, while in [42] and Sect. 3, they are allowed to grow subexponentially. In [14], H. Feichtinger introduced the modulation spaces to measure the timefrequency concentration of a function or distribution on the time-frequency space or the phase space R2d . Nowadays they become popular among mathematicians and engineers because of their numerous applications in signal processing [16, 17], pseudo-differential and Fourier integral operators [5–7, 29, 30, 33–42] and quantum mechanics [8, 11]. The paper is organized as follows. In Sect. 2 we give the main definitions and basic properties of Gelfand-Shilov and modulation spaces and recall some essential results. In Sect. 3 we state our main results on the continuity with anisotropic settings.

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

3

2 Preliminaries In the current section we review basic properties for modulation spaces and other related spaces. More details and proofs can be found in [12–14, 18–21, 23, 39]. In what follows we write f (θ )  g(θ ), θ ∈  ⊆ Rd , if there is a constant c > 0 such that |f (θ )| ≤ c|g(θ )| for all θ ∈ . Moreover, if f (θ )  g(θ ) and g(θ )  f (θ ) for all θ ∈ , we write f g.

2.1 Weight Functions d A function ω on Rd is called a weight or weight function, if ω, 1/ω ∈ L∞ loc (R ) are d positive everywhere. The weight ω on R is called v-moderate for some weight v on Rd , if

ω(x + y)  ω(x)v(y),

x, y ∈ Rd .

(3)

If v is even and satisfies (3) with ω = v, then v is called submultiplicative. Let s, σ > 0. Then we let PE (Rd ) be the set of all moderate weights on Rd , Ps (Rd ) (Ps0 (Rd )) be the set of all ω ∈ PE (Rd ) such that 1 s

ω(x + y)  ω(x)er|y| ,

x, y ∈ Rd ,

0 (R2d )) be the set of all for some r > 0 (for every r > 0), and Ps,σ (R2d ) (Ps,σ 2d ω ∈ PE (R ) such that

ω(x + y, ξ + η)  ω(x, ξ )er(|y|

1 1 s +|η| σ

)

,

x, y, ξ, η ∈ Rd ,

(4)

for some r > 0 (for every r > 0). The following result shows that for any weight in PE , there are equivalent weights that satisfy strong Gevrey regularity. Proposition 1 Let ω ∈ PE (R2d ) and s, σ > 0. Then there exists a weight ω0 ∈ PE (R2d ) ∩ C ∞ (R2d ) such that the following is true: (1) ω0 ω; (2) for every h > 0, |∂xα ∂ξ ω0 (x, ξ )|  h|α+β| α!σ β!s ω0 (x, ξ ) h|α+β| α!σ β!s ω(x, ξ ). β

Proposition 1 is equivalent to [2, Proposition 1.6]. In fact, by Proposition [2, Proposition 1.6] we have that Proposition 1 holds with s = σ . Hence, Proposition 1 implies [2, Proposition 1.6]. On the other hand, let s0 = min(s, σ ). Then [2,

4

A. Abdeljawad and J. Toft

Proposition 1.6] implies that there is a weight function ω0 ω satisfying |∂xα ∂ξ ω0 (x, ξ )|  h|α+β| (α!β!)s0 ω0 (x, ξ ) β

 h|α+β| α!σ β!s ω0 (x, ξ ), giving Proposition 1. 0 (R2d ) are non-increasing with respect to s It is evident that Ps,σ (R2d ) and Ps,σ and σ . On the other hand we have 0 Ps,σ (R2d ) ⊆ Ps,σ (R2d ) = Ps0 ,σ0 (R2d ),

where

s0 = max(1, s), σ0 = max(1, σ ),

(5)

in view of Proposition 1.6 in [1]. It is expected that this may reduce issues in the anisotropic to isotropic situations.

2.2 Gelfand-Shilov Spaces Let 0 < h, s, σ ∈ R be fixed. Then Sσs;h (Rd ) is the Banach space of all f ∈ C ∞ (Rd ) such that f Sσs;h ≡ sup sup

α,β∈Nd x∈Rd

|x α ∂ β f (x)| < ∞, h|α+β| α!s β!σ

(6)

endowed with the norm (6). The Gelfand-Shilov spaces Sσs (Rd ) and sσ (Rd ) are defined as the inductive and projective limits respectively of Sσs;h (Rd ). This implies that Sσs (Rd ) =

 h>0

Sσs;h (Rd ) and

sσ (Rd ) =



Sσs;h (Rd ),

(7)

h>0

and that the topology for Sσs (Rd ) is the strongest possible one such that the inclusion map from Sσs;h (Rd ) to Sσs (Rd ) is continuous, for every choice of h > 0. The space sσ (Rd ) is a Fréchet space with seminorms · Sσs;h , h > 0. Moreover, sσ (Rd ) = {0}, if and only if s + σ ≥ 1 and (s, σ ) = ( 12 , 12 ), and Sσs (Rd ) = {0}, if and only if s + σ ≥ 1. The Gelfand-Shilov distribution spaces (Sσs ) (Rd ) and (sσ ) (Rd ) are the projective and inductive limits respectively of (Sσs;h ) (Rd ). In [28] it is proved that (Sσs ) (Rd ) is the dual of Sσs (Rd ), and (sσ ) (Rd ) is the dual of sσ (Rd ) (also in topological sense), see also [22].

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

5

The Fourier transform F is the linear and continuous map on S (Rd ), given by the formula  d f (x)e−ix,ξ  dx (F f )(ξ ) = f(ξ ) ≡ (2π )− 2 Rd

when f ∈ S (Rd ). Here  · , ·  denotes the usual scalar product on Rd . The Fourier transform extends uniquely to homeomorphisms from (Sσs ) (Rd ) to (Ssσ ) (Rd ), and from (sσ ) (Rd ) to (σs ) (Rd ). Furthermore, it restricts to homeomorphisms from Sσs (Rd ) to Ssσ (Rd ), and from sσ (Rd ) to σs (Rd ). Some considerations later on involve a broader family of Gelfand-Shilov spaces. More precisely, for sj , σj ∈ R+ , j = 1, 2, the Gelfand-Shilov spaces Sσs11,s,σ22 (Rd1 +d2 ) and sσ11,s,σ22 (Rd1 +d2 ) consist of all functions F ∈ C ∞ (Rd1 +d2 ) such that |x1α1 x2α2 ∂xβ11 ∂xβ22 F (x1 , x2 )|  h|α1 +β1 |+|α2 +β2 | α1 !s1 α2 !s2 β1 !σ1 β2 !σ2

(8)

for some h > 0 respectively for every h > 0. The duals (Sσs11,s,σ22 ) (Rd1 +d2 )

and

(sσ11,s,σ22 ) (Rd1 +d2 )

Sσs11,s,σ22 (Rd1 +d2 )

and

sσ11,s,σ22 (Rd1 +d2 ),

of

respectively, and their topologies are defined in analogous ways as for the spaces Sσs (Rd ) and sσ (Rd ) above. The following proposition explains mapping properties of partial Fourier transforms on Gelfand-Shilov spaces, and follows by similar arguments as in analogous situations in [22]. The proof is therefore omitted. Here, F1 F and F2 F are the partial Fourier transforms of F (x1 , x2 ) with respect to x1 ∈ Rd1 and x2 ∈ Rd2 , respectively. Proposition 2 Let sj , σj > 0, j = 1, 2. Then the following is true: (1) the mappings F1 and F2 on S (Rd1 +d2 ) restrict to homeomorphisms d1 +d2 2 ) F1 : Sσs11,s,σ22 (Rd1 +d2 ) → Ssσ11,σ ,s2 (R

and ,s2 F2 : Sσs11,s,σ22 (Rd1 +d2 ) → Sσs11,σ (Rd1 +d2 ); 2

6

A. Abdeljawad and J. Toft

(2) the mappings F1 and F2 on S (Rd1 +d2 ) are uniquely extendable to homeomorphisms d1 +d2 2  ) F1 : (Sσs11,s,σ22 ) (Rd1 +d2 ) → (Ssσ11,σ ,s2 ) (R

and ,s2  F2 : (Sσs11,s,σ22 ) (Rd1 +d2 ) → (Sσs11,σ ) (Rd1 +d2 ). 2

The same holds true if the Sσs11,s,σ22 -spaces and their duals are replaced by corresponding sσ11,s,σ22 -spaces and their duals. The next proposition follows from [10]. The proof is therefore omitted. Proposition 3 Let sj , σj > 0, j = 1, 2. Then the following conditions are equivalent. (1) F ∈ Sσs11,s,σ22 (Rd1 +d2 ) (F ∈ sσ11,s,σ22 (Rd1 +d2 )); (2) for some h > 0 (for every h > 0) it holds |F (x1 , x2 )|  e−h(|x1 |

1 s1

1

+|x2 | s2 )

and

(ξ1 , ξ2 )|  e−h(|ξ1 | |F

1 σ1

1

+|ξ2 | σ2 )

.

We notice that if sj + σj < 1 for some j = 1, 2, then Sσs11,s,σ22 (Rd1 +d2 ) and sσ11,s,σ22 (Rd1 +d2 ) are equal to the trivial space {0}. Likewise, if sj = σj = 12 for some j = 1, 2, then sσ11,s,σ22 (Rd1 +d2 ) = {0}.

2.3 Short Time Fourier Transforms and Gelfand-Shilov Spaces We recall here some basic facts about the short-time Fourier transform. Let φ ∈ Sσs (Rd ) \ 0 (φ ∈ sσ (Rd ) \ 0) be fixed. Then the short-time Fourier transform of f ∈ (Sσs ) (Rd ) (f ∈ (sσ ) (Rd )) is given by d

(Vφ f )(x, ξ ) = (2π )− 2 (f, φ( · − x)ei · ,ξ  )L2 . Here ( · , · )L2 is the unique extension of the L2 -form on Sσs (Rd ) (sσ (Rd )) to a continuous sesqui-linear form on (Sσs ) (Rd ) × Sσs (Rd ) ((sσ ) (Rd ) × sσ (Rd )). In the case f ∈ Lp (Rd ), for some p ∈ [1, ∞], then Vφ f is given by d

Vφ f (x, ξ ) ≡ (2π )− 2

 Rd

f (y)φ(y − x)e−iy,ξ  dy.

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

7

The following characterizations of the Sσs11,s,σ22 (Rd1 +d2 ), sσ11,s,σ22 (Rd1 +d2 ) and their duals follow by similar arguments as in the proofs of Propositions 2.1 and 2.2 in [40]. The details are left for the reader. Proposition 4 Let sj , σj > 0 be such that sj + σj ≥ 1, j = 1, 2, s0 ≤ s and σ0 ≤ σ . Also let φ ∈ Sσs11,s,σ22 (Rd1 +d2 \ 0) (φ ∈ sσ11,s,σ22 (Rd1 +d2 \ 0) and let f be a GelfandShilov distribution on Rd1 +d2 . Then f ∈ Sσs11,s,σ22 (Rd1 +d2 ) (f ∈ sσ11,s,σ22 (Rd1 +d2 )), if and only if |Vφ f (x1 , x2 , ξ1 , ξ2 )|  e−r(|x1 |

1 s1

1

1

1

+|x2 | s2 +|ξ1 | σ1 +|ξ2 | σ2 )

,

(9)

holds for some r > 0 (holds for every r > 0). A proof of Proposition 4 can be found in e. g. [24] (cf. [24, Theorem 2.7]). The corresponding result for Gelfand-Shilov distributions is the following extension of [38, Theorem 2.5]. Proposition 5 Let sj , σj > 0 be such that sj + σj ≥ 1, j = 1, 2, s0 ≤ s and t0 ≤ t. Also let φ ∈ Sσs11,s,σ22 (Rd1 +d2 ) \ 0 and let f be a Gelfand-Shilov distribution on Rd1 +d2 . Then the following is true: (1) f ∈ (Sσs11,s,σ22 ) (Rd1 +d2 ), if and only if |Vφ f (x1 , x2 , ξ1 , ξ2 )|  er(|x1 |

1 s1

1

1

1

+|x2 | s2 +|ξ1 | σ1 +|ξ2 | σ2 )

(10)

holds for every r > 0; (2) if in addition φ ∈ sσ11,s,σ22 (Rd1 +d2 ) \ 0, then f ∈ (sσ11,s,σ22 ) (Rd1 +d2 ), if and only if |Vφ f (x1 , x2 , ξ1 , ξ2 )|  er(|x1 |

1 s1

1

1

1

+|x2 | s2 +|ξ1 | σ1 +|ξ2 | σ2 )

(11)

holds for some r > 0. We omit the proof of Proposition 5, since the result follows by similar arguments as in the proof of [38, Theorem 2.5].

2.4 A Broad Family of Modulation Spaces We recall that a quasi-norm · B of order r ∈ (0, 1] on the vector-space B over C is a nonnegative functional on B which satisfies 1

f + g B ≤ 2 r −1 ( f B + g B ), α · f B = |α| · f B ,

f, g ∈ B, α ∈ C,

f ∈B

(12)

8

A. Abdeljawad and J. Toft

and f B = 0

⇔

f = 0.

The vector space B is called a quasi-Banach space if it is a complete quasinormed space. If B is a quasi-Banach space with quasi-norm satisfying (12) then on account of [3, 31] there is an equivalent quasi-norm to · B which additionally satisfies f + g rB ≤ f rB + g rB ,

f, g ∈ B.

(13)

From now on we always assume that the quasi-norm of the quasi-Banach space B is chosen in such way that both (12) and (13) hold. Definition 1 Let B ⊆ Lrloc (Rd ) be a quasi-Banach of order r ∈ (0, 1], and let v ∈ PE (Rd ). Then B is called a translation invariant Quasi-Banach Function space on Rd (with respect to v), or invariant QBF space on Rd , if there is a constant C such that the following conditions are fulfilled: (1) if x ∈ Rd and f ∈ B, then f ( · − x) ∈ B, and f ( · − x) B ≤ Cv(x) f B ;

(14)

(2) if f, g ∈ Lrloc (Rd ) satisfy g ∈ B and |f | ≤ |g|, then f ∈ B and f B ≤ C g B ; An invariant QBF space B on Rd with respect to v is called a translation invariant Banach Function space or invariant BF-space (on Rd with respect to v), if B is a Banach space and in addition Minkowski’s inequality holds true, i. e. f ∗ ϕ B  f B ϕ L1 , (v)

f ∈ B, ϕ ∈ L1(v) (Rd ).

(15)

If B is an invariant QBF-space or BF-space with respect to v, and v belongs to Ps (Rd ) (Ps0 (Rd )), then B is called an invariant QBF-space and BF-space, respectively, of Roumieu type (Beurling type) of order s. It follows from (2) in Definition 1 that if f ∈ B and h ∈ L∞ , then f · h ∈ B, and f · h B ≤ C f B h L∞ .

(16)

In Definition 1, condition (2) means that a translation invariant BF-space is a solid BF-space in the sense of (A.3) in [15].

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

9 p,q

Example 1 Assume that p, q ∈ [1, ∞], and let L1 (R2d ) be the set of all f ∈ L1loc (R2d ) such that f Lp,q ≡

 

1

p,q

if finite. Then it follows that L1 v = 1.

|f (x, ξ )|p dx

q/p dξ

1/q

is translation invariant BF-spaces with respect to

We refer to [14, 18–21, 23, 32, 39] for more facts about modulation spaces. Next we consider the extended class of modulation spaces which we are interested in. Definition 2 Let B be a translation invariant QBF on R2d , ω ∈ PE (R2d ), and let φ ∈ 1 (Rd ) \ 0. Then the set M(ω, B) consists of all f ∈ 1 (Rd ) such that f M(ω,B ) ≡ Vφ f ω B is finite. p,q

p,q

Obviously, we have M(ω) (Rd ) = M(ω, B) when B = L1 (R2d ) (cf. Example 1). It follows that several properties which are valid for the classical modulation spaces also hold for spaces of the form M(ω, B). We notice that if B is an invariant BF-space, then M(ω, B) is a Banach space which is independent of the choice of φ in Definition 2 (see [27, Proposition 3.2] and [42]).

2.5 Pseudo-Differential Operators Next we recall some facts on pseudo-differential operators. The set of all d × d matrices with entries in the set  is denoted by M(d, ). (Note that at corresponding place in [42, Subsection 2.6] it should stay M(d, ) instead of M(d, ).) Let A ∈ M(d, R) be fixed and let a ∈ 1 (R2d ). Then the pseudo-differential operator OpA (a) is the linear and continuous operator on 1 (Rd ), defined by the formula (OpA (a)f )(x) = (2π )−d



a(x − A(x − y), ξ )f (y)eix−y,ξ  dydξ.

(17)

The definition of OpA (a) extends to any a ∈ 1 (R2d ), and then OpA (a) is continuous from 1 (Rd ) to 1 (Rd ). Moreover, for every fixed A ∈ M(d, R), it follows that there is a one to one correspondence between such operators and pseudo-differential operators of the form OpA (a). (See e. g. [25].) If A = 2−1 I , where I ∈ M(d, R) is the identity matrix, then OpA (a) is equal to the Weyl operator Opw (a) of a. If instead A = 0, then the standard (Kohn-Nirenberg) representation Op(a) = a(x, D) is obtained.

10

A. Abdeljawad and J. Toft

In the next proposition we recall some continuity properties of the operator eiADξ ,Dx  when A ∈ M(d, R) on well-known spaces. Thereafter we present some facts when passing between different pseudo-differential calculi with different values of A in (17). Proposition 6 Let s, σ > 0 be such that s + σ ≥ 1, A ∈ M(d, R), ω ∈ Ps,σ (R2d ) 0 (R2d ). Then eiADξ ,Dx  is homeomorphic on and ω0 ∈ Ps,σ  2d (Sσ,s s,σ ) (R )

2d Sσ,s s,σ (R ),

σ,s s,σ (R2d ),

and on

σ,s  (s,σ ) (R2d ).

(18)

The assertion (1) in Proposition 6 is a special case of [1, Theorem 3.1]. Proposition 7 Let s, σ > 0 be such that s + σ ≥ 1 and A1 , A2 ∈ M(d, R). If σ,s  σ,s   2d 2d 2d a1 ∈ (Sσ,s s,σ ) (R ) (a1 ∈ (s,σ ) (R )), then there is a unique a2 ∈ (Ss,σ ) (R ) σ,s  (a2 ∈ (s,σ ) (R2d )), and OpA1 (a1 ) = OpA2 (a2 )

⇔

eiA1 Dξ ,Dx  a1 (x, ξ ) = eiA2 Dξ ,Dx  a2 (x, ξ ). (19)

Remark 1 Let a1 and a2 be as in Proposition 7, s, σ > 0 be such that s + σ ≥ 1. σ,s 2d 2d Then it follows from Proposition 6 that a1 ∈ Sσ,s s,σ (R ) (a1 ∈ s,σ (R )), if and σ,s σ,s 2d 2d only if a2 ∈ Ss,σ (R ) (a2 ∈ s,σ (R )).

2.6 Symbol Classes Next we introduce function spaces related to symbol classes of the pseudodifferential operators. These functions should obey various conditions of the form |∂xα ∂ξ a(x, ξ )|  h|α+β| α!σ β!s ω(x, ξ ), β

(20)

when a is a function on Rd1 +d2 . For this reason we consider semi-norms of the form  a  σ,s;h ≡ (ω)



sup

sup

(α,β)∈Nd1 +d2

(x,ξ )∈Rd1 +d2

β

|∂xα ∂ξ a(x, ξ )| h|α+β| α!σ β!s ω(x, ξ )

,

(21)

indexed by h > 0, Definition 3 Let s, σ and h be positive constants, ω be a weight on Rd1 +d2 and vr (x, ξ ) ≡ er(|x|

1 1 s +|ξ | σ

)

,

r > 0.

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

11

σ,s;h (1) The set (ω) (Rd1 +d2 ) consists of all a ∈ C ∞ (Rd1 +d2 ) such that a  σ,s;h (ω)

in (21) is finite. The set 0σ,s;h (Rd1 +d2 ) consists of all a ∈ C ∞ (Rd1 +d2 ) such that a  σ,s;h is finite for every r > 0, and the topology is the projective limit (vr )

σ,s;h (Rd1 +d2 ) with respect to r > 0; topology of (v r)

σ,s;0 σ,s (Rd1 +d2 ) and (ω) (Rd1 +d2 ) are given by (2) The sets (ω) σ,s (ω) (Rd1 +d2 ) ≡



σ,s;h (ω) (Rd1 +d2 )

h>0

and σ,s;0 (ω) (Rd1 +d2 ) ≡



σ,s;h (ω) (Rd1 +d2 ),

h>0

and their topologies are the inductive respective the projective topologies of σ,s;h (ω) (Rd1 +d2 ) with respect to h > 0. The following result is a straight-forward consequence of [1, Proposition 2.4] and the definitions. (See also [9].) Proposition 8 Let R > 0, q ∈ [1, ∞], s, σ > 0 be such that s + σ ≥ 1 and σ,s (s, σ ) = ( 12 , 12 ), φ ∈ s,σ (R2d ) \ 0, ω ∈ Ps,σ (R2d ), and let ωr (x, ξ, η, y) = ω(x, ξ )e−r(|y|

1 1 s +|η| σ

)

.

Then σ,s (ω) (R2d ) =



σ,s  { a ∈ (s,σ ) (R2d ) ; ωr−1 Vφ a L∞,q < ∞ },

R>0 σ,s;0 (R2d ) = (ω)



(22) σ,s  { a ∈ (s,σ ) (R2d ) ; ωr−1 Vφ a L∞,q < ∞ }.

R>0 σ,s We have the following analogue of Propositions 6 and 7 for symbols in (ω) (R2d ) σ,s;0 (R2d ). and (ω) 0 (R2d ), Proposition 9 Let s, σ > 0 such that s +σ ≥ 1, ω ∈ Ps,σ (R2d ), ω0 ∈ Ps,σ σ,s   2d A, A1 , A2 ∈ M(d, R), and let a1 , a2 ∈ (s,σ ) (R2d ) (a1 , a2 ∈ (Sσ,s s,σ ) (R )) be such that OpA1 (a1 ) = OpA2 (a2 ). Then the following is true: σ,s;0 σ,s (R2d ) ((ω (R2d )); (1) eiADξ ,Dx  is homeomorphic on (ω) 0)

σ,s;0 σ,s;0 σ,s (R2d ) (a1 ∈ (ω (R2d )), if and only if a2 ∈ (ω) (R2d ) (a2 ∈ (2) a1 ∈ (ω) 0) σ,s 2d (ω0 ) (R )).

12

A. Abdeljawad and J. Toft

σ,s;0 The assertions for (ω) (R2d ) in Proposition 9 follows from [1, Theorem 3.3] and Proposition 7. By similar arguments as in the proof of [1, Theorem 3.3], it σ,s follows that Proposition 9 (1) also holds for the symbol class (ω (R2d ). The 0) σ,s 2d details are left for the reader. Proposition 9 (2) for (ω0 ) (R ) now follows from Proposition 7.

3 Continuity on Modulation Spaces for Pseudo-Differential Operators with Symbols of Infinite Order σ,s;0 σ,s In this section we discuss continuity for operators in Op((ω ) and Op((ω ) when 0) 0) acting on a general class of modulation spaces. In Theorem 1, continuity is treated σ,s where the symbols belong to (ω and in Theorem 2, continuity is treated where 0)

σ,s;0 . This gives an analogy to [37, Theorem 3.2] and [42, the symbols belong to (ω 0) Theorem 3.5] in the framework of operator theory and Gelfand-Shilov spaces. Our main result is stated as follows. 0 (R2d ), a ∈  σ,s (R2d ), Theorem 1 Let A ∈ M(d, R), s, σ ≥ 1, ω, ω0 ∈ Ps,σ (ω0 ) and that B is an invariant BF-space on R2d . Then OpA (a) is continuous from M(ω0 ω, B) to M(ω, B).

We need some preparations for the proof, and start with the following remark. Here we recall that if T is a linear and continuous map from Sσs (Rd ) to (Sσs ) (Rd ) (from sσ (Rd ) to (sσ ) (Rd )), then the (formal) adjoint T ∗ of T is the linear and continuous map from Sσs (Rd ) to (Sσs ) (Rd ) (from sσ (Rd ) to (sσ ) (Rd )), defined by the formula (Tf, g)L2 = (f, T ∗ g)L2 ,

f, g ∈ Sσs (Rd ) (f, g ∈ sσ (Rd )).

σ,s  ) (R2d ), then there Remark 2 Let s, σ > 0 such that s + σ ≥ 1. If a ∈ (s,σ σ,s  is a unique b ∈ (s,σ ) (R2d ) such that, Op(a)∗ = Op(b), where b(x, ξ ) = eiDξ ,Dx  a(x, ξ ) in view of [25, Theorem 18.1.7]. Furthermore, by the latter equality and [4, Theorem 4.1] it follows that σ,s (R2d ) a ∈ (ω)

⇔

σ,s b ∈ (ω) (R2d ).

Lemma 1 Suppose s, σ ≥ 1, ω ∈ PE (Rd0 ) and that f ∈ C ∞ (Rd+d0 ) satisfies 1 s

|∂ α f (x, y)|  h|α| α!σ e−r|x| ω(y), α ∈ Nd+d0

(23)

for some h, r > 0 (for every h, r > 0). Then there are f0 ∈ C ∞ (Rd+d0 ) and ψ ∈ Sσs (Rd ) (ψ ∈ sσ (Rd )) such that (23) holds with f0 in place of f for some h, r > 0 (for every h, r > 0), and f (x, y) = f0 (x, y)ψ(x).

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

13

Proof We only prove the result in the Roumieu case, i. e. when it is assumed that (23) holds true for some h, r > 0. The other case follows by similar arguments and is left for the reader. By Proposition 1, it follows that for any s ≥ 1 and r > 0, there is a submultiplicative weight v0 ∈ Ps (Rd ) ∩ C ∞ (Rd ) such that r

v0 (x) e 2 |x|

1 s

(24)

and |∂ α v0 (x)|  h|α| α!σ v0 (x),

α ∈ Nd

(25)

for some h, r > 0. Since s, σ ≥ 1, a straight-forward application of Faà di Bruno’s formula, for the composed function ψ(x) = g(v0 (x)), where g(t) = 1t , on (25) gives





α 1

 h|α| α!σ · 1 ,

∂

v0 (x)

v0 (x)

(25)

α ∈ Nd

for some h > 0. It follows from (24) and (25) that if ψ = 1/v0 , then ψ ∈ Sσs (Rd ), see [26, Proposition 6.1.7]. Furthermore, if f0 (x, y) = f (x, y)v0 (x), then by an application of Leibnitz formula we get |∂xα ∂yα0 f0 (x, y)|

h

α γ |∂x ∂yα0 f (x, y)| |∂ α−γ v0 (x)|  γ γ ≤α

|α+α0 |

1

α s (γ !α0 !)σ e−r|x| ω(y)(α − γ )!σ v0 (x) γ γ ≤α 1 s

 (2h)|α+α0 | (α!α0 !)σ e−r|x| v0 (x)ω(y) r

1 s

 (2h)|α+α0 | (α!α0 !)σ e− 2 |x| ω(y) for some h > 0, which gives the desired estimate on f0 , since it is clear that f (x, y) = f0 (x, y)ψ(x).   We recall that for any a ∈ 1 (R2d ) there is a unique b ∈ 1 (R2d ) such that Op(a)∗ = Op(b), and that b is given by b(x, ξ ) = eiDξ ,Dx  a(x, ξ ) in view of [25, 0 (R2d ), then it follows from Theorem 18.1.7]. If ω ∈ Ps,σ (R2d ) and ω0 ∈ Ps,σ

14

A. Abdeljawad and J. Toft

σ,s;0 σ,s;0 σ,s Proposition 9 that b ∈ (ω) (R2d ) (b ∈ (ω (R2d )), if and only if a ∈ (ω) (R2d ) 0) σ,s 2d (a ∈ (ω0 ) (R )).

Lemma 2 Let s, σ ≥ 1, 0 (R2d ), ω ∈ Ps,σ

v1 ∈ Ps0 (Rd )



s;0 (v (Rd ) and 1)

v2 ∈ Pσ0 (Rd )



s;0 (v (Rd ) 2)

σ,s (R2d ) is v1 ⊗ v2 -moderate. be such that v1 and v2 are submultiplicative, ω ∈ (ω) σ,s σ 2d d σ d Also let a ∈ (ω) (R ), f ∈ Ss (R ), φ ∈ s (R ), φ2 = φv1 , and choose b ∈ σ,s (ω) (R2d ) such that Op(b) = Op(a)∗ . If

(x, ξ, z, ζ ) =

b(x + z, ξ + ζ ) ω(x, ξ )v1 (z)v2 (ζ )

(26)

and  H (x, ξ, y) =

(x, ξ, z, ζ )φ2 (z)v2 (ζ )eiy−x−z,ζ  dzdζ,

(27)

then Vφ (Op(a)f )(x, ξ ) = (2π )−d (f, ei · ,ξ  H (x, ξ, · ))ω(x, ξ ).

(28)

Furthermore the following is true: (1) H ∈ C ∞ (R3d ) and satisfies |α|

1 s

|∂yα H (x, ξ, y)|  h0 α!σ e−r0 |x−y| ,

(29)

for every α ∈ Nd and some h0 , r0 > 0; (2) there are functions H0 ∈ C ∞ (R3d ) and φ0 ∈ Sσs (Rd ) such that H (x, ξ, y) = H0 (x, ξ, y)φ0 (y − x),

(30)

and such that (29) holds for some h0 , r0 > 0, with H0 in place of H . We observe that the existence of v1 and v2 in Lemma 2 is guaranteed Proposition 1. Lemma 2 follows by similar arguments as in [42]. In order to be self contained we give a different proof. Proof By straight-forward computations we get Vφ (Op(a)f )(x, ξ ) = (2π )−d (f, ei · ,ξ  H1 (x, ξ, · ))ω(x, ξ ),

(31)

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

15

where H1 (x, ξ, y) = (2π )d e−iy,ξ  (Op(a)∗ (φ( · − x) ei·,ξ  ))(y)/ω(x, ξ )  =  =

b(y, ζ ) φ(z − x)eiy−z,ζ −ξ  dzdζ ω(x, ξ )

(x, ξ, z − x, ζ − ξ )φ2 (z − x)v2 (ζ − ξ )eiy−z,ζ −ξ  dzdζ.

If z − x and ζ − ξ are taken as new variables of integrations, it follows that the right-hand side is the same as (27). Hence (28) holds. This gives the first part of the lemma. The smoothness of H is a consequence of the uniqueness of the adjoint (cf. Remark 2) and [42, Lemma 3.7]. To show that (29) holds, let 0 (x, ξ, z, ζ ) = (x, ξ, z, ζ )φ(z), where  is defined as in (26), and let  = F3 0 , where F3 0 is the partial Fourier transform of 0 (x, ξ, z, ζ ) with respect to the z variable. Then it follows from the assumptions and (25) that |∂zα 0 (x, ξ, z, ζ )|



|α|



1

∂zγ b(x + z, ξ + ζ )

s h|α−γ | (α − γ )!σ e−r|z|  ω(x, ξ )v (ζ ) 2 γ ≤α 1

σ −r0 |z| s

h (α − γ )! (γ )! e σ

|α|

 h α!

γ ≤α

σ

 (α − γ )!γ ! σ α!

γ ≤α

e−r0 |z|

 h|α| α!σ e−r0 |z|

1 s

1 s

1.

γ ≤α

Since



γ ≤α

1  2|α| , we get 1 s

|α|

|∂zα 0 (x, ξ, z, ζ )| ≤ C(2h)|α| α!σ e−r0 |z| ≤ Ch0 α!σ e−r0 |z|

1 s

(32)

for some C, h0 > 0 and every r0 > 0. Then z → 0 (x, ξ, z, ζ ) is an element σ,s in Sσs (Rd ). Moreover, { 0 (x, ξ, z, ζ ) }z∈Rd is a bounded set in (1) (Rd × R2d ). d Indeed, for a fixed z ∈ R , then an application of Leibnitz formula gives

 α  β  γ



α β γ ∂ α1 ,β1 ,γ1 (x, ξ, z, ζ ), ∂ ∂  (x, ξ, z, ζ )

≤

x ξ ζ 0 α1 β1 γ1

16

A. Abdeljawad and J. Toft

where α1 ,β1 ,γ1 (x, ξ, z, ζ )



α β1 γ1 1

= ∂x ∂ξ ∂ζ

  φ(z)

1 α0 β0 γ0

∂x ∂ξ ∂ζ b(x + z, ξ + ζ ) · ω(x, ξ )v2 (ζ ) v1 (z)

with α0 = α − α1 , β0 = β − β1 and γ0 = γ − γ1 . Here and in the next, all summations are taken over all α1 ≤ α, β1 ≤ β and γ1 ≤ γ . By Faà di Bruno’s formula, Proposition 1, (25) and the fact that φ ∈ sσ (Rd ), we get α1 ,β1 ,γ1 (x, ξ, z, ζ ) 



h|α1 +β1 +γ1 | α1 !σ (β1 !γ1 !)s

 α0 β0 γ0

∂x ∂ξ ∂ζ b(x + z, ξ + ζ )

ω(x, ξ )v1 (z)v2 (ζ )  h|α+β+γ | (α0 !α1 !)σ (β0 !β1 !)s (γ0 !γ1 !)s ≤ h|α+β+γ | α!σ (β!γ !)s .

A combination of these estimates gives



 α  β  γ

α β γ

|α+β+γ | σ s  h ∂ ∂  (x, ξ, z, ζ ) α! !) (β!γ

∂x ξ ζ 0

α1 β1 γ1 ≤ (4h)|α+β+γ | α!σ (β!γ !)s . In view of Proposition 2 and (32) we have |α|

1 σ

|∂ηα (x, ξ, η, ζ )|  h0 α!s e−r0 |η| , for some h0 , r0 > 0. Hence

1



α |α| σ

∂η ((x, ξ, η, ζ )v2 (η)) η=ζ  h0 α!s e−r0 |ζ | for some h0 , r0 > 0. By letting H2 (x, ξ, · ) be the inverse partial Fourier transform of the function (x, ξ, ζ, ζ ) v2 (ζ ) with respect to the ζ variable, it follows that |α|

|∂yα H2 (x, ξ, y)|  h0 α!σ e−r0 |y|

1 s

(33)

for some h0 , r0 > 0. The assertion (1) now follows from the latter estimate and the fact that H (x, ξ, y) = H2 (x, ξ, x − y).

Anisotropic Gevrey-Hörmander ψdo on Modulation Spaces

17

In order to prove (2) we notice that (33) shows that y → H2 (x, ξ, y) is an (1) element in Sσs (Rd ) with values in s,σ (R2d ). It follows by Lemma 1 that there σ ∞ 3d d exist H3 ∈ C (R ) and φ0 ∈ Ss (R ) such that (33) holds for some h0 , r0 > 0 with H3 in place of H2 , and H2 (x, ξ, y) = H3 (x, ξ, y)φ0 (−y).  

This is the same as (2), and the result follows. Proof of Theorem 1 By Proposition 9 we assume that A = 0. Let g = Op(a)f . In view of Lemma 2 we have d

Vφ g(x, ξ ) = (2π )− 2 F ((f · φ0 ( · − x)) · H0 (x, ξ, · ))(ξ )ω(x, ξ ) = (2π )−d (Vφ0 f )(x, · ) ∗ (F (H0 (x, ξ, · )))(ξ )ω(x, ξ ). 0 (R2d ), then for every r > 0 and x, ξ, η ∈ Rd , we Since ω and ω0 belong to Ps,σ 0 have r0

1 σ

ω(x, ξ )ω0 (x, ξ )  ω(x, η)ω0 (x, η)e 2 |ξ −η| . This inequality and (2) in Lemma 2 give  1 r0 σ (ξ ). |Vφ g(x, ξ )ω0 (x, ξ )|  |(Vφ0 f )(x, · )ω(x, · )ω0 (x, · )| ∗ e− 2 | · | In view of Definition 1, we get for some v ∈ Pσ0 (Rd ), 1 σ

G M(ω0 ,B )  |(Vφ0 f ) · ω · ω0 | ∗ δ0 ⊗ e−r0 | · | B 1 σ

≤ (Vφ0 f ) · ω · ω0 B e−r0 | · | v L1 f M(ω·ω0 ,B ) . This gives the result.

 

By similar arguments as in the proof of Theorem 1 and Lemma 2 we get the following. The details are left for the reader. σ,s;0 Theorem 2 Let A ∈ M(d, R), s, σ ≥ 1, ω, ω0 ∈ Ps,σ (R2d ), a ∈ (ω (R2d ) and 0) B be an invariant BF-space on R2d . Then OpA (a) is continuous from M(ω0 ω, B) to M(ω, B).

Lemma 3 Let s, σ ≥ 1, ω ∈ Ps,σ (R2d ), v1 ∈ Ps (Rd ) and v2 ∈ Pσ (Rd ) be such σ,s;0 (R2d ) is v1 ⊗ v2 -moderate. Also let that v1 and v2 are submultiplicative, ω ∈ (ω)

18

A. Abdeljawad and J. Toft

σ,s;0 a ∈ (ω) (R2d ), f, φ ∈ sσ (Rd ), φ2 = φv1 , and let  and H be as in Lemma 2. Then (28) and the following hold true:

(1) H ∈ C ∞ (R3d ) and satisfies (29) for every h0 , r0 > 0; (2) there are functions H0 ∈ C ∞ (R3d ) and φ0 ∈ s (Rd ) such that (30) holds, and such that (29) holds for every h0 , r0 > 0, with H0 in place of H . Remark 3 Theorems 1 and 2 also hold when B are invariant QBF spaces of Lebesgue spaces (which necessarily do not need to be Banach spaces). This follows σ,s;0 σ,s from [41, Theorem 3.4] and inclusion relations between (ω) and (ω) spaces, and modulation spaces, given in [1]. Remark 4 We recall that Gelfand-Shilov spaces and their distribution spaces can be obtained by suitable intersections and unions of modulation spaces (see [38, 40]). From these characterizations it follows that Theorems 1 and 2 can be used to regain several continuity results in [1, 9] for pseudo-differential operators of infinite orders, acting on Gelfand-Shilov spaces and their distribution spaces.

References 1. A. Abdeljawad, M. Cappiello, J. Toft Pseudo-differential calculus in anisotropic GelfandShilov setting, Integr. Equ. Oper. Theory 91 (2019), 91:26. 2. A. Abdeljawad, S. Coriasco, J. Toft Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey type pseudo-differential operators, (preprint) arXiv:1712.04338 (2017). 3. T. Aoki Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588– 594. 4. M. Cappiello, J. Toft Pseudo-differential operators in a Gelfand-Shilov setting, Math. Nachr. 290 (2017), 738–755. 5. F. Concetti, J. Toft. Trace ideals for Fourier integral operators with non-smooth symbols, “Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis”, Fields Inst. Commun., Amer. Math. Soc., 52 2007, pp.255–264. 6. F. Concetti, G. Garello, J. Toft. Trace ideals for Fourier integral operators with non-smooth symbols II. Osaka J. Math., 47 (2010), 739–786. 7. E. Cordero, K. Gröchenig, F. Nicola, L. Rodino. Wiener algebras of Fourier integral operators, J. Math. Pures Appl., (2013), 219–233. 8. E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Generalized Metaplectic Operators and the Schrödinger Equation with a Potential in the Sjöstrand Class, J. Math. Phys., 55, (2014), 081506:1–17. 9. E. Cordero, F. Nicola and L. Rodino, Gabor Representations of evolution operators, Trans. Amer. Math. Soc. 367 (2015), 7639–7663. 10. J. Chung, S.-Y. Chung, D. Kim, Characterizations of the Gelfand-Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc. 124 (1996), 2101–2108. 11. ] M. A. de Gosson, Symplectic methods in harmonic analysis and in mathematical physics, Pseudo-Differential Operators Theory and Applications 7 Birkhäuser/Springer Basel AG, Basel, 2011. 12. H. G. Feichtinger Banach spaces of distributions of Wiener’s type and interpolation, in: Ed. P. Butzer, B. Sz. Nagy and E. Görlich (Eds), Proc. Conf. Oberwolfach, Functional Analysis and Approximation, August 1980, Int. Ser. Num. Math. 69 Birkhäuser Verlag, Basel, Boston, Stuttgart, 1981, pp. 153–165.

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13. H. G. Feichtinger Banach convolution algebras of Wiener’s type, in: Proc. Functions, Series, Operators in Budapest, Colloquia Math. Soc. J. Bolyai, North Holland Publ. Co., Amsterdam Oxford NewYork, 1980. 14. H. G. Feichtinger Modulation spaces on locally compact abelian groups. Technical report, University of Vienna, Vienna, 1983; also in: M. Krishna, R. Radha, S. Thangavelu (Eds) Wavelets and their applications, Allied Publishers Private Limited, NewDehli Mumbai Kolkata Chennai Hagpur Ahmedabad Bangalore Hyderbad Lucknow, 2003, pp.99–140. 15. H. G. Feichtinger Wiener amalgams over Euclidean spaces and some of their applications, in: Function spaces (Edwardsville, IL, 1990), Lect. Notes in pure and appl. math., 136, Marcel Dekker, New York, 1992, pp. 123–137. 16. H. G. Feichtinger Modulation spaces: Looking back and ahead, Sampl. Theory Signal Image Process. 5 (2006), 109–140. 17. H. G. Feichtinger Choosing function spaces in harmonic analysis, Excursions in harmonic analysis 4, Appl. Numer. Harmon. Anal., 65–101, Birkhäuser/Springer, Cham, 2015. 18. H. G. Feichtinger and K. H. Gröchenig Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307–340. 19. H. G. Feichtinger and K. H. Gröchenig Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math. 108 (1989), 129–148. 20. H. G. Feichtinger and K. H. Gröchenig Gabor frames and time-frequency analysis of distributions, J. Functional Anal. (2) 146 (1997), 464–495. p,q 21. Y. V. Galperin, S. Samarah Time-frequency analysis on modulation spaces Mm , 0 < p, q ≤ ∞, Appl. Comput. Harmon. Anal. 16 (2004), 1–18. 22. I. M. Gelfand, G. E. Shilov Generalized functions, I–III, Academic Press, NewYork London, 1968. 23. K. H. Gröchenig Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001. 24. K. Gröchenig,G. Zimmermann Spaces of test functions via the STFT J. Funct. Spaces Appl. 2 (2004), 25–53. 25. L. Hörmander The Analysis of Linear Partial Differential Operators, vol I, III, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985. 26. F. Nicola, L. Rodino, Global Pseudo-differential Calculus on Eu- clidean Spaces, Birkhäuser Basel, 2010. 27. C. Pfeuffer, J. Toft Compactness properties for modulation spaces, Complex Anal. Oper. Theory (online 2019). 28. S. Pilipovi´c Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J. Math. Anal. 17 (1986), 477–484. 29. S. Pilipovi´c, N. Teofanov On a symbol class of Elliptic Pseudo-differential Operators, Bull. Acad. Serbe Sci. Arts 27 (2002), 57–68. 30. S. Pilipovi´c, N. Teofanov Pseudo-differential operators on ultra-modulation spaces, J. Funct. Anal.208 (2004), 194–228. 31. S. Rolewicz On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astrono. Phys., 5 (1957), 471–473. 32. M. Ruzhansky, M. Sugimoto, N. Tomita, J. Toft Changes of variables in modulation and Wiener amalgam spaces, Math. Nachr. 284 (2011), 2078–2092. 33. K. Tachizawa The boundedness of pseudo-differential operators on modulation spaces, Math. Nachr. 168 (1994), 263–277. 34. N. Teofanov Ultramodulation spaces and pseudo-differential operators, Endowment Andrejevi´c, Beograd, 2003. 35. N. Teofanov Modulation spaces, Gelfand-Shilov spaces and pseudo-differential operators, Sampl. Theory Signal Image Process, 5 (2006), 225–242. 36. N. Teofanov Ultradistributions and Time-Frequency Analysis. In: Boggiatto P., Rodino L., Toft J., Wong M.W. (eds) Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications 164, Birkhäuser 2006, 173192. 37. J. Toft Pseudo-differential operators with smooth symbols on modulation spaces, Cubo, 11 (2009), 87–107.

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38. J. Toft The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. Pseudo-Differ. Oper. Appl. 3 (2012), 145–227. 39. J. Toft Gabor analysis for a broad class of quasi-Banach modulation spaces in: S. Pilipovi´c, J. Toft (eds), Pseudo-differential operators, generalized functions, Operator Theory: Advances and Applications 245, Birkhäuser, 2015, 249–278. 40. J. Toft Images of function and distribution spaces under the Bargmann transform, J. PseudoDiffer. Oper. Appl. 8 (2017), 83–139. 41. J. Toft Continuity and compactness for pseudo-differential operators with symbols in quasiBanach spaces or Hörmander classes, Anal. Appl. 15 (2017), 353–389. 42. J.Toft Continuity of Gevrey-Hörmander pseudo-differential operators on modulation spaces, J. Pseudo-Differ. Oper. Appl. 10 (2019), 337–358.

Hardy Spaces on Weighted Homogeneous Trees Laura Arditti, Anita Tabacco, and Maria Vallarino

Abstract We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space (V, d, μ) is nondoubling and of exponential growth, hence the classical theory of Hardy spaces does not apply in this setting. We construct an atomic Hardy space H 1 (μ) on (V, d, μ) and investigate some of its properties, focusing in particular on real interpolation properties and on boundedness of singular integrals on H 1 (μ). Keywords Hardy spaces · Homogeneous trees · Exponential growth

1 Introduction Let V be an infinite homogeneous tree of order q + 1 endowed with the usual distance d defined on a graph (see Sect. 2 for the precise definitions). Fix a doublyinfinite geodesic g in V and define a mapping N : g → Z such that |N(x) − N(y)| = d(x, y)

∀x, y ∈ g .

(1)

We define the level function  : V → Z as (x) = N(x  ) − d(x, x  ) , where x  is the only vertex in g such that d(x, x  ) = min{d(x, z) : z ∈ g}. Let μ be the measure on V defined by 

f dμ = f (x)q (x) (2) V

x∈V

L. Arditti · A. Tabacco · M. Vallarino () Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Dipartimento di Eccellenza 2018–2022, Politecnico di Torino, Torino, Italy e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_2

21

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for every function f defined on V. Then μ is a weighted counting measure. We shall show in Sect. 2.1 that the space (V, d, μ) is nondoubling and it is of exponential growth. In particular on such space the classical Calderón–Zygmund theory does not hold. Hebisch and Steger [8] developed a new Calderón–Zygmund theory which can be applied also to nondoubling metric measure spaces and showed that such a theory can be applied to the space (V, d, μ). In particular they proved that there exists a family of appropriate sets in V, which are called Calderón–Zygmund sets, which replace the family of balls in the classical Calderón–Zygmund theory. We mention also that some properties of the space (V, d, μ) were investigated in more detail in [1]. The purpose of this work is to develop a theory of Hardy spaces on (V, d, μ), which is a natural development of the Calderón–Zygmund theory introduced in [8]. Following the classical atomic definition of Hardy spaces [5], for each p in (1, ∞] we define an atomic Hardy space H 1,p (μ). Atoms are functions supported in Calderón–Zygmund sets, with vanishing integral and satisfying a certain size condition. We shall prove that all the spaces H 1,p (μ), p ∈ (1, ∞], coincide and we simply denote by H 1 (μ) this atomic Hardy space. We then find the real interpolation spaces between H 1 (μ) and Lq (μ), q ∈ (1, ∞]. The interpolation results which we prove are the analogue of the classical interpolation results (see [7, 10, 12, 13]), but the proofs are different. Indeed, in the classical setting the maximal characterization of the Hardy space is used to obtain the interpolation results, while the Hardy space H 1 (μ) introduced in this paper has an atomic definition. Further, we show that a singular integral operator whose kernel satisfies an integral Hörmander condition, extends to a bounded operator from H 1 (μ) to L1 (μ). As a consequence of this result, we show that spectral multipliers of a distinguished Laplacian L and the first order Riesz transform associated to L extend to bounded operators from H 1 (μ) to L1 (μ). It would be also interesting to characterize the dual space of H 1 (μ) and to obtain complex interpolation results involving H 1 (μ), its dual and the Lq (μ)-spaces. This will be the object of further investigations. All the results described above may be considered as an analogue of the classical theory of Hardy spaces. The classical Hardy space [5, 6, 14] was introduced in (Rn , d, m), where d is the Euclidean metric and m denotes the Lebesgue measure and more generally on a space of homogeneous type, i.e. a metric measure space (X, d, μ) where the doubling condition is satisfied, i.e., there exists a constant C such that     μ B(x, 2r) ≤ C μ B(x, r)

∀x ∈ X ,

∀r > 0.

(3)

Extensions of the theory of Hardy spaces have been considered in the literature on various metric measure spaces which do not satisfy the doubling condition (3). The literature on this subject is huge and we shall only cite here some contributions [3, 4, 11, 15] which are strictly related to our work.

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23

In particular, we mention that Celotto and Meda [4] studied various Hardy spaces on a homogeneous tree V endowed with the metric d and the counting measure, which is not the measure μ that we consider here. Their theory is useful to study the boundedness of singular integral operators related to the standard Laplacian defined on trees which is self-adjoint with respect to the counting measure and not to the measure μ. The theory we develop here instead is useful to study singular integral operators related to a distinguished Laplacian self-adjoint on L2 (μ) (see Sect. 4.3). We mention that in [11, 15] the authors used the Calderón–Zygmund theory introduced by Hebisch and Steger in [8] to construct Hardy spaces on some solvable Lie groups of exponential growth and studied their properties. Our work can be thought as a counterpart in a discrete setting of the results in [15], and some of our proofs are strongly inspired by it. Positive constants are denoted by C; these may differ from one line to another, and may depend on any quantifiers written, implicitly or explicitly, before the relevant formula.

2 Weighted Homogeneous Trees In this section we introduce the infinite homogeneous tree and we define a distance d and and a measure μ on it. We show that the corresponding metric measure space (V, d, μ) does not satisfy the doubling property. We then introduce a family of sets, called trapezoids, which will be fundamental in the construction of Hardy spaces. Definition 1 An infinite homogeneous tree of order q + 1 is a graph T = (V, E), where V denotes the set of vertices and E denotes the set of edges, with the following properties: (i) T is connected and acyclic; (ii) each vertex has exactly q + 1 neighbours. On V we can define the distance d(x, y) between two vertices x and y as the length of the shortest path between x and y. We also fix a doubly-infinite geodesic g in T , that is a connected subset g ⊂ V such that (i) for each element v ∈ g there are exactly two neighbours of v in g; (ii) for every couple (u, v) of elements in g, the shortest path joining u and v is contained in g. We define a mapping N : g → Z such that |N(x) − N(y)| = d(x, y)

∀x, y ∈ g .

(4)

This corresponds to the choice of an origin o ∈ g (the only vertex for which N (o) = 0) and an orientation for g; in this way we obtain a numeration of the vertices in g.

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Fig. 1 Representation of the measure μ (q = 3)

We define the level function  : V → Z as (x) = N(x  ) − d(x, x  ) where x  is the only vertex in g such that d(x, x  ) = min{d(x, z) : z ∈ g}. For x, y ∈ V we say that y lies above x if (x) = (y) − d(x, y). In this case we also say that x lies below y. Let μ be the measure on V such that for each function f : V → C  V

f dμ =

f (x)q (x) .

(5)

x∈V

Then μ is a weighted counting measure such that the weight of a vertex depends only on its level and the weight associated to a certain level is given by q times the weight of the level immediately underneath (see Fig. 1).

2.1 Doubling and Local Doubling Properties Observe that the space (V, d, μ) exhibits exponential volume growth. Indeed given x0 ∈ V and r ≥ 1 consider the sphere Sr (x0 ) = {x ∈ V : d(x, x0 ) = r} and the closed ball Br (x0 ) = {x ∈ V : d(x, x0 ) ≤ r} . A direct computation shows that for

Hardy Spaces on Weighted Homogeneous Trees

25

r ≥ 1 their measures are given by: μ(Sr (x0 )) = q (x0 )+r−1 (1 + q)

and

μ(Br (x0 )) = q (x0 )

q r+1 + q r − 2 . q −1

We notice that they depend on the level of the center x0 and grow exponentially with respect to the radius r. As a consequence we can prove the following. Proposition 1 The space (V, d, μ) is not doubling but it is locally doubling. Proof Fix x0 ∈ V and notice that 2r+1 +q 2r −2 (x ) q q−1 q r+1 +q r −2 q−1

q 0 μ(B2r (x0 )) = lim lim r→∞ μ(Br (x0 )) r→∞ q (x0 )

= lim q r = +∞. r→∞

Thus the doubling property (3) fails. Instead, we show that (V, d, μ) is locally doubling. Indeed, fix x0 ∈ V and R > 0 and consider r ≤ R; one has μ(B2r (x0 )) = q

(x0 )

q 2r+1 + q 2r − 2 ≤ q (x0 ) q −1

q 2R+1 +q 2R −2 q−1 q R+1 +q R −2 q−1

q r+1 + q r − 2 q −1

= CR μ(Br (x0 )) with CR =

q 2R+1 +q 2R −2 q R+1 +q R −2

> 0 independent of x0 and r.

 

2.2 Admissible Trapezoids and Calderón–Zygmund Sets In this subsection we introduce the notion of trapezoid and recall the definition and the main properties of the admissible trapezoids introduced in [8]. Definition 2 We call trapezoid a set of vertices S ⊂ V for which there exist xS ∈ V and a, b ∈ R+ such that S = {x ∈ V : x lies below xS , a ≤ (xS ) − (x) < b} .

(6)

In the following we will refer to xS as the root node of the trapezoid. Among all trapezoids we are mostly interested in those where a and b are related by particular conditions, as specified in the following definitions.

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Fig. 2 Representation of an admissible trapezoid with h(R) = 2 (q = 3)

Definition 3 A trapezoid R ⊂ V is an admissible trapezoid if and only if one of the following occurs: (i) R = {xR } with xR ∈ V, that is R consists of a single vertex ; (ii) ∃xR ∈ V, ∃h ∈ N+ such that R = {x ∈ V : x lies below xR , h ≤ (xR ) − (x) < 2h} . We set h(R) = 1 in the first case and h(R) = h in the second case. In both cases h(R) can be interpreted as the height of the admissible trapezoid, which coincides with the number of levels spanned by R (see Fig. 2). Definition 4 We call width of the admissible trapezoid R the quantity w(R) = q (xR ) . We have that: μ(R) = h(R)q (xR ) = h(R)w(R).

(7)

We now introduce the family of Calderón–Zygmund sets. They are trapezoids, even if not of admissible type; they consist of suitable enlargements of admissible trapezoids, constructed according to the following definition.

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Definition 5 Given an admissible trapezoid R, the envelope of R is the set   h R˜ = x ∈ V : x lies below xR , ≤ (xR ) − (x) < 4h 2

(8)

˜ = h(R). The envelope of an admissible trapezoid is also called a and we set h(R) Calderón–Zygmund set. Proposition 2 Let R be an admissible trapezoid. Then: ˜ ≤ 4μ(R). μ(R)

(9)

˜ = μ(R). In Proof In the degenerate case one has R = {xR } = R˜ and then μ(R) the nondegenerate case

˜ = μ(R)

h (xR )− 2 



h (xR )− 2 

=(xR )−4h+1 x∈R:(x)= ˜

 )

≤ q (xR (xR ) −

h 2

q =

q  q (xR )−

=(xR )−4h+1

− (xR ) + 4h



≤ 4μ(R) ,  

which concludes the proof. Proposition 3 Let R1 and R2 be two admissible trapezoids. If R1 ∩ R2 = ∅ and w(R1 ) ≥ w(R2 ) , then R2 ⊂ R˜1 .

Proof The only nontrivial case is when neither R1 nor R2 is composed of a single vertex. Let xR1 and xR2 be the two root nodes of R1 and R2 , respectively. Then w(R1 ) = q (xR1 ) ≥ q (xR2 ) = w(R2 )

⇒

(xR1 ) ≥ (xR2 ).

Moreover, since R1 ∩ R2 = ∅, xR2 is below xR1 and so is every vertex of R2 . In the following we denote h1 = h(R1 ) and h2 = h(R2 ). Let xˆ ∈ R1 ∩ R2 = ∅. Then we obtain the following constraints: ⎧ ⎨(x ) − 2h + 1 ≤ (x) ˆ ≤ (xR1 ) − h1 R2 2 ⎩(xR ) − 2h1 + 1 ≤ (x) ˆ ≤ (xR ) − h2 1

2

⎧ ⎨(x ) − (x ) ≥ h − 2h + 1 R1 R2 1 2 ⇒ ⎩(xR ) − (xR ) ≤ 2h1 − h2 − 1 . 1

2

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Let x ∈ R2 ; then x lies below xR1 . Moreover     (xR1 ) − (x) = (xR1 ) − (xR2 ) + (xR2 ) − (x) ≤ 2 [2h1 − h2 − 1] + [2h2 − 1] < 4h1 .     (xR1 ) − (x) = (xR1 ) − (xR2 ) + (xR2 ) − (x) ≥ So

h1 2

1 h1 . [h1 − 2h2 + 1] + [h2 ] > 2 2

≤ (xR1 ) − (x) < 4h1 , ∀x ∈ R2 and this shows that R2 ⊂ R˜1 .

 

˜ we have that for all z ∈ R˜ Proposition 4 Given a Calderón–Zygmund set R, ˜ . R˜ ⊂ B(z, 8h(R)) ˜ Every vertex y ∈ R˜ has distance d(z, y) ≤ 8h(R) ˜ −2 < Proof Fix a point z ∈ R. ˜ 8h(R). Indeed,  starting from z it is possible to reach y passing through at most ˜ ˜ (4h(R) − 1) 2 = 8h(R)−2 edges, moving from z to the root node of the trapezoid and then from the root node to y.   ˜ we define the set Definition 6 Given a Calderón–Zygmund set R,   ˜ < h(R)/4 ˜ R˜ ∗ = x ∈ V : d(x, R) .

(10)

It is easy to see that there exists a positive constant C such that for every Calderón– Zygmund set R˜ ˜ . μ(R˜ ∗ ) ≤ Cμ(R)

(11)

See [1, p.75] for a proof of this fact.

3 The Maximal Function In this section we define two maximal functions and describe a way to construct a covering of their level sets which will be useful in the sequel. Definition 7 Given f : V → C, we define the maximal function M as Mf (x) = sup R:x∈R

1 μ(R)

 |f | dμ R

where the supremum is taken over all admissible trapezoids R containing x.

Hardy Spaces on Weighted Homogeneous Trees

29

Consider a function f ∈ Lp (μ) and let λ > 0. We are interested in constructing a covering of the level set λp = {x ∈ V : M(|f |p )(x) > λp } . Define S0 as the family of all admissible trapezoids R such that  |f |p dμ ≥ λp μ(R). R

Since S0 is countable, we can introduce an ordering in S0 . All trapezoids in S0 have bounded measure and bounded width, because ∀R ∈ S0 we have w(R) =

1 μ(R) p ≤ μ(R) ≤ p f Lp . h(R) λ

So it is possible to choose in S0 a trapezoid R0 of largest width (in case of ties, we choose that trapezoid of largest width which occurs earliest in the ordering). Then we proceed inductively: (i) Si+1 is the family of all admissible trapezoids R ∈ Si disjoint from R0 , . . . , Ri ; (ii) Ri+1 is the trapezoid of largest width in Si+1 which occurs earliest in the ordering. Let R ∈ S0 . Then by construction R intersects some Ri with w(Ri ) ≥ w(R). Indeed, there exists a number j ∈ {0, 1, 2, . . . } such that R ∈ Sj and R ∈ / Sj +1 , i.e. in the previous construction there exists a step j in which one of the following occurs: 1. either R is the trapezoid of largest width that occurs earliest in the ordering, and then R is selected and Rj = R, so that R ∩ Ri = ∅ for i = j ; 2. or R is not the trapezoid of largest width that occurs earliest in the ordering and it intersects Rj . Then R is not in Si ∀i ≥ j + 1 and R ∩ Ri = ∅ for i = j . To ensure that there is some j with the stated property it is sufficient to avoid that S0 can contain an infinite number of trapezoids with the same width that do not intersect each other. This possibility is excluded observing that:

i

  1 1 1 p μ(Ri ) ≤ p |f | dμ ≤ p |f |p dμ = p f Lp < ∞ , λ λ λ Ri V i

while if there was among the Ri ’s an infinite number of trapezoids with constant width w we would have

i

μ(Ri ) ≥

∞

n=1

w = ∞.

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In conclusion, ∃i : R ∩ Ri = ∅ and w(Ri ) ≥ w(R).

∀R ∈ S0 ,

By Proposition 3, this implies R ⊂ R˜i . We set E := and μ(E) ≤

μ(R˜i ) ≤ 4

i



μ(Ri ) ≤

i

i

R˜i . We have that λp ⊂ E

4 f Lp . λp

(12)

˜ we define the Definition 8 Given f : V → C and a Calderón–Zygmund set Q, maximal function MQ˜ as follows MQ˜ (f )(x) =

sup ˜ : x∈R R⊂Q

μ(R)−1

 |f | dμ

˜, ∀x ∈ Q

R

where the supremum is taken over all admissible trapezoids R containing x and ˜ When x ∈ ˜ we set M ˜ (f )(x) = 0. contained in Q. /Q Q Consider a function f ∈ Lp (μ) with support contained in a Calderón–Zygmund ˜ and let λ > 0. We define set Q, p p Q,λ ˜ p = {x ∈ V : MQ ˜ (|f | )(x) > λ } .

Arguing as before, we can show that there  exists a family of pairwise disjoint 1 ˜ admissible trapezoids {Ri } such that Ri ⊂ Q, ˜ p ⊂ i μ(Ri ) ≤ λp f Lp and Q,λ  ˜ i Ri .

4 Hardy Spaces In this section we define atomic Hardy spaces replacing balls with Calderón– Zygmund sets in the classical definition of atoms. Definition 9 A function a is a (1, p)-atom, for p ∈ (1, ∞], if it satisfies the following properties: ˜ (i) a is supported in a Calderón–Zygmund set R; 1/p−1 ˜ (ii)  a Lp ≤ μ(R) ; (iii) V a dμ = 0 . Observe that a (1, p)-atom is in L1 (μ) and it is normalized in such a way that its L1 -norm does not exceed 1.

Hardy Spaces on Weighted Homogeneous Trees

31

1,p 1 Definition 10 The  Hardy space H (μ) is the space of all functions h in L (μ) such that h numbers  = j λj aj , where aj are (1, p)-atoms and λj are complex ∞. We denote by h H 1,p the infimum of j |λj | over all such that j |λj | 2[24q(1 + 4p )]1/(p−1) . We shall prove that for all n ∈ N there exist functions aj , hjn and admissible sets R˜ j , j ∈ N ,  = 0, . . . , n, such that b=

n−1

4(6q)1/p α +1

μ(R˜ j ) aj +

fjn ,

(13)

jn ∈Nn

j

=0

where the following properties are satisfied: ˜ (i) aj is a (1, ∞)-atom supported  in the Calderón–Zygmund set Rj ; (ii) fjn is supported in Rjn and fjn dμ = 0;  1/p  1 p dμ (iii) |f | ≤ 21−1/p (6q)1/p (1 + 4p )1/p α n ; j ˜ n Rj ˜ μ(Rjn )

n

(iv) |fjn (x)| ≤ |b(x)| + 4(6q)1/p n α n , ∀x ∈ R˜ jn ;  p n+1 p−1 p ˜ [2 (6q) (1 + 4 )]n α −np b Lp . (v) jn μ(Rjn ) ≤ 4 We first suppose that and we show that a ∈  the decomposition (13) exists 1 (μ) and that its L1 -norm H 1,∞ (μ). Set Fn = f . We prove that F ∈ L j n jn n tends to zero when n tends to ∞. Indeed, by Hölder’s inequality Fn L1 ≤

jn ∈Nn

fjn L1 ≤

jn ∈Nn



μ(R˜ jn )1/p fjn Lp ,

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where p is the conjugate exponent of p. Now by (iii) and (v) we have that Fn L1 ≤



μ(R˜ jn )1/p μ(R˜ jn )1/p 21−1/p (6q)1/p (1 + 4p )1/p α n

jn ∈Nn

≤ 4n+1 [2p−1 6q (1 + 4p )]n α −np b Lp 21−1/p (6q)1/p (1 + 4p )1/p α n , p

which tends to zero when n tends α > 2[24q(1 + 4p )]1/(p−1) . n−1to ∞, since 1/p +1 ˜ This shows that the series =0 4(6q) α j μ(Rj ) aj converges to b in 1 L (μ). Moreover by (v) we deduce that ∞

4(6q)1/p α +1

=0

μ(R˜ j ) ≤

j

∞

4(6q)1/p α +1 4+1 [2p−1 (6q) (1 + 4p )] α −p b Lp p

=0

˜ , ≤ Cp b Lp ≤ Cp μ(Q) p

where Cp depends only on p. ˜ . Thus a = μ(Q) ˜ −1 b It follows that b is in H 1, ∞ (μ) and b H 1,∞ ≤ Cp μ(Q) is in H 1, ∞ (μ) and a H 1,∞ ≤ Cp , as required. It remains to prove that the decomposition (13) exists. We prove it by induction on n. Step n = 1 Define p p Q,α ˜ p = {x ∈ V : MQ ˜ (|b| )(x) > α } .

If  = ∅, then μ({xR })−1 |b(xR )|p μ({xR }) ≤ α p ,

∀xR ∈ V .

˜ j0 , so that and (13) is satisfied It follows that b L∞ ≤ α and we have b = αμ(Q)a −1 −1 ˜ with the (1, ∞)-atom aj0 = α μ(Q) b and fj1 = 0 for every j1 ∈ N. If  = ∅, then we construct a family of trapezoids Ri , i ∈ N, and the corresponding Calderón–Zygmund sets R˜ i , i ∈ N, as in Sect. 3. We then define Ui = R˜ i \ (∪j 0 and for any a ∈ A0 + A1 we define K(t, a; A0 , A1 ) = inf{ a0 A0 + t a1 A1 : a = a0 + a1 , ai ∈ Ai } .   Take q ∈ [1, ∞] and θ ∈ (0, 1). The real interpolation space A0 , A1 θ,q is defined as the set of the elements a ∈ A0 + A1 such that ⎧  ⎨  ∞ t −θ K(t, a; A , A )q dt 1/q if 1 ≤ q < ∞ 0 1 0 t a θ,q = ⎩ t −θ K(t, a; A , A ) if q = ∞ , 0

1

∞

  is finite. The space A0 , A1 θ,q endowed with the norm a θ,q is an exact interpolation space of exponent θ .

Hardy Spaces on Weighted Homogeneous Trees

35

We refer the reader to [9] for an overview of the real interpolation results which hold in the classical setting. Our aim is to prove the same results in our context. Note that in our case a maximal characterization of H 1 (μ) is not avalaible, so that we cannot follow the classical proofs but we shall only use the atomic definition of H 1 (μ) to prove the results. We shall first estimate the K functional of Lp -functions with respect to the couple of spaces (H 1 (μ), Lp1 (μ)), 1 < p1 ≤ ∞. Lemma 1 Suppose that 1 < p < p1 ≤ ∞ and let θ ∈ (0, 1) be such that 1−θ +

θ p1 .

Let f be in

Lp (μ).

1 p

=

The following hold:

(i) for every λ > 0 there exists a decomposition f = g λ + bλ in Lp1 (μ) + H 1 (μ) such that (i’) g λ L∞ ≤ C λ and, if p1 < ∞, then g λ L1p ≤ C λp1 −p f Lp ; p

p

1

p

(i”) bλ H 1 ≤ C λ1−p f Lp ;

(ii) for any t > 0, K(t, f ; H 1 (μ), Lp1 (μ)) ≤ C t θ f Lp ; (iii) f ∈ [H 1 (μ), Lp1 (μ)]θ,∞ and f θ,∞ ≤ C f Lp . Proof Let f be in Lp (μ). We first prove (i). Given a positive λ, let λp = {x ∈ V : M(|f |p )(x) > λp } . Let {Ri } be the collection of trapezoids constructed as in Sect. 3. We now define Ui = R˜ i \ (∪j 0. For any positive λ, let f = g λ + bλ be the decomposition of f in Lp1 (μ) + H 1 (μ) given by (i). Thus   K(t, f ; H 1 (μ), Lp1 (μ)) ≤ inf bλ H 1 + t g λ Lp1 λ>0

 p p/p  ≤ C inf λ1−p f Lp + t λ1−p/p1 f Lp 1 λ>0

p/p1

= C f Lp p(1−1/p1 )

where G(t, λ) = λ1−p f Lp

inf G(t, λ) ,

λ>0

+ t λ1−p/p1 . Since

  p(1−1/p1 ) + (1 − p/p1 )t λ−p/p1 +p , ∂λ G(t, λ) = λ−p (1 − p) f Lp

Hardy Spaces on Weighted Homogeneous Trees

37

we have that if p1 < ∞, then p1 (p−1)   1−p/p inf G(t, λ) = G t, Cp f Lp t p1 /p−pp1 = Cp f Lp 1 t p(p1 −1) .

λ>0

If p1 = ∞, then   inf G(t, λ) = G t, Cp f Lp t −1/p = Cp f Lp t 1−1/p .

λ>0

It follows that K(t, f ; H 1 , Lp1 ) ≤ Cp f Lp t θ , proving (ii). This implies that t −θ K(t, f ; H 1 (μ), Lp1 (μ)) L∞ ≤ Cp f Lp , so that f ∈ [H 1 (μ), Lp1 (μ)]θ,∞ and f θ,∞ ≤ Cp f Lp , as required in (iii).   Following closely the proof of [15, Theorem ] we deduce from Lemma 1 the following result. Theorem 2 Let 1 < p < p1 ≤ ∞ and θ ∈ (0, 1) be such that Then 

H 1 (μ), Lp1 (μ)

 θ,p

1 p

= 1−θ +

θ p1 .

= Lp (μ) .

4.3 Boundedness of Singular Integrals on H 1 (μ) In this subsection we prove that integral operators whose kernels satisfy a suitable integral Hörmander condition are bounded from H 1 (μ) to L1 (μ). Theorem 3 Let T be a linear operator which is bounded on L2 (μ) and admits a locally integrable kernel K off the diagonal that satisfies the condition  sup sup

∗ c R˜ y, z∈R˜ (R˜ )

|K(x, y) − K(x, z)| dμ(x) < ∞ ,

(15)

where the supremum is taken over alla Calderón-Zygmund sets R˜ and R˜ ∗ is defined as in Definition 6. Then T extends to a bounded operator from H 1 (μ) to L1 (μ). Proof Using (15), by [8, Theorem 1.2] it is easy to prove that the operator T is of weak type (1, 1). Then it is enough to show that there exists a constant C such that T a L1 ≤ C for any (1, ∞)-atom a. ˜ Recall that Let a be a (1, ∞)-atom supported in the Calderón–Zygmund set R. ∗ ˜ ˜ ˜ ˜ ˜ R ⊂ B(xR , 8h(R)), and R denote the dilated set {x ∈ V : d(x, R) < h(R)/4}. We estimate the integral V |T a|dμ.

38

L. Arditti et al.

We first estimate the integral on R˜ ∗ by the Cauchy-Schwarz inequality and the size estimate of the atom:  ˜ 1/2 |T a|dμ ≤ T a L2 μ(R˜ ∗ )1/2 ≤ C |T |L2 →L2 a L2 μ(R) R∗

≤ C |T |L2 →L2 .

(16)

We consider the integral on the complementary set of R˜ ∗ by using the fact that a has vanishing integral: 

 R˜ ∗c

|T a|dμ ≤

(R˜ ∗ )c







K(x, y) a(y)dμ(y) dμ(x) R˜







[K(x, y) − K(x, xR )] a(y)dμ(y) dμ(x)

 =  ≤

(R˜ ∗ )c



(R˜ ∗ )c

 =

R˜

R˜

R˜

|a(y)|

|K(x, y) − K(x, xR )| |a(y)|dμ(y)dμ(x)  

(R˜ ∗ )c

≤ a L1 sup

∗ c y∈R˜ (R˜ )

 |K(x, y) − K(x, xR )|dμ(x) dμ(y) |K(x, y) − K(x, xR )|dμ(x)

≤C,  

as required.

Remark The previous result applies to singular integral operators associated with the Laplacian L on the tree defined for every function f : V → C by 1 Lf (x) = f (x) − √ 2 q

q

(y)−(x) 2

f (y)

∀x ∈ V .

(17)

y∈V:d(x,y)=1

The Laplacian L is bounded on Lp (μ) for every p ∈ [1, ∞], it is self-adjoint on L2 (μ) and its spectrum on L2 (μ) is [0, 2]. Suppose that M : R → C is bounded supported in [0, 2) and satisfies the following Mikhlin-Hörmander condition of order s > 3/2 sup (Dt M)φ W2s < ∞ ,

(18)

t>0

for some φ ∈ Cc∞ ([ 12 , 4]), φ =  0, where (Dt M)(λ) = M(tλ) and W2s denotes the Sobolev space of order s modelled on L2 ([0, 2]). Then the operator M(L)

Hardy Spaces on Weighted Homogeneous Trees

39

extends to a bounded operator from H 1 (μ) to L1 (μ). Indeed, it was shown in [8, Theorem 2.3] that the integral kernel of the operator H (L) satisfies condition (15). For every function f : V → C we also define the gradient f by the formula: (∇f )(x) =

|f (y) − f (x)|

∀x ∈ V .

(19)

y∈V:d(x,y)=1

Then the first order Riesz transform ∇L−1/2 extends to a bounded operator from H 1 (μ) to L1 (μ). Indeed, it was shown in [8, Theorem 2.3] that the integral kernel of this operator satisfies condition (15). Acknowledgements Work partially supported by the MIUR project “Dipartimenti di Eccellenza 2018–2022” (CUP E11G18000350001). The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References 1. L. Arditti. Analysis on weighted homogeneous trees. https://webthesis.biblio.polito.it/8359/, (2018). 2. J. Bergh, J. Löfström. Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976. 3. A. Carbonaro, G. Mauceri, S. Meda. H 1 and BMO for certain locally doubling metric measure spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 3, 543–582. 4. D. Celotto, S. Meda. On the analogueue of the Fefferman-Stein theorem on graphs with the Cheeger property. Ann. Mat. Pura Appl.(4) 197 (2018), no. 5, 1637–1677. 5. R.R. Coifman, G. Weiss. Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. 6. G. B. Folland, E. M. Stein. Hardy spaces on homogeneous groups, Princeton University Press, 1982. 7. R. Hanks. Interpolation by the real method between BMO, Lα (0 < α < ∞) and H α (0 < α < ∞), Indiana Univ. Math. J. 26 (1977), 679–689. 8. W. Hebisch, T. Steger. Multipliers and singular integrals on exponential growth groups. Math. Z. 245 (2003), no. 1, 37–61. 9. P.W. Jones. Interpolation between Hardy spaces, Conference on harmonic analysis in honor of Antoni Zygmund I, II, (Chicago, 1981), 437–451, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983. 10. J. L. Journé. Calderón–Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón, Lecture Notes in Mathematics 994, Springer-Verlag, 1983. 11. A. Martini, A. Ottazzi, M. Vallarino. Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups, J. Anal. Math. 136 (2018), 357–397. 12. J. Peetre. Two observations on a theorem by Coifman, Studia Math. 64 (1979), 191–194. 13. N.M. Rivière, Y. Sagher. Interpolation between L∞ and H 1 , the real method, J. Funct. Anal. 14 (1973), 401–409. 14. E. M. Stein. Harmonic Analysis, Princeton University Press, 1993. 15. M. Vallarino. Spaces H 1 and BMO on ax + b-groups. Collect. Math. 60 (2009), no. 3, 277– 295.

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces Alessia Ascanelli

Abstract We consider the initial value problem for the plate equation with (t, x)−depending complex valued lower order terms. Under suitable decay conditions as |x| → ∞ on the imaginary part of the subprincipal term we prove energy estimates in weighted Sobolev spaces. This provides also well posedness of the Cauchy problem in the Schwartz space S(Rn ) and in S (Rn ). Keywords Plate equation · Weighted Sobolev spaces · Pseudodifferential operators

1 Introduction and Main Result In the present paper we focus on the Cauchy problem ⎧ ⎪ ⎪ ⎨Lu(t, x) = f (t, x) u(0, x) = u0 (x) ⎪ ⎪ ⎩∂ u(0, x) = u (x) t

1

(t, x) ∈ (0, T ] × Rn x ∈ Rn x∈

(1)

Rn

for the operator L := ∂t2 + a(t)%2x +

3

aα(k) (t, x)∂xα , a(t) ≥ 0,

(2)

k=0 |α|=k

 where %x = nj=1 ∂x2j as usual. The equation Lu = f with L in (2) is usually called “plate equation” since it models the motion of bending of a thin elastic plate under

A. Ascanelli () Dipartimento di Matematica e Informatica, Università degli studi di Ferrara, Ferrara, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_3

41

42

A. Ascanelli

a force (external force + stress force) which is a linear function of the displacement u = u(t, x) and of its derivatives ∂xα , 1 ≤ |α| ≤ 3, where t is the time and x is the position variable. In such a model, pure imaginary terms of the third order appear in the stress term. From this, it is natural to allow the coefficients ak , 0 ≤ k ≤ 3, to be complex valued. The operator (2) is of non-kowalewskian type, but the assumption √ a(t) real valued guarantees the existence of real characteristic roots τ± = ± a(t)|ξ |2 for the principal part in the Petrowski sense of L, that is L4 = ∂t2 + a(t)%2x . This allows to use some tipically hyperbolic techniques to study existence and uniquenes of the solution to (1), (2) in Sobolev spaces. The lower order terms in (2) have some influence on the well posedness of the Cauchy problem (1). Indeed, it is well known (see [2, 10]) that to have well posedness in Sobolev spaces of the Cauchy problem for a first order non kowalewskian equation some decay conditions are necessary on the lower order terms as |x| → ∞; sufficient conditions for first order equations have been given e.g. in [1, 11, 12]. Well posedness in usual Sobolev spaces H s , s ∈ R, and in H ∞ of the Cauchy problem for equations of the form (2) has been widely studied, see [4, 7] an the reference therein, as far as well posedness in the narrower Gevrey spaces, see [5, 13] and their references. Literature shows that one of the conditions leading to a Cauchy problem (1), (2) well posed respectively in H s , H ∞ , Gs (Gevrey class of index s ≥ (3) 1) is Im aα ∼ x−σ as |x| → ∞, with respectively σ > 1, σ = 1, σ ∈ (0, 1). The energy estimates obtained in these papers give precise information about the regularity of the solution but no information about the behavior as |x| → ∞ of the solution itself. In the present paper we aim to study well posedness of (1), (2) in the weighted Sobolev spaces H s1 ,s2 (Rn ) = {u ∈ S (Rn ) | u s1 ,s2 := xs2 Ds1 u < ∞},

(3)

sj ∈ R, j = 1, 2, where we denote by Ds1 the Fourier multiplier with symbol ξ s1 and · stands for the L2 −norm. We recall that for s2 = 0 we recapture the standard Sobolev spaces and that the following identities hold:  s1 ,s2 ∈R

H s1 ,s2 (Rn ) = S(Rn ),



H s1 ,s2 (Rn ) = S (Rn ).

(4)

s1 ,s2 ∈R

Moreover we recall that S(Rn ) is dense in H s1 ,s2 (Rn ) for any s1 , s2 ∈ R. The study of non kowalewskian evolution equations in H s1 ,s2 spaces started with [6], where first order equations with evolution degree p ≥ 2 in spatial dimension n = 1 have been considered; then, in the very recent paper [3] Schrödinger type equations (evolution degree p = 2) in spatial dimension n ≥ 1 have been studied in the framework of Gelfand-Shilov spaces.

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

43

In this paper we shall prove a well posedness result in weighted Sobolev spaces for (1), (2) by giving an energy estimate that provides information both on the regularity and on the behavior at infinity of the solution. More precisely we are going to give sufficient conditions on the coefficients of (2) such that the Cauchy problem (1) is well posed in H s1 ,s2 (Rn ), i.e. for every given Cauchy data u0 , u1 , f (t) in H s1 ,s2 (Rn ) there exists a unique solution of (1), (2) with both the same regularity and the same behavior at infinity as the data, and with continuous dependence on time. As a direct consequence of this result, by (4) we shall get the well posedness of (1), (2) in S(Rn ), S (Rn ). We are going to prove the following result: Theorem 1 Consider the Cauchy problem (1), (2) with coefficients a ∈ C 1 [0, T ] (k) real valued, aα ∈ C 1 ([0, T ]; C ∞ (Rn )) complex valued for 0 ≤ k = |α| ≤ 3. Assume moreover that for every β ∈ Zn+ and for some constants C¯ > 0, Cβ ≥ 0: a(t) ≥ C¯ |∂xβ

Im aα(3) (t, x)|

∀t ∈ [0, T ] −σ −|β|

≤ Cx

|∂xβ aα(k) (t, x)| ≤ Cx−|β| ,

,

(5)

σ > 1, |α| = 3, (t, x) ∈ [0, T ] × R

(6)

0 ≤ k = |α| ≤ 3, (t, x) ∈ [0, T ] × Rn .

(7)

n

Then, the Cauchy problem (1) is well-posed in H s1 ,s2 (Rn ) for every s1 , s2 ∈ R, in S(Rn ) and in S (Rn ). More precisely, for every (s1 , s2 ) ∈ R2 , u0 ∈ H s1 +2,s2 (Rn ), u1 ∈ H s1 ,s2 (Rn ), f ∈ C([0, T ]; H s1 ,s2 (Rn )), there is a unique solution u ∈ C([0, T ]; H s1 +2,s2 (Rn )) ∩ C 1 ([0, T ]; H s1 ,s2 (Rn )) which satisfies the following energy estimate: u(t, ·) 2s1 +2,s2 + ∂t u(t, ·) 2s1 ,s2   t ≤ Cs1 ,s2 u0 2s1 ,s2 + u1 2s1 ,s2 + f (τ, ·) 2s1 ,s2 dτ

(8)

0

for every t ∈ [0, T ] and for some Cs1 ,s2 > 0. The proof of Theorem 1 will consist of four steps: factorization of (2) into the product of two Schrödinger type operators of the form (9) here below, reduction of (1) to an equivalent first order system, proof of well posedness in H s1 ,s2 of the system, and finally proof of well posedness in H s1 ,s2 of (1) by equivalence. The paper is organized as follows: in Sect. 2 we provide some preliminaries concerning SG-calculus and properties of H s1 ,s2 spaces; in Sect. 3 we shall consider the first order scalar equation of Schrödinger type ∂t u − ia(t)%x u +

n

aj (t, x)∂xj u + b(t, x)u = f (t, x),

(9)

j =1

a(t) ∈ R, aj (t, x), b(t, x) ∈ C, and give sufficient conditions for well posedness in H s1 ,s2 (Rn ) of the associated Cauchy problem; in Sect. 4 we extend the result of

44

A. Ascanelli

Sect. 3 to the case of a first order system; finally, in Sect. 5 we prove Theorem 1 by equivalence of (1) to a first order system of the type studied in Sect. 4.

2 Preliminaries We recall here some basic facts concerning SG classes of pseudo-differential operators and the weighted Sobolev spaces defined by (3). We refer to [8, 9, 14– 16] for proofs and details. Definition 1 For any m1 , m2 ∈ R we shall denote by SGm1 ,m2 (Rn ) the space of all functions p(x, ξ ) ∈ C ∞ (R2n ) such that for every α, β ∈ Zn+





sup ξ −m1 +|α| x−m2 +|β| ∂ξα ∂xβ p(x, ξ ) < +∞.

(10)

(x,ξ )∈R2n

Given any p ∈ SGm1 ,m2 (Rn ), we define the pseudo-differential operator P = Op(p) with symbol p as standard by −n



P u(x) = (2π )

Rn

eix·ξ p(x, ξ )u(ξ ˆ )dξ,

u ∈ S(Rn ),

(11)

where uˆ denotes the Fourier transform of u. We denote by LGm1 ,m2 (Rn ) the space of all operators of the form (11) with symbol in SGm1 ,m2 (Rn ) and by K the space 2n of all operators (11) with symbol in S(R ) = SGm (Rn ). m∈R2

Proposition 1 Given p ∈ SGm1 ,m2 (Rn ), the operator P = Op(p) is linear and continuous from S(Rn ) to S(Rn ) and it extends to a linear continuous map from S (Rn ) to itself. More precisely, P : H s1 ,s2 (Rn ) −→ H s1 −m1 ,s2 −m2 (Rn ) is linear and continuous for every s ∈ R2 . Proposition 2 Every P ∈ K can be extended to a linear and continuous map from S (Rn ) to S(Rn ). 



Proposition 3 Let p ∈ SGm1 ,m2 , q ∈ SGm1 ,m2 . Then: 



1. there exists s ∈ SGm1 +m1 ,m2 +m2 such that P Q = Op(s) + K for some K ∈ K;   2. there exists r ∈ SGm1 +m1 −1,m2 +m2 −1 such that [Op(p), Op(q)] = Op(r) + K  for some K  ∈ K; 3. there exists p ∈ SGm1 ,m2 such that P  = Op(p ) + K  for some K  ∈ K, where P  denotes the L2 −adjoint of P . We remark that we can equivalently define SGm1 ,m2 (Rn ) by asking −m1 +|α|

sup ξ h

(x,ξ )∈R2n





x−m2 +|β| ∂ξα ∂xβ p(x, ξ ) < +∞

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

45

with ξ h := (h2 + |ξ |2 )1/2 , h ≥ 1, instead of (10). This equivalent definition will turn out to be useful in Sect. 3, see (32). Let us now consider for s1 , s2 ∈ R the weighted Sobolev space H s1 ,s2 (Rn ) defined in (3). Proposition 4 The following properties hold: 1. H s1 ,s2 (Rn ) is a Hilbert space endowed with the inner product ! u, vs1 ,s2 = xs2 Ds1 u, xs2 Ds1 v , where  ,  denotes the scalar product in L2 . 2. H s1 ,s2 (Rn ) is the space of all u ∈ S (Rn ) such that Ds1 (xs2 u) ∈ L2 (Rn ) and the norms xs2 Ds1 u and Ds1 (xs2 u) are equivalent. 3. H s1 ,0 (Rn ) coincides with H s1 (Rn ), the space of all u ∈ S (Rn ) such that ξ s1 u(ξ ˆ ) ∈ L2 (Rn ). 4. If sj ≤ tj , j = 1, 2, then H t1 ,t2 (Rn ) ⊆ H s1 ,s2 (Rn ). Moreover, if sj < tj , j = 1, 2, then the embedding H t1 ,t2 (Rn ) → H s1 ,s2 (Rn ) is compact. 5. 

H s1 ,s2 (Rn ) = S(Rn ),

(s1 ,s2 )∈R2



H s1 ,s2 (Rn ) = S (Rn ).

(s1 ,s2 )∈R2

6. If s1 ∈ Z+ , then H s1 ,s2 is the space of all functions u ∈ L2 (Rn ) such that xs2 ∂xα u(x) ∈ L2 (Rn ) for all |α| ≤ s1 and an equivalent norm is given by

xs2 ∂xα u(x) .

|α|≤s1

7. The Fourier transform F : H s1 ,s2 (Rn ) −→ H s2 ,s1 (Rn ) is a linear and continuous bijection.

3 Well-Posedness for a Scalar Schrödinger-Type Equation In this section we focus on the following Cauchy problem on [0, T ] × Rn : "

P (t, x, ∂t , ∂x )u(t, x) = f (t, x) u(0, x) = g(x)

(12)

46

A. Ascanelli n

P (t, x, ∂t , ∂x ) = ∂t − i a(t)% ˜ x +

aj (t, x)∂xj + b(t, x)

(13)

j =1

where a(t) ˜ is a real valued coefficient, while the other coefficients aj and b are complex valued. We are going to prove the following theorem: Theorem 2 Let P (t, x, ∂t , ∂x ) be an operator of the form (13) with a, ˜ aj and b continuous with respect to t and satisfying for all (t, x) ∈ [0, T ] × Rn , β ∈ Nn and 1 ≤ j ≤ n the following conditions: |a(t)| ˜ ≥ C¯ > 0 |∂xβ (Im aj )(t, x)|

(14)

−σ −|β|

≤ Cx

,

σ >1

|∂xβ (Re aj )(t, x)| ≤ Cx−|β| |∂xβ b(t, x)| ≤ Cx−|β| ,

(15) (16) (17)

for some positive constant C independent of β. Let moreover f ∈ C([0, T ]; H s1 ,s2 (Rn )) and g ∈ H s1 ,s2 (Rn ) for some (s1 , s2 ) ∈ R2 . Then the Cauchy problem (12) admits a unique solution u ∈ C([0, T ]; H s1 ,s2 (Rn )) which satisfies:   t f (τ ) 2s1 ,s2 dτ , u(t) 2s1 ,s2 ≤ Cs1 ,s2 g 2s1 ,s2 +

(18)

0

for t ∈ [0, T ] and for some Cs1 ,s2 > 0. In particular, the Cauchy problem (1) is well posed in H s1 ,s2 (Rn ), S(Rn ) and in S (Rn ). The proof of Theorem 2 has already been given in the particular case n = 1 in the paper [6]; there, a more general class of p− evolution equations on [0, T ] × R has been considered. The case of dimension n ≥ 2 is much more technically complicated. Most of the technical problems have been overcome in the very recent paper [3], dealing with well-posedness of the Cauchy problem (12) in GelfandShilov type spaces. We give so the proof of Theorem 2 with x ∈ Rn , relying on [3] for the heavily technical details. Proof The idea is to perform a change of variable of the form v(t, x) = e (x, D)u(t, x),

(19)

where e (x, D) denotes a pseudodifferential operator with symbol e(x,ξ ) and e (x, D) is invertible with inverse (e (x, D))−1 . In this way we are reduced to consider the auxiliary Cauchy problem "

P (x, Dt , Dx )v(t, x) = e (x, D)f (t, x) =: f (t, x) v(0, x) = e (0, x, D)g(x) =: g (x)

(20)

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

47

in the unknown v, where P := e (x, D)P (t, x, ∂t , ∂x )(e (x, D))−1 . By a suitable choice of the symbol , we’ll get that the equivalent Cauchy problem (20) is well-posed in H s1 ,s2 . We are going to prove Theorem 2 under the ¯ assumption a(t) ˜ ≥ C¯ > 0; see Remark 1 for the case a(t) ˜ ≤ −C. Construction of  Let us define a function λ1 (x, ξ ) such that n

(∂xj λ1 )ξj = |ξ |g1 (x),

x ∈ Rn ,

(21)

j =1

where g1 (x) = Mx−σ with M > 0 to be chosen later on. It is easy to check that the desired function λ1 is given by:  λ1 (x, ξ ) =

x·ω

g1 (x − τ ω)dτ,

ω = ξ/|ξ |.

(22)

0

By Lemma 3.1 in [3] we know that λ1 has a symbol-like behavior only in the region R = {(x, ξ ) ∈ R2n | |x · ω| ≤ x/2}, where the estimate |∂ξα ∂xβ λ1 (x, ξ )| ≤ MCx−σ +1−|β| |ξ |−|α|

(23)

holds. So, we introduce a partition of the phase space and in the complementary region |x · ω| ≥ x/2 we define the symbol λ2 (x, ξ ) satisfying the condition n

(∂xj λ2 )ξj = |ξ |g2 (x), x ∈ Rn ,

(24)

j =1

with g2 (x, ξ ) = Mx · ω−σ . As before we can take  λ2 (x, ξ ) :=

x·ω

 g2 (x − τ ω, ξ )dτ =

0

x·ω

Mz−σ dz,

(25)

0

and we get by Lemma 3.2 in [3] that |∂ξα ∂xβ λ2 (x, ξ )| ≤ MCx−σ +1−|β| |ξ |−|α| ,

(x, ξ ) ∈ R2n \ R.

(26)

48

A. Ascanelli

Now, we choose a function χ ∈ Cc∞ (R) such that 0 ≤ χ (t) ≤ 1, tχ  (t) ≤ 0, χ (t) = 1 for |t| ≤ 1/2, χ (t) = 0 for |t| ≥ 1, and define ˜ ˜ ξ )), λ(x, ξ ) = −λ1 (x, ξ )χ˜ (x, ξ ) − λ2 (x, ξ )(1 − χ(x,  2x · ω . χ(x, ˜ ξ) = χ x

(27)

It’s easy to show that (23) and (26) give that for every α, β ∈ Zn+ and for every (x, ξ ) ∈ R2n , |ξ | > 1, we have |∂ξα ∂xβ λ˜ (x, ξ )| ≤ MCx−σ +1−|β| |ξ |−|α| .

(28)

Finally, we cut off the function λ˜ (x, ξ ) near ξ = 0 by defining   ˜ (x, ξ ) = 1 − χ¯ (h−1 |ξ |) λ(x, ξ)

(29)

¯ = 1 for |t| ≤ 1; we have obtained a symbol for a function χ¯ ∈ Cc∞ (Rn ) with χ(t) in the class SG0,−σ +1 (R2n ), which satisfies −|α|

|∂ξα ∂xβ (x, ξ )| ≤ MCx−σ +1−|β| ξ h

,

(x, ξ ) ∈ R2n ,

(30)

h ≥ 1 to be chosen later on, M and C constants independent of h. Moreover, the symbol  satisfies following estimate: n

(∂xj )(x, ξ )ξj ≤ −Mx−σ |ξ |,

(31)

j =1

see [3, Lemma 3.5], that will be crucial in the proof here below. Invertibility of e We now look for an inverse of the operator e , which is a pseudodifferential operator of order (0, 0) thanks to the choice of  in (29). We compute the symbol of the composition of operators e ◦ e− :   σ e e− (x, ξ ) = I − r(x, D), where the symbol of the principal part of r is ∂ξ (x, ξ )Dx (x, ξ ) ∈ SG−1,−2σ +1 , so r satisfies ∀α, β ≥ 0 −1−|α|

|∂ξα Dxβ r(x, ξ )| ≤ Cα,β ξ h

−β x−β ≤ Cα,β · h−1 ξ −α h x .

(32)

This means that for h largeenough, say h ≥ h0 , the operator I −r(x, D) is invertible n by Neumann series and +∞ n=0 R is the inverse operator. Similar considerations

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

49

hold  for the composition e− ◦ e , thus e(x,Dx ) is invertible for every h ≥ h0 and +∞ n − e n=0 R is its left and right inverse. Change of Variable Let us now perform the change of variable (19). We fix h ≥ h0 , and by the rules of composition for operators with symbols in SG-classes we get ⎛ P = e (x, D) ⎝∂t − i a(t)% ˜ x +

n

⎞ aj (t, x)∂xj + b(t, x)⎠ (e (x, D))−1

j =1

= ∂t − i a(t)% ˜ x +

n

aj (t, x)∂xj − 2a(t) ˜

j =1

n

∂xj Dxj + b (t, x),

(33)

j =1

where ∂xj Dxj denotes a pseudodifferential operator with symbol ∂xj (x, ξ )ξj and b (t, is a new term of order (0, 0). The new (and crucial) term x) n −2a(t) ˜ ∂ j =1 xj Dxj in (33) comes from the fact that x e  =

n

∂xj (e ∂xj  + e ∂xj ) =

j =1

n

j =1

  e (∂xj )2 + ∂x2j  + 2(∂xj )∂xj + e x ,

which gives e %x (e )−1 = %x −

n

  e (∂xj )2 + ∂x2j  + 2(∂xj )∂xj (e )−1

j =1

= %x −

n

2(∂xj )∂xj + r0

j =1

with r0 (x, D) a term of order (0, 0). Well-Posedness for the Auxiliary Problem We are now ready to show that the auxiliary problem (20) is well posed in H m (Rn ), S(Rn ), S (Rn ) by means of an energy estimate. We compute so d v(t) 2 = 2 ReP v, v + 2 Rei a(t)% ˜ (34) x v, v dt n

   − 2 Re ˜ iaj (t, x)Dxj − 2a(t)∂ xj Dxj v, v − 2 Reb v, v. j =1

50

A. Ascanelli

We immediately notice that   ∗ ˜ ˜ 2 Rei a(t)% ˜ x v, v =  i a(t)% x + (i a(t)% x ) v, v = 0 since a(t) ˜ is real valued, and n

  2 Re  iaj (t, x)Dxj − 2a(t)∂ ˜ xj Dxj v, v j =1 n

   Im aj (t, x)Dxj + 2a(t)∂ ˜ = 4 Re− xj Dxj v, v + Reb v, v j =1

= 4 ReA(t, x, D)v, v + Reb v, v

(35)

for another term b with symbol b (t, x, ξ ) ∈ SG0,0 , with A(t, x, D) = −

n

  Im aj (t, x)Dxj + 2a(t)∂ ˜ xj Dxj . j =1

Now, we make use of the crucial estimate (31) and, with the constants C¯ and C respectively in (14) and (15), we get: n

j =1

Im aj (t, x)ξj + 2a(t) ˜

n

−σ ∂xj (x, ξ )ξj ≤ Cx−σ |ξ | − 2a(t)Mx ˜ |ξ |

j =1 −σ ¯ ≤ (C − 2CM)x |ξ | ≤ 0

(36)

¯ By (36) we see that, if we if we take M large enough, precisely M ≥ C/(2C). choose M large enough, then the operator A(t, x, D) has symbol A(t, x, ξ ) ∈ SG1,−σ such that A(t, x, ξ ) ≥ 0 ∀(x, ξ ) ∈ R2n ; by the sharp-Garding inequality we know that there exists a constant c > 0 such that ReA(t, x, D)v, v ≥ −c v 2 . Substituting this into (35) and (34) we come finally to   d v(t) 2 ≤ C P v 2 + v 2 dt for a new constant C, and an application of Gronwall’s inequality gives   t v(t, ·) 2 ≤ C v(0, ·) 2 + P v(τ, ·) 2 dτ , 0

∀t ∈ [0, T ].

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

51

By standard arguments from the energy method we deduce that the same estimate holds in H s1 ,s2 spaces for all (s1 , s2 ) ∈ R2 , and that for every f ∈ C([0, T ], H s1 ,s2 (Rn )), g ∈ H s1 ,s2 (Rn ), the Cauchy problem (20) has a unique solution v(t, x) ∈ C([0, T ], H s1 ,s2 (Rn )) satisfying   t v(t, ·) 2s1 ,s2 ≤ C g 2s1 ,s2 + f (τ, ·) 2s1 ,s2 dτ ,

∀t ∈ [0, T ].

0

By the relation (19) between u and v and since the operator e± has order (0, 0), we get existence and uniqueness of a solution u of (12) satisfying estimate (18).   ¯ Remark 1 In the case of a leading coefficient a(t) ˜ such that a(t) ˜ ≤ −C, the proof is the same, with the only difference that we define (x, ξ ) := − 1 − χ˜ (h−1 |ξ |) λ˜ (x, ξ ) instead of (29). Formula (31) trivially changes into  n −σ |ξ |, and (36) is true thanks to the negative sign j =1 (∂xj )(x, ξ )ξj ≥ Mx of a. ˜

4 Well-Posedness for First Order Systems In this section we are going extend the results of Sect. 3 for a scalar equation to an ν × ν system of the form 

P (t, x, D)W (t, x) = F (t, x), W (0, x) = W0 (x)

(t, x) ∈ [0, T ] × Rn x ∈ Rn

(37)

with ⎛ ⎜ ⎜ ⎜ P = ∂t − i ⎜ ⎜ ⎝

⎞

λ1 (t, x, D)

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

λ2 (t, x, D) ..

. λν−1 (t, x, D)

(38)

λν (t, x, D) + R(t, x, D), λj (t, x, ξ ) = αj (t)|ξ |2 + βj (t, x, ξ ) + γj (t, x), j = 1, . . . , ν,

(39)

αj real valued, βj , γj complex valued, j = 1, . . . , ν, and R(t, x, ξ ) ∈ SG0,0 . We are going to prove the following:

52

A. Ascanelli

Theorem 3 Consider the Cauchy problem (37), (38), (39), and assume that for 1 ≤ j ≤ ν the following conditions hold: |αj (t)| ≥ C¯ > 0, ∀t ∈ [0, T ], Im βj ∈ C([0, T ]; SG1,−σ ), Re βj ∈ C([0, T ]; SG

1,0

(40) σ > 1,

(41) (42)

),

γj ∈ C([0, T ]; SG0,0 ).

(43)

Then, the Cauchy problem (37) is well-posed in H s1 ,s2 (Rn ), S(Rn ), S (Rn ). More precisely, for all s1 , s2 ∈ R, F ∈ C([0, T ]; H s1 ,s2 (Rn )) and W0 ∈ H s1 ,s2 (Rn ) there is a unique solution W ∈ C([0, T ]; H s1 ,s2 (Rn )) which satisfies the following energy estimate:   t |F (τ, ·) |2s1 ,s2 dτ ∀t ∈ [0, T ], (44) |W (t, ·) |2s1 ,s2 ≤ C |W0 |2s1 ,s2 + 0

for some C = C(s1 , s2 ) > 0, where for a given vector W = (W1 , · · · , Wν ) we ν

2 denote |W |s1 ,s2 := Wj 2s1 ,s2 . j =1

Proof We perform the change of variable e W = Z with  in (29), and we obtain the (equivalent to (37)) Cauchy problem "

P Z(t, x) = F (t, x)

(t, x) ∈ [0, T ] × Rn

Z(0, x) = Z (x)

x ∈ Rn

(45)

for F = e F , Z = e W0 and ⎛ ⎜ P = (e )L(e )−1 = e ∂t (e )−1 − e ⎝ ⎛ ⎜ = ∂t − i ⎝

e λ1 (e )−1

⎟  −1   −1 ⎠ (e ) + e R(e )

. iλν

⎟ ⎠ + R

. e λν (e )−1

=: ∂t − iA + R .

.. ⎞

..

⎞

iλ1

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

53

with R (t, x, ξ ) ∈ SG0,0 For a solution Z of (45) we have: d |Z |2 = 2 ReZ  , Z = 2 ReF , Z + 2 ReiA Z, Z − 2 ReR Z, Z dt ≤ C( |F |2 + |Z |2 ) + 2 ReiA Z, Z

(46)

for some C > 0, where for given vectors U = (U1 , . . . , Uν ) and V = (V1 , . . . , Vν ) ν

Uj , Vj . We have so to estimate the term we denote U, V  := j =1

ReiA Z, Z =

ν

Re(iA )j Zj , Zj ,

j =1

where (iA )j are for 1 ≤ j ≤ ν the entries of the diagonal matrix iA . But we note that (iA )j are of the form   (iA )j = ie λj (e )−1 = e −iαj (t)%x + iβj (t, x, D) + iγj (t, x) (e )−1

(47)

˜ in (13), iβj (t, x, D) for j = 1, . . . , ν, where the αj correspond to −a(t) are pseudodifferential operators of order (1, 0), complex valued and satisfying the conditions (41) and (42) corresponding to (15) and (16) in (13), iγ under condition (43) corresponds to b in (13) under assumption (17). Thus, following (33), we get (iA )j = −iαj (t)%x + iβj (t, x, D) − 2αj (t)

n

∂ξh Dxh + iγj (t, x)

h=1

for a new term γj ∈ SG0,0 . So, as in (34), (35), we come to Re(iA )j Zj , Zj  = Re−iαj (t)%x Zj , Zj  ⎛ − ⎝Im βj (t, x, D) + 2αj (t)

n

⎞ ∂ξh Dxh ⎠ Zj , Zj  + Reγ  Zj , Zj 

h=1

and taking M large enough (see (36)) we get   Im βj (t, x, D) + 2αj (t)

n

∂ξh Dxh Zj , Zj  ≥ −c Zj 2 .

h=1

This gives, possibly enlarging the constant C in (46), d |Z |2 ≤ C |F |2 + |Z |2 . dt

54

A. Ascanelli

Now, by standard arguments, we get energy estimates in L2 and in H s1 ,s2 for Z, and by the change of variable Z = e W with e of order (0, 0) the energy estimate (44) straightly follows.  

5 Proof of Theorem 1 We first rewrite the operator L as L = −Dt2 + a(t)Dx4 −

iaα(3) (t, x)Dxα +

2

i k aα(k) (t, x)Dxα

k=0 |α|=k

|α|=3

and compute its principal symbol in the Petrowski sense P4 (t, τ, ξ ) = −τ 2 + a(t)|ξ |4 ; the equation P4 (t, τ, ξ ) = 0 admits the two real and distinct roots √ λ± (t, ξ ) = ± a(t) |ξ |2 ∈ C 1 ([0, T ]; SG2,0 ). By assumption (5), the difference (λ+ − λ− )(t, ξ ) is elliptic. We use these two roots to define two symbols λ˜ ± (t, x, ξ ) = λ± (t, ξ ) ± λ1 (t, x, ξ ),

λ1 (t) ∈ SG1,0

(48)

such that (τ − λ˜ + (t, x, ξ ))(τ − λ˜ − (t, x, ξ )) = τ 2 − a(t)|ξ |4 +

iaα(3) (t, x)ξ α + lower order terms.

|α|=3

The correct way to choose λ1 is given by λ1 (t, x, ξ ) = −

iaα(3) (t, x)ξ α . √ 2 a(t)|ξ |2 |α|=3

(49)

Now we bring this factorization to operators level by computing    Dt − λ˜ + (t, x, D) Dt − λ˜ − (t, x, D)

iaα(3) (t, x)Dx3 + T2,0 (t, x, D) = Dt2 − a(t)Dx4 + |α|=3

= −L +

2

k=0 |α|=k

i k aα(k) (t, x)Dxα + T2,0 (t, x, D)

(50)

The Global Cauchy Problem for the Plate Equation in Weighted Sobolev Spaces

55

where λ˜ ± (t, x, D) are pseudodifferential operators with symbol λ˜ ± (t, x, ξ ), and with T2,0 (t, x, ξ ) ∈ SG2,0 . Equation (50) provides a factorization of the operator L modulo terms of the second order; by means of this factorization we perform the following reduction of the Cauchy problem (1) to an equivalent first order system. We define the vector V = (v0 , v1 ) by 

v0 = D2x u v1 = (Dt − λ˜ − (t, x, D))u.

(51)

Computations give: (Dt − λ˜ − )v0 = D2 v1 − [λ˜ − , D2 ]D−2 v0 , where [λ˜ − , D2 ]D−2 is an operator of order (0, 0), and   Dt − λ˜ + v1 = (Dt − λ˜ + )(Dt − λ˜ − )u ⎛ ⎞ 2

= ⎝−L + i k aα(k) (t, x)Dxα + T2,0 (t, x, D)⎠ u k=0 |α|=k

= −f − T0,0 (t, x, D)v0 where T0,0 (t, x, ξ ) ∈ SG0,0 . We have so )

 Dt −

λ˜ − D2 0 λ˜ +



* + R0,0 V =



0 −f

,

(52)

where R0,0 is a matrix with all the entries of order (0, 0). At symbol’s level, the  λ˜ − (t, x, ξ ) ξ 2 diagonalizer for the matrix is given by 0 λ˜ + (t, x, ξ )  K(t, x, ξ ) =

1 k(t, x, ξ ) 0 1

k(t, x, ξ ) =

,

ξ 2 , (λ˜ + − λ˜ − )(t, x, ξ )

which is by (5) a matrix of order (0, 0), invertible with inverse given by K −1 (t, x, ξ ) =



1 0

−k(t, x, ξ ) 1

.

At operator’s level this factorization gives, for the new variable W K −1 (t, x, D)V the following system: 

P W = F˜ W (0) = W0

=

(53)

56

A. Ascanelli

where F˜ = iK(t, x, D)−1 (0, −f )T , W0 = W (0) = K −1 (t, x, D)V (0) and * )  λ˜ − D2 + iR K(t, x, D) P = K −1 (t, x, D) ∂t − i 0,0 0 λ˜ +  λ˜ − (t, x, D) 0 = ∂t − i + R(t, x, D), 0 λ˜ + (t, x, D)

(54)

R a matrix of remainders of order (0, 0). Notice that the system (53), (54) has the structure (37), (38), and that by (48), (49) we have that

iaα(3) (t, x)ξ α + λ˜ ± (t, x, ξ ) = ± a(t)|ξ |2 ∓ √ 2 a(t)|ξ |2 |α|=3

(55)

fulfill the assumptions of Theorem 3. Thus, for every F˜ ∈ C([0, T ]; H s1 ,s2 (Rn )) and W0 ∈ H s1 ,s2 (Rn ) there exists a unique W ∈ C([0, T ]; H s1 ,s2 (Rn )) satisfying the energy estimate (44). In the present case, given u0 , u1 , f as in Theorem 1, we use the changes of variable W (t, x) = K −1 (t, x, D)V (t, x) = K −1 (t, x, D)(D2 u(t, x), (Dt − λ˜ + (t, x, D))u(t, x)) F˜ (t, x) = iK −1 (t, x, D)(0, −f )T

with K, K −1 of order (0, 0) and, by the energy estimate (44) for the system (53), we obtain for the scalar Cauchy problem (1) the following energy estimate: u(t, ·) 2s1 +2,s2 = D−2 v0 (t, ·) 2s1 +2,s2 ≤ |V (t, ·) |2s1 ,s2   t ≤ C |W (t, ·) |2s1 ,s2 ≤ C |W0 |2s1 ,s2 + |F (τ, ·) |2s1 ,s2 dτ  ≤ C |V0 |2s1 ,s2 +

 t 0

0

f (τ, ·) 2s1 ,s2 dτ

  t f (τ, ·) 2s1 ,s2 dτ , ≤ C u0 2s1 +2,s2 + u1 2s1 ,s2 + 0

t ∈ [0, T ]

with a positive constant C possibly enlarging line by line. This concludes the proof.

References 1. A.Ascanelli, C.Boiti, L.Zanghirati. Well-posedness of the Cauchy problem for p-evolution equations. J. Differential Equations 253 (2012), 2765–2795. 2. A.Ascanelli, C.Boiti, L.Zanghirati. A Necessary condition for H ∞ Well-Posedness of pevolution equations. Advances in Differential Equations 21, n.12 (2016), 1165–1196.

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3. A.Ascanelli, M.Cappiello. Schrödinger type equations in Gelfand-Shilov spaces. Journal de Mathématiques Pures et Appliquées 132 (2019), 207–250. 4. A.Ascanelli, M.Cicognani, F.Colombini. The global Cauchy problem for a vibrating beam equation. J. Differential Equations 247 (2009) 1440–1451. 5. A.Ascanelli, M.Cicognani. Gevrey solutions for a vibrating beam equation. Rend. Sem. Mat. Univ. Pol. Torino 67 (2009), n.2, 151–164. 6. A.Ascanelli, M.Cappiello. Weighted energy estimates for p-evolution equations in SG classes. Journal of Evolution Equations 15, n.3 (2015), 583–607. 7. M.Cicognani, M.Reissig. On Schrödinger type evolution equations with non-Lipschitz coefficients. Ann. Mat. Pura Appl. 190, n.4 (2011), 645–665. 8. H.O. Cordes. The technique of pseudodifferential operators. Cambridge Univ. Press, 1995. 9. Y.V. Egorov, B.-W. Schulze. Pseudo-differential operators, singularities, applications. Operator Theory: Advances and Applications, 93 Birkhäuser Verlag, Basel, 1997. 10. W. Ichinose. Some remarks on the Cauchy problem for Schrödinger type equations. Osaka J. Math. 21 (1984), 565–581. 11. W.Ichinose. Sufficient condition on H ∞ well-posedness for Schrödinger type equations. Comm. Partial Differential Equations, 9, n.1 (1984), 33–48. 12. K. Kajitani, A. Baba. The Cauchy problem for Schrödinger type equations. Bull. Sci. Math. 119 (1995), 459–473. 13. T. Kinoshita, H. Nakazawa. On the Gevrey wellposedness of the Cauchy problem for some non-Kowalewskian equations. J. Math. Pures Appl. 79 (2000), 295–305. 14. C. Parenti. Operatori pseudodifferenziali in Rn e applicazioni. Ann. Mat. Pura Appl. 93, 359– 389 (1972). 15. E. Schrohe. Spaces of weighted symbols and weighted Sobolev spaces on manifolds. “Pseudodifferential Operators”, Proceedings Oberwolfach 1986. H. O. Cordes, B. Gramsch and H. Widom editors, Springer LNM, 1256 New York, 360–377 (1987). 16. B.-W. Schulze. Boundary value problems and singular pseudodifferential operators. J. Wiley & sons, Chichester, 1998.

Cone-Adapted Shearlets and Radon Transforms Francesca Bartolucci, Filippo De Mari, and Ernesto De Vito

Abstract We show that the cone-adapted shearlet coefficients can be computed by means of the limited angle horizontal and vertical (affine) Radon transforms and the one-dimensional wavelet transform. This yields formulas that open new perspectives for the inversion of the Radon transform. Keywords Cone-adapted shearlets · Wavelets · Radon transforms

1 Introduction The inversion of the Radon transform is a classical ill-posed inverse problem and consists in reconstructing an unknown signal f on R2 from its line integrals [14]. The Radon transform of a signal f is a function on P1 × R = { |  line of R2 } whose value at a line is the integral of f along that line. We label lines in the plane by pairs (v, t) ∈ R2 as x + vy = t and we define the horizontal (affine) Radon transform Rf : R2 → C of any f ∈ L1 (R2 ) by  Rf (v, t) =

R

f (t − vy, y) dy,

a.e. (v, t) ∈ R2 .

This version of the Radon transform is proved to be particularly well-adapted to the structure of the classical shearlet transform, see [5] and [11]. We recall that the key idea in shearlet analysis is to construct a family of analyzing functions −1 2 × {Sb,s,a ψ(x) = |a|−3/4 ψ(A−1 a Ns (x − b)) : b ∈ R , s ∈ R, a ∈ R }

F. Bartolucci () · F. De Mari · E. De Vito Department of Mathematics, University of Genova, Genova, Italy e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_4

59

60

F. Bartolucci et al.

by translating, shearing and dilating a fixed initial function ψ ∈ L2 (R2 ), called mother shearlet. Once we have this family of analyzing functions, we define the shearlet transform of any f ∈ L2 (R2 ) by Sψ f (b, s, a) = f, Sb,s,a ψ. If ψ satisfies the admissible condition (4) we can recover any signal f ∈ L2 (R2 ) from its shearlet transform through the reconstruction formula  f =

  R×

R

R2

Sψ f (b, s, a) Sb,s,a ψ dbds

da , |a|3

(1)

where the integral converges in the weak sense. In [5] we have shown that the classical shearlet transform can be realised by applying first the horizontal (affine) Radon transform, then by computing a one-dimensional wavelet transform and, finally, performing a one-dimensional convolution. This relation opens the possibility to recover a signal from its Radon transform by using the shearlet inversion formula (1), where the coefficients Sψ f (b, s, a) depend on f only through its Radon transform. Thus, formula (1) allows to reconstruct an unknown signal f from its Radon transform Rf by computing the family of coefficients {Sψ f (b, s, a)}b∈R2 ,s∈R,a∈R× . Equation (1) has a disadvantage if one wants to use it in applications since the shearing parameter s is allowed to vary over a non-compact set. This gives rise to problems in the reconstruction of signals mostly concentrated on the x-axis since the energy of such signals is mostly concentrated in the coefficients Sψ f (b, s, a) as s → ∞. The standard way to address this problem is the so-called “shearlets on the cone” construction introduced by Kutyniok and Labate [6] for classical admissible shearlets ψ and then generalized by Grohs [11] requiring weaker conditions on ψ. The basic idea in this construction is to decompose the signals as f = PC f + PC v f previous,to the analysis, where PC is - the frequency projection on the horizontal cone C = (ξ1 , ξ2 ) ∈ R2 : |ξ2 /ξ1 | ≤ 1 and PC v is the , projection on the vertical cone C v = (ξ1 , ξ2 ) ∈ R2 : |ξ1 /ξ2 | ≤ 1 . Then, chosen a suitable window function g, the following reconstruction formula holds true:  f =



2

 +

|f, Tb g| db + 2

R2 1



2

−1 −2

 R2

1



2

−1 −2

 R2

|Sψ [PC f ](b, s, a)|2 dbds

|Svψ v [PC v f ](b, s, a)|2 dbds

da , |a|3

da |a|3 (2)

where Fψ v (ξ1 , ξ2 ) = Fψ(ξ2 , ξ1 ) and the so-called vertical shearlet transform Svψ v f (b, s, a) is obtained from the classical shearlet transform by switching the roles of the x-axis and the y-axis. In formula (2), PC f is reconstructed via the classical shearlet transform and PC v f via the vertical shearlet transform and this allows to restrict the shearing parameter s over a compact interval. In this paper, applying the “shearlets on the cone” construction to our results presented in [5], we obtain for any f ∈ L1 (R2 ) ∩ L2 (R2 ) a reconstruction formula of the form (2), i.e. where both the scale parameter a and the shearing parameter s range over compact intervals, and where the coefficients depend on f only through its

Cone-Adapted Shearlets and Radon Transforms

61

Radon transform. Precisely, we show that the shearlet coefficients Sψ [PC f ](b, s, a) depend on f through its (affine) horizontal Radon transform Rf (v, t) and the action of the projection PC on f turns into the restriction of the directional parameter v over the compact interval [-1,1]. Analogously, the vertical shearlet coefficients Svψ v [PC v f ](b, s, a) depend on the limited angle (affine) vertical Radon transform Rv f (v, t), |v| ≤ 1, obtained by switching the roles of the x-axis and the y-axis in the affine parametrization. Therefore, equation (2) allows to reconstruct an unknown signal f ∈ L1 (R2 ) ∩ L2 (R2 ) from its Radon transform by computing the family of coefficients {f, Tb g, Sψ [PC f ](b, s, a), Svψ v [PC v f ](b, s, a)}b∈R2 ,s∈R,a∈R× by means of Theorem 3. The different contributions Rf (v, t) and Rv f (v, t), |v| ≤ 1, reconstruct the frequency projections PC f and PC v f , respectively. Finally, in Sect. 4 we generalize reconstruction formula (2) by applying to f localization operators different from PC and PC v in order to avoid artificial singularities in the reconstructed signal. The paper is organised as it follows. In Sect. 2 we recall the notion of wavelet transform, shearlet transform and Radon transform and part of the results in [5]. In Sect. 3 we present the main results. Finally, in Sect. 4 we generalize the results presented in Sect. 3.

2 Preliminaries In this section we introduce the notation and we recall the definition and the main properties of the three main ingredients, namely the wavelet transform, the shearlet transform and the Radon transform. Then, we recall part of the results in [5] which show how these three classical transforms are related.

2.1 Notation We briefly introduce the notation. We set R× = R \ {0}. The Euclidean norm of a vector v ∈ Rd is denoted by |v| and its scalar product with w ∈ Rd by v · w. For any p ∈ [1, +∞] we denote by Lp (Rd ) the Banach space of functions f : Rd → C that are p-integrable with respect to the Lebesgue measure dx and, if p = 2, the corresponding scalar product and norm are ·, · and · , respectively. The Fourier transform is denoted by F both on L2 (Rd ) and on L1 (Rd ), where it is defined by  Ff (ξ ) =

Rd

f (x)e−2π i ξ ·x dx,

f ∈ L1 (Rd ).

If G is a locally compact group, we denote by L2 (G) the Hilbert space of squareintegrable functions with respect to a left Haar measure on G. If A ∈ Md (R), the vector space of square d × d matrices with real entries, tA denotes its transpose and we denote the (real) general linear group of size d × d by GL(d, R). Finally, the

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translation operator acts on a function f : Rd → C as Tb f (x) = f (x − b), for any b ∈ Rd .

2.2 The Wavelet Transform The one-dimensional affine group W is the semidirect product R  R× with group operation (b, a)(b , a  ) = (b + ab , aa  ) and left Haar measure |a|−2 dbda. It acts on L2 (R) by means of the square-integrable representation  1 x−b . Wb,a f (x) = |a|− 2 f a The wavelet transform is then Wψ f (b, a) = f, Wb,a ψ, which is a multiple of an isometry provided that ψ ∈ L2 (R) satisfies the admissibility condition, namely the Calderón equation,  0
1

(20)

.

From now on we consider the case |v| ≤ 1. Since f ∈ L1 (R2 )∩L2 (R2 ), Corollary 1 and equation (20) imply that for almost all v ∈ R, Rf (v, ·) ∈ L2 (R) and 1

1

F(Q[PC f ](v, ·))(τ ) = |τ | 2 Ff (τ, τ v) = |τ | 2 FRf (v, ·)(τ ).

(21)

Since τ → F(Q[PC f ](v, ·))(τ ) ∈ L2 (R) for almost all v ∈ R, equality (21) implies that Rf (v, ·) is in the domain of the differential operator J0 : L2 (R) → L2 (R) defined as 1

FJ0 g(τ ) = |τ | 2 Fg(τ ),

(22)

and, by the definition of J0 , Q[PC f ](v, ·) = J0 Rf (v, ·). Since J0 is a self-adjoint operator, the wavelet coefficients in (18) become Wφ1 (Q[PC f ](v, ·))(x + vy, a) = Q[PC f ](v, ·), Wx+vy,a φ1 2 = J0 Rf (v, ·), Wx+vy,a φ1 2 = Rf (v, ·), J0 Wx+vy,a φ1 2 1

= |a|− 2 Rf (v, ·), Wx+vy,a J0 φ1 2 1

= |a|− 2 WJ0 φ1 (Rf (v, ·))(x + vy, a), by taking into account that 1

J0 Wb,a = |a|− 2 Wb,a J0 , for any (b, a) ∈ R × R× . We set χ1 = J0 φ1 = J20 ψ1 , that is Fχ1 (τ ) = |τ |Fψ1 (τ ), which is well-defined since ψ1 satisfies (17). From the above calculations, we can conclude that for almost every v ∈ R " Wφ1 (Q[PC f ](v, ·))(x + vy, a) =

1

|a|− 2 Wχ1 (Rf (v, ·))(x + vy, a)

|v| ≤ 1

0

|v| > 1

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and formula (18) becomes Sψ [PC f ](x, y, s, a) = |a|

− 34



1

−1

 Wχ1 (Rf (v, ·))(x + vy, a)φ2

v−s |a|1/2

dv. (23)

Using the same arguments as in the case of the horizontal cone, we obtain the following formula for the vertical shearlet transform 3

Svψ v [PC v f ](x, y, s, a) = |a|− 4



1 −1

 Wχ1 (Rv f (v, ·))(vx + y, a)φ2

v−s |a|1/2

dv. (24)  

This completes the proof.

It is worth observing that formulas (23) and (24) turn the action of the frequency projections PC and PC v on f into the restriction of the interval over which we integrate the directional variable v and so, (23) and (24) eliminate the need to perform a frequency projection on f prior to the analysis. Furthermore, as a consequence, the shearlet coefficients Sψ [PC f ](b, s, a) and Svψ v [PC v f ](b, s, a) depend on f through its limited angle (affine) horizontal and vertical Radon transforms Rf (v, t) and Rv f (v, t), with |v| ≤ 1, respectively. Finally, let us show that also the first integral in the right hand side of reconstruction formula (9) may be expressed in terms of Rf only. Proposition 3 For any f ∈ L1 (R2 ) ∩ L2 (R2 ) and for any smooth function g in L1 (R2 ) ∩ L2 (R2 ) we have that  f, Tb g = Rf (v, ·), Tn(v)·b ζ (v, ·)dv, R

for any b ∈ R2 , where ζ = J20 Rg and n(v) = t(1, v). Proof We take a function f ∈ L1 (R2 ) ∩ L2 (R2 ) and we consider a smooth function g ∈ L1 (R2 ) ∩ L2 (R2 ). We readily derive  f, Tb g =Qf, QTb g =

R

Qf (v, ·), QTb g(v, ·)dv.

Since f and g are in L1 (R2 ) ∩ L2 (R2 ), Corollary 1 implies that for almost all v ∈ R, Rf (v, ·) and Rg(v, ·) are square-integrable functions and  f, Tb g =

R

J0 Rf (v, ·), J0 RTb g(v, ·)dv,

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where we recall that J0 is the differential operator defined by (22). By the behavior of the horizontal Radon transform under translations [14] and since the operator J0 commutes with translations, denoting n(v) = t(1, v), we have that J0 RTb g(v, t) = J0 (I ⊗ Tn(v)·b )Rg(v, t) = (I ⊗ Tn(v)·b )J0 Rg(v, t). We need to choose g in such a way that J0 Rg(v, ·) is in the domain of the operator J0 for almost every v ∈ R. Assuming this, the same property holds true for Tn(v)·b J0 Rg(v, ·) by the translation invariance of dom J0 and we obtain  f, Tb g =  =

R

R

J0 Rf (v, ·), Tn(v)·b J0 Rg(v, ·)dv Rf (v, ·), Tn(v)·b J20 Rg(v, ·)dv.

(25)

It is worth observing that the extra assumption that J0 Rg(v, ·) is in the domain of J0 for almost every v ∈ R is always satisfied. Indeed, by the definition of J0 and Corollary 1 

 |τ ||FJ0 Rg(v, ·)(τ )| dτ = 2

R

 =

R

R

|τ |2 |FRg(v, ·)(τ )|2 dτ |τ |2 |Fg(τ, τ v)|2 dτ < +∞

since by definition g is a smooth function. We set ζ (v, τ ) = J20 Rg(v, τ ), that is Fζ (v, ·)(τ ) = |τ |Fg(τ, τ v), so that (25) becomes  f, Tb g =

R

Rf (v, ·), Tn(v)·b ζ (v, ·)dv.

(26)

Furthermore, if possible, we choose g of the form  Fg(ξ1 , ξ2 ) = Fg1 (ξ1 )Fg2

ξ2 ξ1

,

with g1 ∈ L1 (R2 ) ∩ L2 (R2 ) satisfying the condition  R

|τ |2 |Fg1 (τ )|2 dτ < +∞

(27)

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and g2 ∈ L1 (R2 ) ∩ L2 (R2 ). Under these hypotheses, (26) becomes  f, Tb g =

R

Rf (v, ·), Tn(v)·b ζ1 ζ2 (v)dv,

where ζ1 = J2o g1 , which is well-defined by (27), and ζ2 = Fg2 .

 

Theorem 1 and formulas (23), (24) and (26) give our main result. We recall that ψ is an admissible vector of the form (13) satisfying conditions (14) and such that ψ1 satisfies (17). Furthermore we require that ψ is smooth with all directional vanishing moments in the x1 -direction [11]. Theorem 3 For any f ∈ L1 (R2 ) ∩ L2 (R2 ), we have the reconstruction formula  f =



2

 +

|f, Tb g| db + 2

R2 1



2

−1 −2

 R2

1



2

−1 −2

 R2

|Sψ [PC f ](b, s, a)|2 dbds

|Svψ v [PC v f ](b, s, a)|2 dbds

da |a|3

da , |a|3

(28)

where g is a smooth function in L1 (R2 ) ∩ L2 (R2 ) such that (10) holds true and for any b = (x, y) ∈ R2 , s ∈ R, a ∈ R× v−s dv, Sψ [PC f ](x, y, s, a) = |a| Wχ1 (Rf (v, ·))(x + vy, a)φ2 |a|1/2 −1   1 3 v−s Svψ v [PC v f ](x, y, s, a) = |a|− 4 dv, Wχ1 (Rv f (v, ·))(vx + y, a)φ2 |a|1/2 −1  f, T(x,y) g = Rf (v, ·), Tx+vy ζ (v, ·)dv, − 43



1



R

where ζ = J20 Rg, Fχ1 (τ ) = |τ |Fψ1 (τ ), φ2 = Fψ2 . Proof The proof follows immediately by Theorem 1 and Propositions 2 and 3.

 

This theorem gives an alternative reproducing formula for any f ∈ in which, by the “shearlets on the cone” construction, the scale and shearing parameters range over compact sets and, by Propositions 2 and 3, the coefficients depend on f only through its Radon transform. Therefore equation (28) allows to reconstruct an unknown signal f from its Radon transform by computing the family of coefficients {f, Tb g, Sψ [PC f ](b, s, a), Svψ v [PC v f ](b, s, a)}b∈R2 ,s∈R,a∈R× by means of Theorem 3. It is worth observing that the different contributions in (28) with Rf (v, t) and Rv f (v, t), |v| ≤ 1, reconstruct the frequency projections PC f and PC v f , respectively. L1 (R2 )∩L2 (R2 )

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4 Generalizations A disadvantage in formula (9), and therefore in formula (28), is that the frequency projections PC and PC v performed on f can lead to artificially slow decaying shearlet coefficients. In order to avoid this problem we consider an open cover {U, U v } of the unit circle in the plane S 1 ) (−π, π ] (Fig. 3), where 3 π π 3 U =(−π, − π + !) ∪ (− − !, + !) ∪ ( π − !, π ], 4 4 4 4 3 π π 3 U v = (− π − !, − + !) ∪ ( − !, π + !). 4 4 4 4 Then, there exist even functions ϕ, ϕ v ∈ C ∞ ((−π, π ]) such that supp ϕ ⊆ U , supp ϕ v ⊆ U v and ϕ(θ )2 + ϕ v (θ )2 = 1 for all θ ∈ (−π, π ], see [10]. For any ξ ∈ R2 \ {0}, we denote by θξ ∈ (−π, π ] the angle corresponding to ξ/|ξ | ∈ S 1 by the canonical isomorphism S 1 ) (−π, π ]. Then, we define the functions , v ∈ C ∞ (R2 \ {0}) by v (ξ ) = ϕ v (θξ ).

(ξ ) = ϕ(θξ ),

! 3 π− 4 3 π+ 4

" "

π

#

3 − π− 4

"!

!

"3 4

Uv

π + 4

"

π 4

π

3 − π 4

! !

3 − π+ 4

−

−

π 4

""" ""

π − 4

"

! −

U

π + 4

π − 4

Fig. 3 The open cover {U, U v } of the unit circle in the plane S 1 ) (−π, π ]

Cone-Adapted Shearlets and Radon Transforms

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, It is easy to verify that supp  = (ξ1 , ξ2 ) ∈ R2 : |ξ2 /ξ1 | ≤ tan ( π4 + !) , , supp v = (ξ1 , ξ2 ) ∈ R2 : |ξ1 /ξ2 | ≤ cot ( π4 − !) and (ξ )2 + v (ξ )2 = 1 for all ξ ∈ R2 \ {0}. We define the operators L : L2 (R2 ) → L2 (R2 ) and Lv : L2 (R2 ) → L2 (R2 ) as follows F(Lf )(ξ ) = Ff (ξ )(ξ ) and F(Lv f )(ξ ) = Ff (ξ )v (ξ ). We recall that ψ is an admissible vector of the form (13) satisfying conditions (14) and such that ψ1 satisfies (17). Using analogous computations as in Sect. 3, it is possible to show that for any f ∈ L1 (R2 ) ∩ L2 (R2 ) and (x, y, s, a) ∈ R2 × R × R× Sψ [Lf ](x, y, s, a)   v−s − 34 = |a| Wχ1 (Rf (v, ·))(x + vy, a)φ2 ϕ(arctan v) dv, |a|1/2 R (29) Svψ v [Lv f ](x, y, s, a)    3 v−s 1 v ϕ arctan dv, = |a|− 4 Wχ1 (Rv f (v, ·))(vx + y, a)φ2 v |a|1/2 R (30) where Fχ1 (τ ) = |τ |Fψ1 (τ ) and φ2 = Fψ2 . Furthermore, following the proof of Theorem 3 in [7, Chapter 2], it is possible to derive a reconstruction formula of the form (28) in this new setup. Theorem 4 For any f ∈ L1 (R2 ) ∩ L2 (R2 ), we have the reconstruction formula   1 2 da 2 2 f = |f, Tb g| db + |Sψ [Lf ](b, s, a)|2 dbds 3 2 2 |a| R −1 −2 R  1 2 da + |Svψ v [Lv f ](b, s, a)|2 dbds 3 , (31) 2 |a| −1 −2 R where g is a smooth function in L1 (R2 ) ∩ L2 (R2 ) such that for all ξ ∈ R2  |Fg(ξ )| + (ξ ) 2

1

2

+  (ξ )

2

−1 −2

 v



2

1 −1



|Fψ(Aa tNs ξ )|2 ds

da |a|3/2

2

da |Fψ v (A˜a Ns ξ )|2 ds 3/2 = 1 |a| −2

and the coefficients in (31) are given by (29), (30) and (26).

(32)

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Proof Consider a smooth function g ∈ L1 (R2 ) ∩ L2 (R2 ) such that (32) holds true. By Plancherel theorem, we have that

2   



2π ib·ξ

db |f, Tb g|2 db = Ff (ξ )Fg(ξ )e dξ



R2 R2 R2  = |F−1 (Ff Fg)(b)|2 db  =

R2

|Ff (ξ )|2 |Fg(ξ )|2 dξ.

R2

(33)

Using an analogous computation, by Plancherel theorem and Fubini’s theorem we have  1 2 da |Sψ [Lf ](b, s, a)|2 dbds 3 |a| −1 −2 R2  1 2 da = |Lf, Sb,s,a ψ|2 dbds 3 2 |a| −1 −2 R

2  1  2  



da 2π iξ b t

= Ff (ξ )(ξ )Fψ(Aa Ns ξ )e dξ

dbds 3/2

2 2 |a| R −1 −2 R  1 2 da = |F−1 (Ff Fψ(Aa tNs ·))(b)|2 dbds 3/2 2 |a| −1 −2 R  1 2 da = |Ff (ξ )|2 (ξ )2 |Fψ(Aa tNs ξ )|2 dξ ds 3/2 2 |a| −1 −2 R   1 2 da = |Ff (ξ )|2 (ξ )2 |Fψ(Aa tNs ξ )|2 ds 3/2 dξ. (34) 2 |a| R −1 −2 Similarly, we have that 



1



2

−1 −2 R2

|Svψ v [Lv f ](b, s, a)|2 dbds



=

R2



|Ff (ξ )|2 v (ξ )2

1



2

−1 −2

da |a|3

|Fψ v (A˜a Ns ξ )|2 ds

da dξ. |a|3/2

(35)

Thus, combining equations (33), (34) and (35) we obtain the reconstruction formula  f 2 =  +



R2 1

|f, Tb g|2 db + 

2

−1 −2

 R2

1



2



−1 −2 R2

|Sψ [Lf ](b, s, a)|2 dbds

|Svψ v [Lv f ](b, s, a)|2 dbds

da , |a|3

da |a|3 (36)

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for any f ∈ L1 (R2 ) ∩ L2 (R2 ), where the shearlet coefficients are given by (29) and (30). It is worth observing that there always exists a function g satisfying (32) provided that the admissible vector ψ is smooth and possesses all vanishing moments in the x1 -direction [11]. Indeed, we have that  z(ξ ) := 1 − (ξ ) 

1

2



2

−1 −2

1



|Fψ(Aa tNs ξ )|2 ds

da |a|3/2

2

da |Fψ v (A˜a Ns ξ )|2 ds 3/2 |a| −1 −2  1 2 da = (ξ )2 (1 − |Fψ(Aa tNs ξ )|2 ds 3/2 ) |a| −1 −2  1 2 da + v (ξ )2 (1 − |Fψ v (A˜a Ns ξ )|2 ds 3/2 ). |a| −1 −2

− v (ξ )2

Following the proof of Lemma 3 in [7, Chapter 2] it is possible to prove that  1−

1



ξ2

π da −N |Fψ(Aa Ns ξ )| ds 3/2 = O(|ξ | ),



≤ tan ( + !), ξ1 4 |a| −2



−1

2

t

2

for all N ∈ N. Analogously,  1−

1



2

−1 −2

|Fψ v (A˜a Ns ξ )|2 ds



ξ1

da −N

≤ cot ( π − !), = O(|ξ | ),

ξ

3/2 4 |a| 2

for all N ∈ N. Therefore, there exists a smooth function g ∈ L1 (R2 ) ∩ L2 (R2 ) √ such that Fg(ξ ) = z(ξ ), so that (32) holds true. Finally, by Proposition 3, we can express the coefficients f, Tb g in reconstruction formula (36) in terms of Rf only.   Acknowledgements F. Bartolucci, F. De Mari and E. De Vito are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References 1. D. Labate, W.-Q. Lim, G. Kutyniok, G. Weiss. Sparse multidimensional representation using shearlets. Optics & Photonics 2005, International Society for Optics and Photonics (2005), 59140U-59140U. 2. E. Cordero, and A. Tabacco. Triangular subgroups of Sp(d, R) and reproducing formulae. J. Fourier Anal. Appl. 264 (2013), no. 9, 2034–2058.

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3. E. Cordero, F. De Mari, K. Nowak, and A. Tabacco. Analytic features of reproducing groups for the metaplectic representation. J. Fourier Anal. Appl. 12 (2006), no. 2, 157–180. 4. E. Cordero, F. De Mari, K. Nowak, and A. Tabacco. Dimensional upper bounds for admissible subgroups for the metaplectic representation. Math. Nachr. 283 (2010), no. 7, 982–993. 5. F. Bartolucci, F. De Mari, E. De Vito, and F. Odone. The Radon transform intertwines wavelets and shearlets. Appl. Comput. Harmon. Anal. 47 (2019), no. 3, 822–847 (available on line https://doi.org/10.1016/j.acha.2017.12.005). 6. G. Kutyniok and D. Labate. Resolution of the wavefront set using continuous shearlets. Trans. Amer. Math. Soc. 361 (2009), no. 5, 2719–2754. 7. G. Kutyniok and D. Labate. Shearlets. Appl. Numer. Harmon. Anal. Birkhäuser/Springer, New York, 2012. 8. H. Führ. Continuous wavelet transforms with abelian dilation groups. J. Math. Phys. 39 (1998), no. 8, 3974–3986. 9. H. Führ and R. R. Tousi. Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups. Appl. Comput. Harmon. Anal. (2016). 10. L. Borup and M. Nielsen. Frame decomposition of decomposition spaces. J. Fourier Anal. Appl. 13 (2007), no. 1, 39–70. 11. P. Grohs. Continuous shearlet frames and resolution of the wavefront set. Monatsh. Math. 164 (2011), no. 4, 393–426. 12. S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H. Stark, and G. Teschke. The uncertainty principle associated with the continuous shearlet transform. Int. J. Wavelets Multiresolution Inf. Process. 6 (2008), no. 2, 157–181. 13. S. Dahlke, G. Steidl, and G. Teschke. The continuous shearlet transform in arbitrary space dimensions. J. Fourier Anal. Appl. 16 (2010), no. 3, 340–364. 14. S. Helgason. The Radon transform. Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 2nd edition, 1999.

Linear Perturbations of the Wigner Transform and the Weyl Quantization Dominik Bayer, Elena Cordero, Karlheinz Gröchenig, and S. Ivan Trapasso

Abstract We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal’s formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen’s class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus. Keywords Time-frequency analysis · Wigner distribution · Cohen’s class · Modulation space · Pseudodifferential operator · Quantization 2010 Mathematics Subject Classification 42A38, 42B35, 46F10, 46F12, 81S30

D. Bayer Universität der Bundeswehr München, München, Germany e-mail: [email protected] E. Cordero () Dipartimento di Matematica, Università di Torino, Torino, Italy e-mail: [email protected] K. Gröchenig Faculty of Mathematics, University of Vienna, Vienna, Austria e-mail: [email protected] S. I. Trapasso Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Torino, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_5

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1 Introduction The Wigner transform is a key concept lying at the heart of pseudodifferential operator theory and time-frequency analysis. It was introduced by Wigner [44] as a quasi-probability distribution in order to extend the phase-space formalism of classical statistical mechanics to the domain of quantum physics. Subsequently, this line of thought led to the phase-space formulation of quantum mechanics. In engineering, the Wigner transform was considered the ideal tool for the simultaneous investigation of temporal and spectral features of signals, because it enjoys all properties desired from a good time-frequency representation (except for positivity) [40]. To be precise, the (cross-)Wigner distribution of two signals f, g ∈ L2 (Rd ) is defined as   y  y dy . (1) e−2π iy·ω f x + g x− W (f, g)(x, ω) = 2 2 Rd If f = g we write Wf (x, ω). This is an example of a quadratic time-frequency representation, and heuristically Wf (x, ω) is interpreted as a measure of the energy content of the signal f in a “tight” spectral band around ω during a “short” time interval near x. The Wigner transform plays a key role in the Weyl quantization and the corresponding pseudodifferential calculus. Quantization is a formalism that associates a function on phase space (an observable) with an operator on a Hilbert space. The standard quantization rule in physics is the Weyl correspondence: given the phase-space observable σ ∈ S (R2d ) (called symbol in mathematical language) the corresponding Weyl transform opW (σ ) : S(Rd ) → S (Rd ) is (formally) defined by 

 opW (σ )f (x) =

R2d

e2π i(x−y)·ω σ

x+y , ω f (y)dydω. 2

(2)

A formal computation reveals the role of the Wigner transform in this definition, because opW (σ )f, g = σ, W (g, f ),

f, g ∈ S(Rd ).

(3)

Whereas the rigorous interpretation of (2) is subtle and requires oscillatory integrals, the weak formulation (3) is easy to handle and works without problems for distributional symbols σ . The bracket f, g denote the extension to S (Rd )×S(Rd ) of the inner product on L2 (Rd ). Unfortunately, not all properties which are desired from a time-frequency representation are compatible. The Wigner transform is real-valued, but it may take negative values. This is a serious obstruction to the interpretation of the Wigner transform as a probability distribution or as an energy density of a signal. By

Linear Perturbations of the Wigner Transform and the Weyl Quantization

81

Hudson’s Theorem [34] only generalized Gaussian functions have positive Wigner transforms. To obtain time-frequency representations that are positive for all functions, one takes local averages of the Wigner transform in order to tame the sign oscillations. This is usually done by convolving Wf with a suitable kernel θ . This idea yields a class of quadratic time-frequency representations, which is called Cohen’s class [8]. The time-frequency representations in Cohen’s class are parametrized by a kernel θ ∈ S (R2d ), and the associated representation is defined as Qθ (f, g) := W (f, g) ∗ θ,

f, g ∈ S(Rd ).

(4)

Most time-frequency representations proposed so far belong to Cohen’s class [7, 33], and the correspondence between properties of θ and Qθ is well understood [33, 37]. In many examples Qθ can be interpreted as a perturbation of the Wigner distribution, while retaining some of its important properties. Thus Cohen’s class provides a unifying framework for the study of time-frequency representations appearing in signal processing. See for instance [7, 9, 32, 33]. Next, for every time-frequency representation in Cohen’s class one can introduce a quantization rule in analogy to the Weyl quantization (3), namely, opθ (σ )f, g = σ, Qθ (g, f ) = σ ∗ θ ∗ , W (g, f ),

f, g ∈ S(Rd ) ,

(5)

whenever the expressions make sense. Although the new operator opθ (σ ) is just a Weyl operator with the modified symbol σ ∗ θ ∗ (whenever defined in S (Rd )), the variety of pseudodifferential calculi given by definition (5) adds flexibility and a new flavor to the description and analysis of operators. For example, a first variation of the Wigner transform are the τ -Wigner transforms  Wτ (f, g)(x, ω) = e−2π iy·ω f (x + τy)g(x − (1 − τ )y) dy, f, g ∈ S(Rd ) . Rd

(6)

These belong to Cohen’s class and possess the kernel θτ ∈ S(R2d ) with Fourier transform −2π i θτ (ξ, η) = e



 τ − 12 ξ ·η

,

(ξ, η) ∈ R2d .

(7)

The corresponding pseudodifferential calculi are the Shubin τ -pseudodifferential operators opθτ (σ ) in formula (5). For the parameter τ = 1/2 this is the Weyl calculus, for τ = 0 this is the Kohn-Nirenberg calculus. The important Born-Jordan quantization rule is obtained as an average over τ ∈ [0, 1], see [5] or the textbook [18].

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From (7) we see that the deviation from the Wigner transform is measured by μ = τ − 1/2. Hence a natural generalization of the τ -Wigner transform is the replacement of the scalar parameter μ with a matrix expression M = T − (1/2)I , with T ∈ Rd×d . This gives the family of matrix-Wigner distributions  WM (f, g)(x, ω) =

Rd

e

−2πiy·ω

 f

   1 1 x + M + I y g x + M − I y dy . 2 2

(8) Again these are members of the Cohen class in (4) with a kernel θM given by its Fourier transform −2π iξ ·Mη θ. . M (ξ, η) = e

(9)

An even more general definition in the  spirit of (8) uses an arbitrary linear A11 A12 be an invertible, real-valued mapping of the pair (x, y) ∈ R2d . Let A = A21 A22 2d × 2d-matrix. We define the matrix-Wigner transform BA of two functions f, g by  BA (f, g) (x, ω) = e−2π iy·ω f (A11 x + A12 y) g (A21 x + A22 y)dy . (10) Rd

Clearly, WM in (8) is a special case by choosing  A = AM =

I M + (1/2)I I M − (1/2)I

.

(11)

Once again, every matrix Wigner transform BA is associated with a pseudodifferential calculus or a quantization rule. Given an invertible 2d × 2d -matrix A and a symbol σ ∈ S (R2d ), we define the operator σ A by / 0 σ A f, g ≡ σ, BA (g, f ) ,

f, g ∈ S(Rd ) .

(12)

This is then a continuous operator from S(Rd ) to S (Rd ) and the mapping σ → σ A is a form of quantization similar to the Weyl quantization. The class of matrix Wigner transforms has already a sizeable history. To our knowledge they were first introduced in [23] in dimension d = 1 for a different purpose, but the original contribution to the subject went by unnoticed. The first thorough investigation of the matrix Wigner transforms BA is contained in the unpublished Ph.D. thesis [1] of the first-named author who studied the general properties of this class of time-frequency representations and the associated pseudodifferential operators. Independently, [3] introduced and studied these “Wigner representations associated with linear transformations of the time-frequency plane” in dimension d = 1. In [24] matrix Wigner transforms were used for a signal

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estimation problem. Recently, in [43] Toft discusses “matrix parametrized pseudodifferential calculi on modulation spaces”, which correspond to the time-frequency representations in (8). Finally, two of us [11] took up and reworked and streamlined several results of [1] and determined the precise intersection between matrix Wigner transforms and Cohen’s class. The goal of this chapter is a systematic survey of the accumulated knowledge about the matrix Wigner transforms and their pseudodifferential calculi. In the first part (Sect. 3) we discuss the general properties of the matrix Wigner transforms. (i) We state the main formulas for covariance, the behavior with respect to the Fourier transform, the analog of Moyal’s formulas, and the inversion formula. (ii) It is well-known that, up to normalization, the ambiguity function and the short-time Fourier transform are just different versions and names for the Wigner transform. This is no longer true for the matrix Wigner transforms, so we give precise conditions on the parametrizing matrix A so that BA can be expressed as a short-time Fourier transform, up to a phase factor and a change of coordinates. (iii) Of special importance is the intersection of the class of matrix Wigner transforms with Cohen’s class. After a partial result in [1],it was proved  in [11] I M+I /2 that BA belongs to Cohen’s class, if and only if A = I M−I /2 for some d × d-matrix M. Thus in general a matrix Wigner transform does not belong to Cohen’s class. This fact explains why for certain results we have to impose assumptions on the parametrizing matrix A. (iv) A further item is the boundedness of the bilinear mapping (f, g) → BA (f, g) on various function spaces. These results are quite useful in the analysis of the mapping properties of the pseudodifferential operators σ A . In the second part (Sect. 4) we study the pseudodifferential calculi defined by the rule (12). (i) Based on Feichtinger’s kernel theorem (see Theorem 1), we first show that every “reasonable” operator can be represented as a pseudodifferential operator σ A . We remark that the map (σ, A) → σ A is highly non-injective and, given two matrices A and B and two symbols σ, ρ, we obtain formulas characterizing the condition σ A = ρ B . (ii) A large section is devoted to the mapping properties of the pseudodifferential operator σ A on various function spaces, in particular on Lp -spaces and on modulation spaces. (iii) Finally, we extend the boundedness results for symbols in the Sjöstrand class to those pseudodifferential operators for which BA is in Cohen’s class. For most results we will include proofs, but we will omit those proofs that only require a formal computation. We hope that the self-contained and comprehensive presentation of matrix Wigner distributions and their pseudodifferential calculi will offer some added value when compared with the focussed, individual publications.

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2 Preliminaries Notation We define t 2 = t · t, for t ∈ Rd , and x · y is the scalar product on Rd . The Schwartz class is denoted by S(Rd ), the space of temperate distributions by S (Rd ). The bracketsf, g denote the extension to S (Rd ) × S(Rd ) of the inner product f, g = f (t)g(t)dt on L2 (Rd ). The conjugate exponent p of p ∈ [1, ∞] is defined by 1/p +1/p = 1. The symbol λ means that the underlying inequality holds up to a positive constant factor C = C(λ) > 0 that depends on the parameter λ: f g

⇒

∃C > 0 : f ≤ Cg.

If f  g and g  f we write f g. The Fourier transform of a function f ∈ S(Rd ) is normalized as  Ff (ω) =

Rd

e−2π ix·ω f (x) dx,

ω ∈ Rd .

For any x, ω ∈ Rd , the modulation Mω and translation Tx operators are defined as Mω f (t) = e2π it·ω f (t) ,

Tx f (t) = f (t − x) .

Their composition π(x, ω) = Mω Tx is called a time-frequency shift. Given a complex-valued function f on Rd , the involution f ∗ is defined as f ∗ (t) := f (−t),

t ∈ Rd .

The short-time Fourier transform (STFT) of a signal f ∈ S (Rd ) with respect to the window function g ∈ S(Rd ) is defined as  Vg f (x, ω) = f, π(x, ω)g = F(f · Tx g)(ω) =

Rd

f (y) g(y − x) e−2π iy·ω dy.

The group of invertible, real-valued 2d × 2d matrices is denoted by   GL (2d, R) = M ∈ R2d×2d | det M = 0 , and we denote the transpose of an inverse matrix by M # ≡ (M −1 )* = (M * )−1 ,

M ∈ GL (2d, R) .

(13)

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Let J denote the canonical symplectic matrix in R2d , namely  0d Id . J = −Id 0d Observe that, for z = (z1 , z2 ) ∈ R2d , we have J z = J (z1 , z2 ) = (z2 , −z1 ) , J −1 z = J −1 (z1 , z2 ) = (−z2 , z1 ) = −J z, and J 2 = −I2d×2d .

2.1 Function Spaces Recall that C0 (Rd ) denotes the class of continuous functions on Rd vanishing at infinity. Modulation Spaces Fix a non-zero window g ∈ S(Rd ) and 1 ≤ p, q ≤ ∞. (i) The modulation space M p,q (Rd ) consists of all temperate distributions f ∈ S (Rd ) such that Vg f ∈ Lp,q (R2d ) (mixed-norm Lebesgue space). The norm on M p,q is  f M p,q = Vg f Lp,q =

 Rd

1/q

q/p |Vg f (x, ω)| dx p

Rd

dω

,

(with obvious modifications for p = ∞ or q = ∞). If p = q, we write M p instead of M p,p . (ii) The modulation space W (FLp , Lq )(Rd ) can be defined as the space of distributions f ∈ S (Rd ) such that   1/q q/p

f W (FLp ,Lq )(Rd ) :=

Rd

Rd

|Vg f (x, ω)|p dω

dx

0.

Two special transformations deserve a separate notation. The first is the flip operator F˜ (x, y) ≡ TI˜ F (x, y) = F (y, x) ,

I˜ =



0d Id Id 0d

∈ GL (2d, R) ,

while the other one is the reflection operator: IF (x, y) ≡ T−I F (x, y) = F (−x, −y) . Sometimes we will also write I = −I ∈ GL (2d, R).

2.4 Partial Fourier Transforms Given F ∈ L1 (R2d ), we use the symbols F1 and F2 to denote the partial Fourier transforms  . e−2π iξ ·t F (t, y) dt, ξ, y ∈ Rd F1 F (ξ, y) = Fy (ξ ) = Rd

.x (ω) = F2 F (x, ω) = F

 Rd

e−2π iω·t F (x, t) dt,

x, ω ∈ Rd ,

where · denotes the Fourier transform on L1 (Rd ) and Fx (y) = F (x, y) ,

Fy (x) = F (x, y)

x, y ∈ Rd

are the sections of F at fixed x and y, respectively. The definition is well-posed  d 1 thanks to Fubini’s theorem, which implies that Fx ∈ L Ry for a.e. x ∈ Rd and   Fy ∈ L1 Rdx for a.e. y ∈ Rd , thus F1 F and F2 F are indeed well defined. The Fourier transform F is therefore related to the partial Fourier transforms as F = F1 F2 = F2 F1 . Using Plancherel’s theorem and properties of modulation spaces (Proposition 1, item (iv)), the following extension of the partial Fourier transform is routine.

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Lemma 3 (i) The partial Fourier transform F2 is a unitary operator on L2 (R2d ). In particular, F∗2 F (x, y) = F−1 2 F (x, y) = F2 F (x, −y) = TI2 F2 F (x, y) , 

I 0 . 0 −I (ii) The partial Fourier transform F2 is an isomorphism on M 1 (R2d ) and on M ∞ (R2d ). where I2 =

3 Matrix-Wigner Distributions Let us define the main characters of this survey.  A11 A12 ∈ GL (2d, R) . The time-frequency distribution Definition 1 Let A = A21 A22 of Wigner type for f and g associated with A (in short: matrix-Wigner distribution, MWD) is defined for suitable functions f, g as BA (f, g) (x, ω) = F2 TA (f ⊗ g) (x, ω) .

(16)

When g = f , we write BA f for BA (f, f ). Explicitly, BA is given by  BA (f, g) (x, ω) =

Rd

e−2π iω·y f (A11 x + A12 y) g (A21 x + A22 y)dy.

This definition is meaningful on many function spaces. A first result is for the triple (M 1 , L2 , M ∞ ). Proposition 3 Assume A ∈ GL (2d, R). (i) If f, g ∈ L2 (Rd ), then BA (f, g) ∈ L2 (R2d ) and the mapping BA ,: L2 (Rd ) × L2 (Rd ) -→ L2 (R2d ) is continuous. Furthermore, span BA (f, g) | f, g ∈ L2 (Rd ) is a dense subset of L2 (R2d ). (ii) If f, g ∈ M 1 (Rd ), then BA (f, g) ∈ M 1 (R2d ) and the mapping BA : M 1 (Rd )× M 1 (Rd ) → M 1 (R2d ) is continuous. (iii) If f, g ∈ M ∞ (Rd ), then BA (f, g) ∈ M ∞ (R2d ) and the mapping BA : M ∞ (Rd ) × M ∞ (Rd ) → M ∞ (R2d ) is continuous.

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The standard time-frequency representations covered within this framework include for instance: • the short-time Fourier transform:  Vg f (x, ω) = e−2π iω·y f (y) g (y − x)dy = BAST (f, g) (x, ω) , Rd

where

 AST =

0 I −I I

(17)

;

• the cross-ambiguity function:  Amb (f, g) (x, ω) =

Rd

 x  x e−2πiω·y f y + g y− dy = BAAmb (f, g) (x, ω) , 2 2

(18) where  AAmb =

1 2I − 12 I

I I

;

• the Wigner distribution:  W (f, g) (x, ω) =

Rd

 y  y dy; e−2π iω·y f x + g x− 2 2

(19)

• the Rihaczek distribution:  R (f, g) (x, ω) = e−2π iω·y f (x) g (x − y)dy = e−2π ix·ω f (x) gˆ (ω). Rd

(20)

The latter two distributions are special cases of the τ -Wigner distribution defined in (6). For any τ ∈ [0, 1], we have Wτ (f, g) (x, ω) = BAτ (f, g) (x, ω) , where  Aτ =

I τI I − (1 − τ ) I

.

(21)

The list of elementary properties is in line of those for the short-time Fourier transform or the Wigner distribution. Mostly the proof is a straightforward computation, and we refer to [1, 11] for the details. The interesting aspect is how the parametrizing matrix A intervenes in the formulas for BA .

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Proposition 4 Let A ∈ GL (2d, R) and f, g ∈ M 1 (Rd ). The following properties hold: (i) Interchanging f and g: BA (g, f ) (x, ω) = BC1 (f, g) (x, ω),

(x, ω) ∈ R2d ,

where C1 = I˜AI2 =



0I I 0



A11 A12 A21 A22



I 0 0 −I



 =

A21 −A22 A11 −A12

.

In particular, BA f is a real-valued function if and only if A = C, namely A11 = A21 ,

A12 = −A22 .

(ii) Behaviour of Fourier transforms:   BA fˆ, gˆ (x, ω) = |det A|−1 BC2 (f, g) (−ω, x) ,

(x, ω) ∈ R2d ,

where C2 = I2 A I˜ = #



I 0 0 −I

 A

−1

*  0 I . I 0

(iii) Fourier transform of a MWD: FBA (f, g) (ξ, η) = BAJ (f, g) (η, ξ ) ,

(22)

where  AJ =

A11 A12 A21 A22



0 I −I 0



 =

−A12 A11 −A22 A21

.

3.1 Connection to the Short-Time Fourier Transform We first investigate the relation of the time-frequency representations BA to the ordinary STFT. Whereas the Wigner distribution and the ambiguity transform coincide with the STFT up to normalization, the time-frequency representation BA can be written as a STFT only under an extra condition.

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A11 A12 ∈ R2d×2d is called left-regular Definition 2 A block matrix A = A21 A22 (resp. right-regular), if the submatrices A11 , A21 ∈ Rd×d (resp. A12 , A22 ∈ Rd×d ) are invertible.  A11 A12 ∈ GL (2d, R) is left-regular It is not difficult to prove that A = A21 A22  # *  (A )11 (A# )12 (resp. right-regular) if and only if the matrix2 A# = A−1 = (A# )21 (A# )22 is right-regular (resp. left-regular). As a matter of fact, the right-regularity of the matrix AST in (17) stands out at a first glance and one might guess that this is an essential condition to express BA (f, g) as a short-time Fourier transform. In fact, this characterization is very strong, as stated in the subsequent results. 

Theorem 2 ([1, Thm. 1.2.5]) Assume that A ∈ GL (2d, R) is right-regular. For every f, g ∈ M 1 (Rd ) the following formula holds: BA (f, g) (x, ω) = |det A12 |−1 e2π iA12 ω·A11 x Vg˜ f (c (x) , d (ω)) , #

x, ω ∈ Rd , (23)

where   x, c (x) = A11 − A12 A−1 A 21 22

d (ω) = A#12 ω,

  g˜ (t) = g A22 A−1 t . 12

For the sake of clarity, one might use the following formulation: Theorem 3 Given matrices M, N, P ∈ Rd×d and Q, R ∈ GL(R, d), set  Q# Q# N * M   . A= R Q# N * M − P RQ# Then A is right-regular and BA (f, g) (x, ω) = |det Q| e2π iMx·N ω Vg◦R (P x, Qω) , for any f, g ∈ M 1 (Rd ). Proof The proof is by computation: |det Q| e2π iMx·N ω Vg◦R (P x, Qω)  2π iMx·N ω = |det Q| e e−2π iQω·y f (y) g (Ry − RP x)dy Rd

2 Beware

−1 that (A# )ij = A#ij = (A* ij ) , i, j = 1, 2.

x, ω ∈ Rd ,

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 = |det Q|  = |det Q|  =

Rd

Rd

Rd

e−2π iQω·



y−Q# N * Mx



f (y) g (Ry − RP x)dy

      e−2π iQω·y f y + Q# N * Mx g Ry + R Q# N * M − P x dy

      e−2π iω·y f Q# y + Q# N * Mx g RQ# y + R Q# N * M − P x dy

=BA (f, g) (x, ω) ,  

where A = AM,N,P ,Q,R is as claimed. Remark 1 The peculiar way the blocks of A are combined in   c(x) = A11 − A12 A−1 22 A21 x

is a well-known construction in linear algebra and is usually called Schur complement. The Schur complement comes up many times in our results, ultimately because of its distinctive role in the inversion of block matrices (cf. for instance [39, Thm. 2.1]). For distributions associated with right-regular matrices, most results about the short-time Fourier transform can be formulated for BA : for instance, one can easily produce orthogonality formulae or a reconstruction formula for BA f . We will study these issues in more generality in the subsequent sections.

3.2 Main Properties of the Transformation BA Having in mind that the entire knowledge on the uncertainty principles could be easily transposed here, we give a qualitative result in the spirit of Benedick’s theorem for the Fourier transform - which is based on the corresponding uncertainty principle for the STFT [27, Thm. 2.4.2]. Theorem 4 ([1, Thm. 1.4.3]) Let A ∈ GL (2d, R) be a right-regular matrix. If the support of BA (f, g) has finite Lebesgue measure, then necessarily f ≡ 0 or g ≡ 0. Next we characterize the boundedness of BA (f, g) on Lebesgue spaces - which is a completely established issue for the STFT, cf. [4]. Proposition 5 Assume that A ∈ GL (2d, R) is right-regular. For any 1 ≤ p ≤ ∞  and q ≥ 2 such that q  ≤ p ≤ q, f ∈ Lp (Rd ) and g ∈ Lp (Rd ), we have (i) BA (f, g) ∈ Lq (R2d ), with BA (f, g) Lq ≤

f Lp g Lp 1 q

1

1

1

|det A| |det A12 | p − q |det A22 | p

− q1

.

(24)

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(ii) If 1 < p < ∞ then BA (f, g) ∈ C0 (R2d ). In particular, BA (f, g) ∈ L∞ (R2d ). Furthermore, if 1 ≤ p, q ≤ ∞ such that p < q  or p > q, the map BA (f, g) :  Lp (Rd ) × Lp (Rd ) → Lq (R2d ) is not continuous. Proof We refer to the proof of [11, Prop. 3.9] for the details concerning the first part. Item (ii) is a direct application of [4, Prop. 3.2].   The right-regularity of A is not only a technical condition required for (23) to hold, but also has unexpected effects on the continuity of BA . Theorem 5 ([1, Theorem 1.2.9]) Assume A ∈ GL (2d, R) such that det A22 = 0 but det A12 = 0. Then there exist f, g ∈ L2 (Rd ) such that BA (f, g) is not a continuous function on R2d . Let us exhibit the orthogonality relations, which extend the Parseval identity to time-frequency distributions. The generalization of the orthogonality relations was one of the main motivations for introducing BA in [1]. Theorem 6 ([1, Thm. 1.3.1]) Let A ∈ GL (2d, R) and f1 , f2 , g1 , g2 ∈ L2 (Rd ). Then BA (f1 , g1 ) , BA (f2 , g2 )L2 (R2d ) =

1 f1 , f2 L2 (Rd ) g1 , g2 L2 (Rd ) . |det A|

(25)

In particular, BA (f, g) L2 (R2d ) =

1 |det A|1/2

f L2 (Rd ) g L2 (Rd ) .

Thus, the representation BA,g : L2 (Rd ) - f → BA (f, g) ∈ L2 (R2d ) is a nontrivial constant multiple of an isometry whenever g ≡ 0. The proof follows directly from the definition in (16), since F2 is unitary and TA is a multiple of a unitary operator. Corollary 1 If {en }n∈N is an orthonormal basis for L2 (Rd ), then   |det A|1/2 BA (em , en ) | m, n ∈ N is an orthonormal basis for L2 (R2d ). While the relevance of the orthogonality relations for signal processing or physics purposes has been sometimes debated [6], they are in fact a useful tool for proving several properties of the time-frequency distributions that satisfy them. In particular, orthogonality relations are the main ingredients of a general procedure for reconstructing a signal from the knowledge of its (cross-)time-frequency distribution with a given window.

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Theorem 7 ([11, Cor. 3.1.7]) Assume A ∈ GL (2d, R) and fix g, γ ∈ L2 (Rd ) such that g, γ  = 0. Then, for any f ∈ L2 (Rd ), the following inversion formula holds: f =

|det A| g, γ 

B∗A,γ BA,g f,

where B∗A,γ : L2 (R2d ) → L2 (Rd ) is the adjoint operator of BA,γ ≡ BA (·, γ ), defined as B∗A,γ H

1 (x) = |det A|

 Rd

TA F2 H (x, y) γ (y) dy,

with A = I2 A−1 ∈ GL (2d, R) . For right-regular matrices the reconstruction can be made more explicit. Proposition 6 ([1, Thm. 1.3.3]) Let A ∈ GL (2d, R) be a right-regular matrix, and g, γ ∈ L2 (Rd ) such that g, γ  = 0. The following inversion formula (to be interpreted as vector-valued integral in L2 (Rd )) holds for any f ∈ L2 (Rd ): f =

1 g, γ 

 R2d

e−2π iA12 ω·A11 x ˜ Md(ω) Tc(x) gdxdω, |det A12 | #

BA,γ f (x, ω)

where   c (x) = A11 − A12 A−1 22 A21 x,

d (ω) = A#12 ω,

  g˜ (t) = g A22 A−1 12 t .

Another property that is expected to hold for a time-frequency representation is the covariance under phase-space shifts. Theorem 8 ([1, Thm. 1.5.1]) Let A ∈ GL (2d, R). For any f, g ∈ M 1 (Rd ) and a, b, α, β ∈ Rd , the following formula holds:   BA Mα Ta f, Mβ Tb g (x, ω) = e2π iσ ·s M(ρ,−s) T(r,σ ) BA (f, g) (x, ω) =e

2π iσ ·s 2π i(x·ρ−ω·s)

e

BA (f, g) (x − r, ω − σ ) , (27)

where   a r , = A−1 b s

(26)

  α ρ . = A* −β σ

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Of course, this result encompasses the covariance formula for the τ -Wigner distribution with A = Aτ as in (21), cf. [15, Prop. 3.3] and also for the STFT with A = AST , cf [26, Lem. 3.1.3]. We now cite an amazing representation result for the STFT of a MWD, sometimes called the magic formula for other distributions (cf. [28]). This relation allows to painlessly extend our results to general function spaces tailored for the purposes of time-frequency analysis, namely modulation spaces. Theorem 9 ([1, Thm. 1.7.1]) Assume A ∈ GL (2d, R) and f, g, ψ, φ ∈ M 1 (Rd ), and set z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ R2d . Then, VBA (φ,ψ) BA (f, g) (z, ζ ) = e−2π iz2 ·ζ2 Vφ f (a, α) Vψ g (b, β),

(28)

where    z A11 z1 − A12 ζ2 a , = AI2 1 = ζ2 A21 z1 − A22 ζ2 b    α ζ (A# )11 ζ1 + (A# )12 z2 . = I2 A# 1 = z2 −(A# )21 ζ1 − (A# )22 z2 β As a concluding remark, we want to underline that the benefits of linear algebra should be appreciated in view of the very short and simple proofs. This aspect should not be underestimated: the proof of similar results for certain special members has lead to quite cumbersome computations (cf. the proofs for the τ -Wigner distributions in [15]).

3.3 Cohen Class Members as Perturbations of the Wigner Transform We already described the heuristics behind the Cohen class of distributions in the introduction. Definition 3 ([26]) A time-frequency distribution Q belongs to the Cohen’s class if there exists a tempered distribution θ ∈ S (R2d ) such that Q (f, g) = W (f, g) ∗ θ,

∀f, g ∈ S(Rd ).

Although the Wigner distribution was the main inspiration for the MWDs studied so far, the connection to the Cohen class is by no means clear. This question is the point of departure of the paper [11] and the following result completely characterizes the intersection between these families. We rephrase [11, Thm. 1.1] using the Feichtinger algebra as follows.

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Theorem 10 Let A ∈ GL (2d, R). The distribution BA belongs to the Cohen class if and only if  A = AM =

I M + (1/2)I I M − (1/2)I



as in (11), for some M ∈ Rd×d . Furthermore, in this case we have WM (f, g) ≡ BAM (f, g) = W (f, g) ∗ θM ,

f, g ∈ M 1 ,

(29)

where the Cohen’s kernel θM ∈ S (Rd ) is with $M (ξ, η) = e−2π iξ ·Mη ,

θM = F$M ,

(ξ, η) ∈ R2d .

(30)

If M is invertible, the kernel θM is explicitly θM (x, ω) =

1 −1 e2π ix·M ω , |det M|

(x, ω) ∈ R2d .

(31)

We say that A = AM is a Cohen-type matrix associated with M ∈ Rd×d . Remark 2 We mention that, according to the proof of the necessity part in the previous result, a Cohen-type matrix A should be defined by the following conditions on the blocks: A11 = A21 = I,

A12 − A22 = I.

(32)

The choice A22 = M − (1/2)I with M ∈ Rd×d is thus a suitable parametrization, but by no means the only possible one - and in fact neither the most natural one. The reason underlying our choice appears if one writes down the explicit formula for BAM as  WM (f, g)(x, ω) =

Rd

e

−2πiω·y

    1 1 f x + M + I y g x + M − I y dy, 2 2

(33) which reveals the similarity with the Wigner distribution. A sort of symmetry with respect to the Wigner distribution (corresponding to M = 0) immediately stands out. We interpret these representations as a family of “linear perturbations” of the Wigner distribution and M as the control parameter, exactly as τ controls the degree of deviation of τ -Wigner distributions. For this reason, we will refer to A = AM as the perturbative form of a Cohen-type matrix. The analogy with the τ -Wigner distributions naturally leads to another representation, hence another choice of A22 in (32). A closer inspection of the kernel (7) and also of (6) reveals that the role of

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perturbation parameter is not played by τ , rather by the deviation μ = τ − 1/2. In this analogy one chooses A21 = T ∈ Rd×d and A22 = −(I − T ) and obtains  WT (f, g)(x, ω) ≡ BAT (f, g)(x, ω) =

Rd

e−2πiω·y f (x + T y) g (x − (I − T )y)dy,

(34) which should be compared to (6) (see also (21)). Occasionally, we refer to AT as the affine form of the Cohen-type matrix A. It is clear that the two forms of a Cohen-type matrix are perfectly equivalent, the connection being M = T − (1/2)I.

(35)

Therefore, the choice of a form is just a matter of convenience: when studying the properties of BA as a time-frequency representation, it seems better to explicitly see the effect of the perturbation M (which could be easily turned off setting M = 0) and use the perturbative form accordingly. As an example of this, the perturbed representation of a Gaussian signal is provided. Lemma 4 ([11, Lem. 4.1]) Consider A = AM ∈ GL (2d, R) as in (11) and 2 ϕλ (t) = e−π t /λ , λ > 0. Then, WM ϕλ (x, ω) = (2λ)d/2 det (S)−1/2 e−2π x · e8π



2 /λ

 M * x·S −1 M * x /λ 8π iS −1 ω·M * x −2π λω·S −1 ω

e

e

,

(36)

where S = I + 4M * M ∈ Rd×d .

3.3.1

Main Properties of the Cohen Class

The properties of a time-frequency distribution belonging to the Cohen class are intimately related to the structure of the Cohen kernel. There is an established list of correspondences between the kernel and the properties, which can be used to deduce the following results. See [7, 11, 36, 38]. Proposition 7 Assume that BA belongs to the Cohen’s class. For any f, g ∈ M 1 (Rd ), the following properties are satisfied: (i) Correct marginal densities: 

 Rd

BA f (x, ω) dω = |f (x)|2 ,

Rd



2



BA f (x, ω) dx = fˆ (ω) ,

x, ω ∈ Rd .

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In particular, the energy is preserved:  BA f (x, ω) dxdω = f 2L2 . R2d

(ii) Moyal’s identity: BA f, BA gL2 (R2d ) = |f, g|2 . (iii) Symmetry: for all x, ω ∈ Rd , BA (If ) (x, ω) = IBA f (x, ω) = BA f (−x, −ω) ,   BA f (x, ω) = I2 BA f (x, ω) = BA (x, −ω). (iv) Convolution properties: for all x, ω ∈ Rd , BA (f ∗ g) (x, ω) = BA f ∗1 BA g, BA (f · g) (x, ω) = BA f ∗2 BA g. Here ∗1 (and ∗2 ) denotes the convolution with respect to the first (second) variable. (v) Scaling invariance: setting Uλ f (t) := |λ|d/2 f (λt), λ ∈ R \ {0}, t ∈ Rd ,   BA (Uλ f ) (x, ω) = BA f λx, λ−1 ω . (vi) Strong support property3 : the only MWDs in the Cohen class satisfying the strong correct support properties are Rihaczek and conjugate-Rihaczek distributions. (vii) Weak support property4 : the only MWDs in Cohen’s class satisfying the weak correct support properties are the τ -Wigner distributions with τ ∈ [0, 1].

d Qf : R2d (x,ω) → C be a time-frequency distribution associated with the signal f : Rt → C in a suitable function space. Recall that Q is said to satisfy the strong support property if

3 Let

f (x) = 0 ⇔ Qf (x, ω) = 0,

∀ω ∈ Rd ,

fˆ (ω) = 0 ⇔ Qf (x, ω) = 0,

∀x ∈ Rd .

4 With

the notation of the previous footnote, we say that Q satisfies the weak support property if, for any signal f :   πω (suppQf ) ⊂ C suppfˆ , πx (suppQf ) ⊂ C (suppf ) ,

2d d d where πx : R2d (x,ω) → Rx and πω : R(x,ω) → Rω are the projections onto the first and second 2d d d factors (R(x,ω) ) Rx × Rω ) and C (E) is the closed convex hull of E ⊂ Rd .

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We now give a few hints on several aspects of interests for both theoretical problems and applications; extensive discussions on these issues may be found in [1, 11]. Real-Valuedness In view of Proposition 4 (i), BA0 = W (the Wigner distribution) is the only real-valued member of the family BAM . More on Marginal Densities The marginal densities for a general distribution BA can be easily computed. For f, g ∈ M 1 (Rd ),  Rd



Rd

BA f (x, ω) dω = f (A11 x) f (A21 x),     BA f (x, ω) dx = |det A|−1 fˆ (A# )12 ω fˆ −(A# )22 ω .

The correct marginal densities are thus recovered if and only if A11 = A21 = I and (A# )12 = −(A# )22 = I . These conditions force both |det A| = 1 and the block structure of A as that of Cohen’s type. This fact provides an equivalent characterization of the distributions BA belonging to the Cohen class: these are exactly those satisfying the correct marginal densities. Relation Between Two Distributions Let A1 = AM1 and A2 = AM2 be two Cohen-type matrices as in (11). The two distributions WM1 and WM2 are connected by a Fourier multiplier as follows [11, Lem. 4.2]. For f, g ∈ M 1 (Rd ), FWM2 (f, g) (ξ, η) = e−2π iξ ·(M2 −M1 )η FWM1 (f, g) (ξ, η) .

(37)

Furthermore, if M2 − M1 ∈ GL (d, R), BA2 (f, g) (x, ω) =

1 −1 e2π ix·(M2 −M1 ) ω ∗ BA1 (f, g) (x, ω) . |det (M2 − M1 )|

The proofs follow at once from Theorem 10. Regularity of the Cohen Kernel Let A = AM ∈ GL (2d, R) be a Cohen-type matrix and assume in addition that M ∈ GL (d, R). The Cohen kernel associated with BAM is therefore given by (31), and we can study its regularity with respect to the scale of modulation spaces: we have [11, Prop. 4.8]   θM , FθM ∈ M 1,∞ (R2d ) ∩ W FL1 , L∞ (R2d ). This result has an interesting counterpart on the regularity of WM on all modulation  spaces, in view of the boundedness of Fourier multipliers with symbols in W FL1 , L∞ (cf. [2, Lem. 8]): Theorem 11 ([11, Thm. 4.10]) Let A = AM ∈ GL (2d, R) be a Cohen-type matrix with M ∈ GL (d, R) and f ∈ M ∞ (Rd ) be a signal. Then, for any 1 ≤ p, q ≤ ∞,

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we have Wf ∈ M p,q (R2d ) ⇐⇒ WM f ∈ M p,q (R2d ). Unfortunately, one cannot go too far if the non-singularity of M is dropped; as a trivial instance, notice  that for M = 0 one has θ0 = δ, and it is easy to verify that δ ∈ M 1,∞ (R2d )\W FL1 , L∞ (R2d ), cf. [13]. Perturbation and Interferences The emergence of unwanted artefacts is a wellknown drawback of any quadratic representation. The signal processing literature is full of strategies to mitigate these effects (see for instance [7, 32, 33]). For what concerns the Cohen class, it is folklore that the severity of interferences is somewhat related to the decay of the Cohen kernel. In fact, a precise formulation of this principle is rather elusive and recent contributions unravelled further nontrivial fine points [14, Prop. 4.4 and Thm. 4.6]. We remark that the chirp-like kernel $M = FθM does not decay at all, and thus no smoothing effect should be expected for the perturbed representations. This is confirmed by the experiments in dimension d = 1 in [3, 11]. The only effect of the perturbation consists of a distortion and relocation of interferences, but there is no damping. Following the engineering literature, we suggest that convolution with suitable decaying distributions may provide some improvement, probably at the price of loosing other nice properties. Covariance Formula For any z = (z1 , z2 ) , w = (w1 , w2 ) ∈ R2d , the covariance formula (26) now reads 

WM (π (z) f, π (w) g) (x, ω) = e

2π i



1 2 (z2 +w2 )+M(z2 −w2 )

·(z1 −w1 )

× MJ (z−w) TTM (z,w) WM (f, g) (x, ω) ,

(38)

where (1/2) (z1 + w1 ) + M (w1 − z1 ) TM (z, w) = (1/2) (z2 + w2 ) + M (z2 − w2 )  1 −M 0 = (z + w) + (z − w) . 0 M 2 

Alternatively, adopting the affine representation of A (35):  PT =

−T 0 , 0 −(I − T )

 I + PT =

I −T 0 0 T

,

(39)

we can also write  TT (z, w) =

(I − T )z1 + T w1 T z2 + (I − T )w2

= (I + PT ) z − PT w.

(40)

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Boundedness on Modulation Spaces We cite here some results on the continuity of the distributions BAM on the aforementioned spaces. For the sake of clarity, we report a simplified, unweighted form of [11, Thm. 4.12]. Theorem 12 Let A = AT ∈ GL (2d, R) be a Cohen-type matrix. Let 1 ≤ pi , qi , p, q ≤ ∞, i = 1, 2, such that pi , qi ≤ q,

i = 1, 2,

(41)

1 1 1 1 + ≥ + . q1 q2 p q

(42)

and 1 1 1 1 + ≥ + , p1 p2 p q

  (i) If f1 ∈ M p1 ,q1 (Rd ) and f2 ∈ M p2 ,q2 (Rd ), then WT (f1 , f2 ) ∈ M p,q R2d , and the following estimate holds: WT (f1 , f2 ) M p,q T f1 M p1 ,q1 f2 M p2 ,q2 . (ii) Assume further that both T and I − T are invertible (equivalently: AT is rightp1 ,q1 (Rd ) and f ∈ M p2 ,q2 (Rd ), regular, or PT is invertible, cf. (39)). If f 2 1 ∈ M  2d p q then WT (f1 , f2 ) ∈ W (FL , L ) R , and the following estimate holds: WT (f1 , f2 ) W (FLp ,Lq ) T (CT )1/q−1/p f1 M p1 ,q1 f2 M p2 ,q2 , where CT = |det T | |det (I − T )| > 0.

(43)

Sharp estimates and continuity results of this type have been given by some of the authors for the case of τ -Wigner distributions in [16, Lem. 3.1] and [15]. To conclude this section we remark that one can specialize Proposition 5 in order to characterize the boundedness on Lebesgue spaces at the price of assuming rightregularity of AM , see [11, Thm. 4.14].

4 Pseudodifferential Operators In this section we discuss the formalism of pseudodifferential operators that is associated with every time-frequency representation BA . Imitating the timefrequency analysis of Weyl pseudodifferential operators, we introduce the following general calculus for pseudodifferential operators.

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  Theorem 13 ([1, Prop. 2.2.1]) Let A ∈ GL (2d, R) and σ ∈ M ∞ R2d . The mapping opA (σ ) ≡ σ A defined by duality as / 0 σ A f, g ≡ σ, BA (g, f ) ,

f, g ∈ M 1 (Rd )

is a well-defined linear continuous map from M 1 (Rd ) to M ∞ (Rd ). The proof easily follows from the continuity of the distribution BA : M 1 (Rd ) × M 1 (Rd ) → M 1 (R2d ), from Proposition 3.   Definition 4 Let A ∈ GL (2d, R) and σ ∈ M ∞ R2d . The mapping defined in Theorem 13, namely / 0 σ A : M 1 (Rd ) - f → σ A f ∈ M ∞ (Rd ) : σ A f, g = σ, BA (g, f ) ,

∀g ∈ M 1 (Rd ),

is called quantization rule with symbol σ associated with the matrix-Wigner distribution BA or pseudodifferential operator with symbol σ associated with the matrix-Wigner distribution BA . Using Feichtinger’s kernel theorem (Theorem 1), we now provide a number of equivalent representations for σ A f . ∞ d ) be a continuous Theorem 14 Let A ∈ GL (2d, R). Let T : M 1 (Rd ) → M  2d(R  ∞ such that T admits linear operator. There exist distributions k, σ, F ∈ M R the following representations: ! 1. as an integral operator with kernel k: Tf, g = k, g ⊗ f for any f, g ∈ M 1 (Rd ); 2. as pseudodifferential operator with symbol σ associated with BA : T = σ A ; 3. as a superposition (in weak sense) of time-frequency shifts (also called spreading representation):

 T =

R2d

F (x, ω) Tx Mω dxdω.

The relations among k, σ, F and A are the following: σ = |det A| F2 TA k,

F = F2 TAST k.

(44)

Proof The first representation   is exactly the claim of kernel theorem. Now set σ = |det A| F2 TA k ∈ M ∞ R2d : this is a well-defined distribution, since F2 and TA are   isomorphisms on M ∞ R2d . In particular, for any f, g ∈ M 1 (Rd ) we have ! Tf, g = k, g ⊗ f 0 / −1 = |det A|−1 T−1 A F2 σ, g ⊗ f

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105

 ! = σ, F2 TA g ⊗ f = σ, BA (g, f ) / 0 = σ A f, g . This proves that Tf = σ A f in M ∞ (Rd ). The relation between the kernel representation in 1 and the spreading representation in 3 is well-known, e.g.  [26]. It 0 I can also be deduced from item 2 from the special matrix AST = and −I I ! ! Tf, g = F, Vf g = F, BAST (g, f ) ,

f, g ∈ M 1 (Rd ).  

−1 Remark 3 Since k = |det A|−1 TA−1 F−1 TI2 A−1 F−1 σ , one can 2 σ = |det A| 1  formally obtain another representation of the third type with a special spreading function:

σ A f (x) =

1 |det A|

 R2d

 σ (ξ, −(A−1 )21 x − (A−1 )22 y)e2πiξ ·[(A

−1 )

11 x+(A

−1 )

22 y]

f (y)dξ dy.

(45) Notice that the inverse of a Cohen-type matrix A = AT has the form A−1 T

 =

−(I − T ) T I −I

,

thus the previous formula becomes  σ f (x) = A

R2d

 σ (ξ, u)e−2π i(I −T )u·ξ T−u Mξ f (x)dξ du.

(46)

This should be compared with [26, Eq. 14.14] and [15, Eq. 20]. We now study the relations among pseudodifferential operators associated with MWDs and the corresponding symbols.   Proposition 8 Let A, B ∈ GL (2d, R) and σ, ρ ∈ M ∞ R2d . Then, σ A = ρB

⇐⇒

σ =

|det A| F2 TB −1 A F−1 2 ρ |det B|

Proof Assume that T = σ A = ρ B . According to Theorem 14, T has a distributional kernel k such that σ = |det A| F2 TA k,

ρ = |det B| F2 TB k.

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Therefore, σ = |det A| F2 TA k =

|det A| −1 F2 TA T−1 B F2 ρ |det B|

=

|det A| F2 TB −1 A F−1 2 ρ. |det B|

On the other side, if σ = |det A| |det B|−1 F2 TB −1 A F−1 2 ρ, then for any f, g ∈ M 1 (Rd ) / 0 σ A f, g = σ, F2 TA (f ⊗ g) / 0 −1 = |det A| |det B|−1 F2 TA T−1 F ρ, F T g) ⊗ (f 2 A B 2 = ρ, F2 TB (f ⊗ g) / 0 = ρ B f, g .   When the operators are associated with Cohen-type matrices, we have a more explicit relation that covers the usual rule for τ -Shubin operators (cf. [42, Rem. 1.5]). The proof is a straightforward application of (37). Proposition9 Let  A1 = AT1 , A2 = AT2 be Cohen-type invertible matrices, and σ, ρ ∈ M ∞ R2d . Then, σ1T1 = σ2T2

⇐⇒

σ2 (ξ, η) = e−2π iξ ·(T2 −T1 )η σ1 (ξ, η) .

It is also interesting to characterize the matrices yielding self-adjoint operators.   Proposition 10 ([1, Prop. 2.2.3]) Let A ∈ GL (2d, R) and σ ∈ M ∞ R2d . Then  ∗ σ A = ρB , where ρ = σ,

B = I˜AI2 =



A21 −A22 A11 −A12

.

In particular, σ A is self-adjoint if and only if σ = σ (real symbol) and B = A, hence A21 = A11 ,

A12 = −A22 .

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P Q Remark 4 Thus, only matrices of the form , with P , Q ∈ GL (d, R), P −Q give rise to pseudodifferential operators which are self-adjoint for real symbols. This occurs for Weyl calculus but not for Kohn-Nirenberg operators (T = 0).

4.1 Boundedness Results 4.1.1

Operators on Lebesgue Spaces

The boundedness of a pseudodifferential operator σ A associated with BA is intimately related to the boundedness of the distribution BA on certain function spaces, in view of the duality in the definition of σ A . Let us start this section with two easy results. Proposition  11 ([1, Theorems 2.2.6, 2.2.7 and 2.2.9]) Let A ∈ GL (2d, R) and σ ∈ M ∞ R2d .   1. If A is right-regular and σ ∈ L1 R2d , then σ A is a bounded operator on L2 (Rd ) such that 1 1 1 A1 1σ 1

≤

L2 →L2

σ L1 |det A12 |1/2 |det A22 |1/2

.

  2. If σ ∈ L2 R2d , then σ A is a bounded operator on L2 (Rd ) such that 1 1 1 A1 1σ 1

L2 →L2

≤

σ L2 |det A|1/2

.

  If σ ∈ L2 R2d , then σ A is actually a Hilbert-Schmidt operator, and every HilbertSchmidt operator T possesses a symbol σ ∈ L2 (R2d ) such that T = σ A . The study on Lebesgue spaces can be largely expanded thanks to the following result.   Theorem 15 Let A ∈ GL (2d, R) be right-regular and σ ∈ Lq R2d . The quantization mapping   σ ∈ Lq (R2d ) → σ A ∈ L Lp (Rd ) is continuous if and only if q ≤ 2 and q ≤ p ≤ q  , with norm estimate 1 1 1 A1 1σ 1

Lp →Lp

≤

σ Lq |det A|

1 q

1

|det A12 | p

− q1

1

|det A22 | p

− q1

.

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Proof Assume f ∈ Lp (Rd ) and g ∈ Lp (Rd ), with p = 1 nor p = ∞. Therefore, by (24) (switch q and q  ) and Hölder inequality:

/ 0



A

σ f, g = |σ, BA (g, f )| ≤ σ Lq BA (g, f ) Lq  ≤

σ Lq |det A|

1 q

1

|det A12 | p

− q1

1

|det A22 | p

− q1

f Lp g Lp .

The non-continuity is a consequence of [4, Prop. 3.2].

 

Remark 5 Note that the closed graph theorem implies non-continuity of the quantization map. This means that there exists a symbol σ ∈ Lq for which the operator σ A is not bounded on Lp (R2d ), cf. [4, Prop. 3.4]. One could also study compactness and Schatten class properties for these operators. We confine ourselves to prove a result for symbols in the Feichtinger algebra.   Theorem 16 ([1, Thm. 2.2.8]) Let A ∈ GL (2d, R) and σ ∈ M 1 R2d . The     operator σ A ∈ L L2 (Rd ) belongs to the trace class S1 L2 (Rd ) , and the following estimate holds for the trace class norm: 1 1 1 A1 1σ 1

S1

 |det A|1/2 σ M 1 .

Proof The proof follows the outline   of [25]. Proposition   11 immediately yields  σ A ∈ L L2 (Rd ) , since M 1 R2d ⊆ L1 R2d ∩ L2 R2d . In line with the paradigm of time-frequency analysis of operators (cf. [26, Sec. 14.5]), let us decompose the action of σ A into elementary pseudodifferential operators with timefrequency shifts of a suitable function as symbols. The inversion formula for the STFT allows to write   V σ (z, ζ ) Mζ Tz dzdζ, σ = R2d

R2d

  for any window function  ∈ M 1 R2d with  L2 = 1. Therefore, for any f, g ∈ M 1 (Rd ): / 0 σ A f, g = σ, BA (g, f )   ! = V σ (z, ζ ) Mζ Tz , BA (g, f ) dzdζ  =

R2d

R2d



R2d

R2d

/ 0 A V σ (z, ζ ) Mζ Tz  f, g dzdζ.

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This shows that σ A acts (in an operator-valued sense on L2 ) as a continuous weighted superposition of elementary operators:  σA =

 R2d

R2d

A  V σ (z, ζ ) Mζ Tz  dzdζ.

 A The action of the building blocks Mζ Tz  can be unwrapped by means of the magic formula (28) provided one takes  = BA ϕ for some ϕ ∈ M 1 (Rd ) with ϕ L2 = |det A|1/4 (see the orthogonality relations (25)): / 0 A ! Mζ Tz  f, g = Mζ Tz , BA (g, f ) = VBA ϕ BA (g, f ) (z, ζ ) = e2π iz2 ·ζ2 Vϕ f (b, β) Vϕ g (a, α), where a, α, b, β are continuous functions of z and ζ . In particular, we have  A Mζ Tz  : L2 (Rd ) → L2 (Rd )

:

! f → e2π iz2 ·ζ2 f, Mβ Tb ϕ Mα Ta ,

 A hence M T  is a rank-one operator with trace class norm given by ζ z 1 A 1 1 1 1 Mζ Tz  1 1 = ϕ 2L2 = |det A|1/2 , independent of z, ζ . To conclude, we S

reconstruct the operator σ A and compute its norm by means of the estimates for the pieces: 1 1 1 A1 1σ 1

S1

 ≤

R2d

 R2d

1 A 1 1 1 |V σ (z, ζ )| 1 Mζ Tz  1

S1

dzdζ ≤ CA σ M 1 .  

4.1.2

Operators on Modulation Spaces

We now study the boundedness on modulation spaces of pseudodifferential operators associated with Cohen-type representations. Theorem 17 (Symbols in M p,q ) Let A = AT ∈ GL (2d, R) be a Cohen-type matrix and consider indices 1 ≤ p, p1 , p2 , q, q1 , q2 ≤ ∞, satisfying the following relations: p1 , p2 , q1 , q2 ≤ q  ,

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and 1 1 1 1 +  ≥  + , p1 p2 p q

1 1 1 1 +  ≥  + . q1 q2 p q

  For any σ ∈ M p,q R2d , the pseudodifferential operator σ A is bounded from M p1 ,q1 (Rd ) to M p2 ,q2 (Rd ). In particular, 1 1 1 A1 1σ 1

M p1 ,q1 →M p2 ,q2

A σ M p,q .

Proof Under the given assumptions on the indices, Theorem 12 implies that     BAT (g, f ) ∈ M p ,q (Rd ) for any f ∈ M p1 ,q1 (Rd ) and g ∈ M p2 ,q2 (Rd ). Therefore, by the duality of modulation spaces we obtain

/ 0



A

σ f, g = |σ, BA (g, f )| 1 1 ≤ σ M p,q 1BAM (g, f )1M p ,q  A σ M p,q f M p1 ,q1 g M p2 ,q2 .   For symbols in W (FLp , Lq ) spaces, we can extend [16, Thm. 1.1] to the matrix setting. Theorem 18 (Symbols in W (FLp , Lq )) Let A = AT ∈ GL (2d, R) a rightregular Cohen-type matrix and consider indices 1 ≤ p, q, r1 , r2 ≤ ∞, satisfying the following relations: q ≤ p ,

r1 , r1 , r2 , r2 ≤ p.

  For any σ ∈ W (FLp , Lq ) R2d , the pseudodifferential operator σ A is bounded on M r1 ,r2 (Rd ); in particular, 1 1 1 A1 1σ 1

M r1 ,r2 →M r1 ,r2

A σ W (FLp ,Lq ) .

Proof We isolate two special cases of Theorem 12. First, for any f ∈ M p1 ,p2 (Rd )   and g ∈ M p1 ,p2 (Rd ) we have WT (g, f ) W (FL1 ,L∞ ) T f M p1 ,p2 g

M

p1 ,p2

,

1 ≤ p1 , p2 ≤ ∞.

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111

p1 ,p2 (Rd ), for any 1 ≤ p , p ≤ ∞ and that This yields that σ A isbounded on 1 2  M  2d ∞ 1 the symbol σ is in W FL , L R , because

/ 0



A

σ f, g = |σ, WT (g, f )| ≤ σ W (FL∞ ,L1 ) WT (g, f ) W (FL1 ,L∞ ) T σ W (FL∞ ,L1 ) f M p1 ,p2 g

M

p1 ,p2

.

The second case requires f, g ∈ M 2 (Rd ), then WT (g, f ) W (FL2 ,L2 ) T f M 2 g M 2 .    If σ ∈ W FL2 , L2 R2d = L2 (R2d ), then σ A is bounded on M 2 (Rd ) = L2 (Rd ) by similar arguments: 1 1 1 A1 1σ 1

M 2 →M 2

T σ W (FL2 ,L2 ) .

We proceed now by complex interpolation of the continuous mapping opA on modulation spaces; in particular, we are dealing with    opA : W FL1 , L∞ R2d × M p1 ,p2 (Rd ) → M p1 ,p2 (Rd ),    opA : W FL2 , L2 R2d × M 2 (Rd ) → M 2 (Rd ). For θ ∈ [0, 1], we have         W FL1 , L∞ , W FL2 , L2 = W FLp , Lp , θ



M p1 ,p2 , M 2,2

 θ

2 ≤ p ≤ ∞,

= M r1 ,r2 ,

with 1−θ 1 1−θ θ 1 = + = + , ri pi 2 pi p

i = 1, 2.

From these estimates we immediately derive the condition r1 , r1 , r2 , r2 ≤ The inclusion relations for modulation spaces allow to extend the result  W (FLp , Lq ) R2d for any q ≤ p . Finally, exchanging the role of p and allows us to cover any p ∈ [1, ∞].

p. to p  

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4.2 Symbols in the Sjöstrand Class Another important space of symbols is the Sjöstrand class. It has been introduced by 0 and later recognized Sjöstrand [41] to extend the well-behaved Hörmander class S0,0 to coincide with the modulation space M ∞,1 (R2d ). Accordingly, it consists of bounded symbols with low regularity in general, namely temperate distributions σ ∈ S (R2d ) such that  sup |σ, π(z, ζ )g|dζ < ∞. R2d z∈R2d

Nevertheless, they lead to L2 -bounded pseudodifferential operators. In fact, much more is true: the family of Weyl operators with symbols in the Sjöstrand class is an inverse-closed Banach *-subalgebra of L(L2 )(Rd ), in the following sense. Theorem 19

  (i) (Boundedness) If σ ∈ M ∞,1 R2d , then opW (σ ) is a bounded operator on L2 (Rd ).   (ii) (Algebra property) If σ1 , σ2 ∈ M ∞,1 R2d and opW (ρ) = opW (σ1 )opW (σ2 ),   then ρ ∈ M ∞,1 R2d .   (iii) (Wiener property) If σ ∈ M ∞,1 R2d and opW (σ ) is invertible on L2 (Rd ),   −1  = opW (ρ) for some ρ ∈ M ∞,1 R2d . then opW (σ ) These results have been put into the context of time-frequency analysis in [29] and have been generalized, see [15, 30, 31]. In this section we extend Theorem 19 with respect to MWDs. Let us first provide some conditions on the matrices for which the associated pseudodifferential operators with symbols in M ∞,1 (R2d ) are bounded on modulation spaces. Theorem 20 ([1, Thm. 2.3.1]) Let σ ∈ M ∞,1 (R2d ) and assume A ∈ GL (2d, R) is a left-regular matrix. The pseudodifferential operator σ A is bounded on all modulation spaces M p,q (Rd ), 1 ≤ p, q ≤ ∞, with 1 1 1 A1 1σ 1

M p,q →M p,q

A

1 

|det A11 |1/p |det A21 |1/p

1 σ M ∞,1 . ·





det(A12 )# 1/q det(A22 )# 1/q

(47)   Proof Let f, g ∈ M 1 (Rd ) and  ∈ M 1 R2d \ {0}. Then,

/ 0



A

σ f, g = |σ, BA (g, f )| = |V σ, V BA (g, f )| ≤ V σ L∞,1 V BA (g, f ) L1,∞ ,

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where in the last line we used Hölder inequality for mixed-norm Lebesgue spaces. Let us choose for instance  = BA ϕ where ϕ ∈ M 1 (Rd ) is the Gaussian function. We introduce the affine transformations     z1 0 −A12 ζ1 A11 0 + , Pζ (z1 , z2 ) = P1 z + P2 ζ = # # z2 0 ζ2 0 (A12 ) (A11 )  Qζ (z1 , z2 ) = Q1 z+Q2 ζ =

0 A21 0 − (A22 )#



  z1 0 −A22 ζ1 + , z2 − (A21 )# 0 ζ2

in according with the magic formula (28), and using again Hölder’s inequality we get  V BA (g, f ) L1,∞ =

sup (ζ1 ,ζ2 )∈R2d

≤

sup

R2d

  



Vϕ g Pζ (z1 , z2 ) Vϕ f Qζ (z1 , z2 ) dz1 dz2

1 1  1 V ϕ f ◦ Qζ 1

(ζ1 ,ζ2 )∈R2d

p,q

Lz

1 1  1 Vϕ g ◦ Pζ 1

p  ,q 

Lz

1 1 1 1 1Vϕ f 1 p,q 1Vϕ g 1 p ,q  L L =

1/q

1/q  

|det A21 |1/p det (A22 )# |det A11 |1/p det (A12 )#

≤C

|det A21

g M p ,q  f M p,q





,

det (A22 )# 1/q |det A11 |1/p det (A12 )# 1/q

|1/p

where the constant C does not depend on f , g or A. On the other hand, V σ L∞,1 ≤ C σ M ∞,1 ,

(48)

where the constant C depends on  = BA ϕ, hence on A. We conclude by duality and get the claimed result.   Remark 6 This result broadly generalizes [26, Thm. 14.5.2] and confirms again that the Sjöstrand’s class is a well-suited symbol class leading to bounded operators on modulation spaces. It is worth to mention that the left-regularity assumption for A covers any Cohen-type matrix A = AM . Remark 7 Unfortunately, it is not easy to sharpen the estimate (47) neither in the case of Cohen-type matrices, the main obstruction being the estimate in (48). The usual strategy consists of finding a suitable alternative window function  ∈ M 1 (R2d ) and then estimate V BA φ L1 , in order to apply [26, Lem. 11.3.3] and [26, Prop. 11.1.3(a)]. This require cumbersome computations even in the case of τ -distributions with  Gaussian function, but the result is a uniform estimate for any τ ∈ [0, 1], cf. [16, Lem. 2.3].

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We now prove a similar boundedness result on W (FLp , Lq ) spaces. We require a very special case of the symplectic covariance for the Weyl quantization which is stable under matrix perturbations—cf. [15, Lem. 5.1] for the τ -Wigner case.   Lemma 5 For any symbol σ ∈ M ∞ R2d and any Cohen-type matrix A = AT ∈ GL (2d, R):   FopAT (σ ) F−1 = opAI −T σ ◦ J −1 . Proof We use a formal argument and leave to the reader the discussion on the function spaces on which it is well defined. Recall the spreading representation of the operator opAT from (46)  opAT f (x) =

R2d

 σ (ξ, u) e−2π i(I −T )u·ξ T−u Mξ f (x) dudξ.

Since FT−u Mξ F−1 = e2π iu·ξ Tξ Mu , we obtain FopAT F−1 =

 R2d

 σ (ξ, u) e−π i(2T −I )u·ξ Tξ Mu dudξ

  = opAI −T σ ◦ J −1 .   Remark 8 A comprehensive account of the symplectic covariance for perturbed representation is out of the scope of this paper. This would require to investigate how the metaplectic group should be modified in order to accommodate the perturbations. As a non-trivial example of this issue, we highlight the contribution of de Gosson in [18] for τ -pseudodifferential operators. Theorem 21  For  any Cohen-type matrix A = AT ∈ GL (2d, R) and any symbol σ ∈ M ∞,1 R2d , the operator opA (σ ) is bounded on W (FLp , Lq ) (Rd ) with ∞,1 opA (σ ) W (FLp ,Lq )→W (FLp ,Lq ) ≤ CA σ M ,

for a suitable CA > 0. Proof The proof follows the pattern of [15, Thm. 5.6]. Set σJ = σ ◦ J and consider the following commutative diagram: M p,q (Rd )

opI −T (σJ )

F −1

W (FLp , Lq ) (Rd )

M p,q (Rd ) F

opτ (σ )

W (FLp , Lq ) (Rd )

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From the formula VG (σJ )(z, ζ ) = VG◦J  −1 σ (J z, J ζ ), for any suitable  window G, it follows easily that σ ∈ M ∞,1 R2d implies σJ ∈ M ∞,1 R2d . The operator opI −T (σJ ) is bounded on M p,q (Rd ) as a consequence of Theorem 20 and the claim follows at once thanks to the previous lemma.   The significance of the Sjöstrand class as space of symbols comes in many shapes. In [29] an alternative characterization of the Sjöstrand class was given in terms of a quasi-diagonalization property satisfied by the Weyl operators with symbol in M ∞,1 . Let us briefly recall the main ingredients of this result. First, fix a non-zero window ϕ ∈ M 1 (Rd ) and a lattice  = AZ2d ⊆ R2d , where A ∈ GL(2d, R), such that G (ϕ, ) is a Gabor frame for L2 (Rd ). The action of pseudodifferential operators on time-frequency shifts is described by the entries of the so-called channel matrix, that is opW (σ )π(z)ϕ, π(w)ϕ,

z, w ∈ R2d ,

or M(σ )λ,μ := opW (σ )π(λ)ϕ, π(μ)ϕ,

λ, μ ∈ ,

if we restrict to the discrete lattice . We say that opW is almost diagonalized by the Gabor frame G(ϕ, ) if the associated channel matrix exhibits some sort of offdiagonal decay - equivalently, if the time-frequency shifts are almost eigenvectors for opW . Theorem 22 ([29]) Let ϕ ∈ M 1 (Rd )(Rd ) be a non-zero window function such that G (ϕ, ) be a Gabor frame for L2 (Rd ). The following properties are equivalent: (i) σ ∈ M ∞,1 (R2d ). (ii) σ ∈ M ∞ (R2d ) and there exists a function H ∈ L1 (R2d ) such that |opW (σ ) π (z) ϕ, π (w) ϕ| ≤ H (w − z) ,

∀w, z ∈ R2d .

  (iii) σ ∈ M ∞ R2d and there exists a sequence h ∈ 1 () such that |opW (σ ) π (μ) ϕ, π (λ) ϕ| ≤ h (λ − μ) ,

∀λ, μ ∈ .

Corollary 2 Under the hypotheses of the previous theorem, assume that T : M 1 (Rd ) → M ∞ (Rd ) is continuous and satisfies one of the following conditions: (i) |T π (z) ϕ, π (w) ϕ| ≤ H (w − z) , (ii) |T π (μ) ϕ, π (λ) ϕ| ≤ h (λ − μ) ,

∀w, z ∈ R2d for some H ∈ L1 . ∀λ, μ ∈  for some h ∈ 1 .   Then T = opW (σ ) for some symbol σ ∈ M ∞,1 R2d .

The backbone of this result is the interplay between the entries of the channel matrix of opW and the short-time Fourier transform of the symbol. Theorem 22

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can be extended without difficulty to τ -pseudodifferential operators [15]. We now indicate a further generalization to operators associated with Cohen-type matrices. Assume that A = AM ∈ GL (2d, R) is a matrix of Cohen’s type. The following result easily follows from the covariance formula (38). Lemma 6 Let A = AT ∈ GL (2d, R) be a matrix of Cohen’s type andfix a nonzero window ϕ ∈ M 1 (Rd ), then set A = WT ϕ. Then, for any σ ∈ M ∞ R2d

/ 0







A

σ π (z) ϕ, π (w) ϕ = VA σ (TT (w, z) , J (w − z)) = VA σ (x, y) , and 0



/

V σ (x, y) =

σ A π (z (x, y)) ϕ, π (w (x, y)) ϕ

, A for any x, y, z, w ∈ R2d , where TT is defined in (40) and z (x, y) = x + (I + PT ) Jy,

w (x, y) = x + PT Jy.

(49)

Proof We have

/ 0

A

σ π (z) ϕ, π (w) ϕ = |σ, BA (π (w) ϕ, π (z) ϕ)|

!

= σ, MJ (w−z) TTT (w,z) BA ϕ (x, ω)



= VA σ (TT (w, z) , J (w − z)) . Now, setting x = TT (w, z) and y = J (w − z) we immediately get Eq. (49).

 

Theorem 23 Let ϕ ∈ be a non-zero window function and assume that  is a lattice such that G (ϕ, ) is a Gabor frame for L2 (Rd ). For any Cohen-type matrix A = AT ∈ GL (2d, R), the following properties are equivalent:   (i) σ ∈ M ∞,1 R2d .   (ii) σ ∈ M ∞ R2d and there exists a function H = HT ∈ L1 (Rd ) such that M 1 (Rd )

/ 0



A

σ π (z) ϕ, π (w) ϕ ≤ HT (w − z) ,

∀w, z ∈ R2d .

  (iii) σ ∈ M ∞ R2d and there exists a sequence h = hT ∈ 1 () such that

/ 0



A

σ π (μ) ϕ, π (λ) ϕ ≤ hT (λ − μ) ,

∀λ, μ ∈ R2d .

Proof The proof faithfully mirrors the one for Weyl operators [29, Theorem 3.2]. We detail here only the case (i) ⇒ (ii), the discrete case for hT is similar. For ϕ ∈ M 1 (Rd ) the T -Wigner distribution T = WT ϕ is in M 1 (R2d ) by Theorem 12.

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This implies   that the short-time Fourier transform VT σ is well-defined for σ ∈ M ∞,1 R2d (cf. [26, Theorem 11.3.7]). The main insight here is that the controlling function HT ∈ L1 (Rd ) can be provided by the so-called grand symbol associated   with σ , which is H˜ T (v) = supu∈R2d |VT σ (u, v)|. By definition of M ∞,1 R2d , we have H˜ T ∈ L1 (R2d ), so that Lemma 6 implies

/ 0



A

σ π (z) ϕ, π (w) ϕ = VT σ (TT (z, w) , J (w − z))



≤ sup VT σ (u, J (w − z))

u∈R2d

= H˜ T (J (w − z)) . Setting HT = H˜ T ◦ J yields the claim.

 

Let us further discuss the main trick of the proof. While the choice of the grand symbol as controlling function is natural in view of the M ∞,1 norm, the effect of the perturbation matrix T is confined to the window function T and to the time variable of the short-time Fourier transform of the symbol. A natural question is then the following: what happens if we try to control the time dependence of Vτ σ ? Following the pattern of [15, Thm. 4.3], this remark provides a similar characterization for symbols belonging to the modulation space W (FL∞ , L1 ) = FM ∞,1 . Theorem 24 Let ϕ ∈ M 1 (Rd ) be a non-zero window function. For any rightregular Cohen-type matrix A = AT ∈ GL (2d, R), the following properties are equivalent:   (i) σ ∈ W (FL∞ , L1 ) R2d .   (ii) σ ∈ M ∞ R2d and there exists a function H = HT ∈ L1 (Rd ) such that

/ 0



A

σ π (z) ϕ, π (w) ϕ ≤ HT (w − UT z),

∀w, z ∈ R2d ,

where 0 (I − T )−1 T . UT = − 0 T −1 (I − T ) 

Proof The proof is a straightforward adjustment of the one provided for [15, Thm. 4.3]. Again, we detail here only the case (i) ⇒ (ii) for the purpose of tracking the origin of UT . If φ ∈ M 1(Rd ), φ = 0, then T = WT φ ∈ M 1 = W (FL1 , L1 ) by Theorem 12. For σ ∈ W FL∞ , L1 , we have that VT σ is well defined and  



H˜ T (x) = sup VT σ (x, y) ∈ L1 R2d . y∈R2d

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From Lemma 6 we infer



!

op (σ ) π (z) ϕ, π (w) ϕ = V σ (Tt (w, z) , J (w − z))

T T



≤ sup VT σ (TT (w, z) , y)

y∈R2d

= H˜ T (TT (w, z)) . Notice that if AT is right-regular, then I + PT from (39) is invertible (and the converse holds, too). In particular, we have (I + PT )−1 (TT (w, z)) = w − (I + PT )−1 PT z = w − Uτ z,   −1 ˜ and thus H˜ T (TT (w, z)) = H˜ T B−1 T (w − UT z) . Define HT = HT ◦ (I + PT ) ;  then HT ∈ L1 (R2d ) since HT L1 = H˜ T ◦ (I + PT )−1 L1 H˜ T L1 < ∞.  Remark 9 (i) The almost diagonalization of the (continuous) channel matrix does not survive the perturbation, but the new result can be interpreted as a measure of the concentration of the time-frequency representation of opT (σ ) along the graph of the map UT . (ii) We recover [15, Thm. 4.3], where the same problem was first studied for τ operators with τ ∈ (0, 1). One may push further the analogy and generalize the results for τ = 0 or τ = 1. (iii) As already remarked for τ -operators in [15], the discrete characterization via Gabor frames is lost: for a given lattice , the inclusion Uτ  ⊆  exclusively holds for τ = 1/2 (the Weyl transform). (iv) The boundedness and algebraic properties of AT -operators with symbols in W (FL∞ , L1 ) proved in [15] rely on the characterization as generalized metaplectic operators according to [12, Def. 1.1]. In order to benefit from this framework, it is necessary for UT to be a symplectic matrix, the latter condition being realized if and only if T is a symmetric matrix (cf. [17, (2.4) and (2.5)] and notice that T −1 (I − T ) = (I − T )T −1 ). Acknowledgements E. Cordero and S. I. Trapasso are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). K. Gröchenig acknowledges support from the Austrian Science Fund FWF, project P31887-N32.

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References 1. Bayer, D.: Bilinear Time-Frequency Distributions and Pseudodifferential Operators. PhD Thesis, University of Vienna (2010) 2. Bényi, A., Gröchenig, K., Okoudjou, K., and Rogers, L. G.: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246 (2007), no. 2, 366–384 3. Boggiatto, P., Carypis, E., and Oliaro, A.: Wigner representations associated with linear transformations of the time-frequency plane. In Pseudo-Differential Operators: Analysis, Applications and Computations (275–288), Springer (2011) 4. Boggiatto, P., De Donno, G., and Oliaro, A.: Weyl quantization of Lebesgue spaces. Math. Nachr. 282 (2009), no. 12, 1656–1663 5. Boggiatto, P., De Donno, G., and Oliaro, A.: Time-frequency representations of Wigner type and pseudo-differential operators. Trans. Amer. Math. Soc. 362 (2010), no. 9, 4955–4981 6. Cohen, L.: Time-frequency distributions – A review. Proc. IEEE 77 (1989), no. 7, 941–981 7. Cohen, L.: Time-frequency Analysis. Prentice Hall (1995) 8. Cohen, L.: Generalized phase-space distribution functions. J. Math. Phys. 7 (1966), no. 5, 781– 786 9. Cohen, L.: The Weyl Operator and its Generalization. Springer (2012) 10. Cordero, E., and Nicola, F.: Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation. J. Funct. Anal. 254 (2008), no. 2, 506–534 11. Cordero, E., and Trapasso, S. I.: Linear Perturbations of the Wigner Distribution and the Cohen Class. Anal. Appl. - DOI: https://doi.org/10.1142/S0219530519500052 (2018) 12. Cordero, E., Gröchenig, K., Nicola, F., and Rodino, L.: Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class. J. Math. Phys. 55 081506 (2014) 13. Cordero, E., de Gosson, M., and Nicola, F.: Time-frequency analysis of Born-Jordan pseudodifferential operators. J. Funct. Anal. 272 (2017), no. 2, 577–598 14. Cordero, E., de Gosson, M., Dörfler, M., and Nicola, F.: On the symplectic covariance and interferences of time-frequency distributions. SIAM J. Math. Anal. 50 (2018), no. 2, 2178– 2193 15. Cordero, E., Nicola, F., and Trapasso, S. I.: Almost diagonalization of τ -pseudodifferential operators with symbols in Wiener amalgam and modulation spaces. J. Fourier Anal. Appl. DOI: https://doi.org/10.1007/s00041-018-09651-z (2018) 16. Cordero, E., D’Elia, L., and Trapasso, S. I.: Norm estimates for τ -pseudodifferential operators in Wiener amalgam and modulation spaces. J. Math. Anal. Appl. 471 (2019), no. 1–2, 541–563 17. de Gosson, M.: Symplectic Methods in Harmonic Analysis and in Mathematical Physics. Springer (2011) 18. de Gosson, M.: Born-Jordan quantization. Fundamental Theories of Physics, Vol. 182, Springer [Cham], (2016) 19. Feichtinger, H. G.: On a new Segal algebra. Monatsh. Math. 92 (1981), no. 4, 269–289 20. Feichtinger, H. G.: Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, (1983) and also in Wavelets and Their Applications, M. Krishna, R. Radha, S. Thangavelu, editors, Allied Publishers (2003), 99–140. 21. Feichtinger, H. G.: Generalized amalgams, with applications to Fourier transform, Canad. J. Math., 42 (1990), 395–40 22. Feichtinger, H. G., and Gröchenig, K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146 (1997), no. 2, 464–495. 23. Feig, E., and Micchelli, C. A.: L2 -synthesis by ambiguity functions. In Multivariate Approximation Theory IV, 143–156, International Series of Numerical Mathematics. Birkhäuser, Basel, 1989. 24. Goh, S. S., and Goodman, T. N.: Estimating maxima of generalized cross ambiguity functions, and uncertainty principles. Appl. Comput. Harmon. Anal. 34 (2013), no. 2, 234–251.

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25. Gröchenig, K.: An uncertainty principle related to the Poisson summation formula. Studia Math. 121 (1996), no. 1, 87–104. 26. Gröchenig, K.: Foundations of Time-frequency Analysis. Appl. Numer. Harmon. Anal., Birkhäuser (2001) 27. Gröchenig, K.: Uncertainty principles for time-frequency representations. In Advances in Gabor analysis, 11–30, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2003 28. Gröchenig, K.: A pedestrian’s approach to pseudodifferential operators. In Harmonic analysis and applications, 139–169, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2006 29. Gröchenig, K.: Time-frequency analysis of Sjöstrand’s class. Rev. Mat. Iberoam. 22 (2006), no. 2, 703–724 30. Gröchenig, K., and Rzeszotnik, Z.: Banach algebras of pseudodifferential operators and their almost diagonalization. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2279–2314 31. Gröchenig, K., and Strohmer, T.: Pseudodifferential operators on locally compact abelian groups and Sjöstrand’s symbol class. J. Reine Angew. Math. 613 (2007), 121–146 32. Hlawatsch, F., and Auger, F. (Eds.).: Time-frequency Analysis. John Wiley & Sons (2013) 33. Hlawatsch, F., and Boudreaux-Bartels, G. F.: Linear and quadratic time-frequency signal representations. IEEE Signal Proc. Mag. 9 (1992), no. 2, 21–67 34. Hudson, R. L.: When is the Wigner quasi-probability density non-negative? Rep. Mathematical Phys. 6 (1974), no. 2, 249–252 35. Jakobsen, M. S.: On a (no longer) new Segal algebra: a review of the Feichtinger algebra. J. Fourier Anal. Appl. 24 (2018), no. 6, 1579–1660 36. Janssen, A. J. E. M.:A note on Hudson’s theorem about functions with nonnegative Wigner distributions. SIAM J. Math. Anal. 15 (1984), no. 1, 170–176 37. A. J. E. M. Janssen: Bilinear time-frequency distributions. In Wavelets and their applications (Il Ciocco, 1992), 297–311, Kluwer Acad. Publ., Dordrecht, 1994 38. Janssen, A. J. E. M.: Positivity and spread of bilinear time-frequency distributions. In The Wigner distribution, 1–58, Elsevier Sci. B. V., Amsterdam, 1997 39. Lu, T., and Shiou, S.: Inverses of 2 × 2 block matrices. Comput. Math. Appl. 43 (2002), no. 1–2, 119–129 40. W. Mecklenbräuker and F. Hlawatsch, editors. The Wigner distribution. Elsevier Science B.V., Amsterdam, 1997. Theory and applications in signal processing. 41. Sjöstrand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1 (1994), no. 2, 185–192 42. Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I. J. Funct. Anal., 207 (2004), no. 2, 399–429 43. Toft, J.: Matrix parameterized pseudo-differential calculi on modulation spaces. In Generalized Functions and Fourier Analysis, 215–235, Birkhäuser, 2017 44. Wigner, E.: On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev., 40 (1932), no. 5, 749–759

About the Nuclearity of S(Mp ) and Sω Chiara Boiti, David Jornet, and Alessandro Oliaro

Dedicated to Prof. Luigi Rodino on the occasion of his 70th birthday.

Abstract We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space S(Mp ) to be nuclear. As a consequence, we obtain that for a weight function ω satisfying the mild condition: 2ω(t) ≤ ω(H t) + H for some H > 1 and for all t ≥ 0, the space Sω in the sense of Björck is also nuclear. Keywords Nuclear spaces · Weighted spaces of ultradifferentiable functions of Beurling type

1 Introduction and Preliminaries For a sequence (Mp )p∈N0 which satisfies Komatsu’s standard condition (M2) (stability under differential operators) and, moreover, the condition that there is

C. Boiti () Dipartimento di Matematica e Informatica, Università di Ferrara, Ferrara, Italy e-mail: [email protected] D. Jornet Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, Valencia, Spain e-mail: [email protected] A. Oliaro Dipartimento di Matematica, Università di Torino, Torino, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_6

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H > 0 such that for any C > 0 there is B > 0 with s s/2 Mp ≤ BC s H s+p Ms+p ,

for any s, p ∈ N0 ,

(1)

where N0 := N ∪ {0}, Langenbruch [7] proves that the Hermite functions are a Schauder basis in the spaces of ultradifferentiable functions of (Beurling type):  S(Mp ) (Rd ) := f ∈ C ∞ (Rd ) : for any j ∈ N,  sup sup |x α D β f (x)|j |α+β| /M|α+β| < +∞ .

α,β∈Nd0 x∈Rd

Moreover, in [7] it is also established an isomorphism between S(Mp ) and the Köthe sequence space:   1/2 (Mp ) := (ck )k∈N0 : for any j ∈ N0 , sup |ck |eM(j k ) < +∞ , k∈N0

where M(t) = sup log p

t p M0 , Mp

t > 0,

(2)

is the associated function of (Mp ). In this paper we use Grothendieck-Pietsch criterion to characterize when the space (Mp ) is nuclear under the assumption that (Mp /M0 )1/p is bounded below by a positive constant and, hence, M(t) is increasing and convex in log t (see [6, p. 49]). Indeed, we prove in Theorem 1 that (Mp ) is nuclear if and only if there is H > 1 such that for any t > 0 we have M(t) + log t ≤ M(H t) + H.

(3)

By [6, Prop. 3.4], condition (M2) implies (3). Therefore, conditions (M2) and (1) imply that S(Mp ) is nuclear (see Corollary 1). This should be compared with [9], where the authors prove that S(Mp ) is nuclear under Komatsu’s conditions (M1) and (M2). As a consequence of Theorem 1 we give a simple proof of the nuclearity of the space Sω in the sense of Björck [1] given in Definition 1 under the following condition of Bonet, Meise and Melikhov [5] on the weight function ω: (BMM)

∃H > 1 s.t. 2ω(t) ≤ ω(H t) + H,

t ≥ 0.

In fact, in this case the space Sω is isomorphic to the space S(Mp ) for some suitable sequence (Mp ).

About the Nuclearity of S(Mp ) and Sω

123

2 Results for the Space S(Mp ) In this section we characterize the nuclearity of (Mp ) and give sufficient conditions for the nuclearity S(Mp ) . We consider a sequence (Mp )p satisfying the condition that (Mp /M0 )1/p is bounded from below by a positive constant, so that the associated function defined by (2) is increasing and convex in log t. From Grothendieck-Pietsch criterion it is easy to obtain the following Lemma 1 The Köthe sequence space (Mp ) is nuclear if and only if for every j ∈ N there exists m ∈ N with m ≥ j such that +∞

eM(j k

1/2 )−M(mk 1/2 )

< +∞.

(4)

k=0

 

Proof It follows from Proposition 28.16 of [8]. Theorem 1 The space (Mp ) is nuclear if and only if (3) holds. Proof Let us first remark that (3) implies M(t) + 2 log t = M(t) + log t + log t ≤ M(H t) + H + log(H t) − log H ≤ M(H 2 t) + 2H − log H and, more in general, M(t) + N log t ≤ M(H N t) + CN,H ,

∀N ∈ N,

(5)

for some constant CN,H > 0 depending on N and H . Let us now assume that (3) is satisfied and prove the nuclearity of (Mp ) , using (5) for a fixed N > 2. By Lemma 1, it’s enough to prove the convergence of the series (4). Indeed, for every fixed j ∈ N, choosing m ≥ H N j , eM(j k

1/2 )−M(mk 1/2 )

≤ eM(j k

1/2 )−M(H N j k 1/2 )

≤ eM(j k

1/2 )−M(j k 1/2 )−N

= eCN,H j −N

log(j k 1/2 )+CN,H

1 k N/2

 1 and the series +∞ k=1 k N/2 converges since N > 2. Let us now assume that the series (4) converges and prove (3). To this aim, let us first remark that, for m > j , k −→ M(j k 1/2 ) − M(mk 1/2 )

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is decreasing, because M(et ) is convex by our assumptions (see [6, p. 49]), and t )−M(es ) is increasing with respect to both therefore its difference quotient M(e t−s variables t and s; this implies that

M(mk 1/2 ) − M(j k 1/2 ) =

    1/2 1/2 − M elog j k M elog mk log mk 1/2

− log j k 1/2

log

m j

is increasing with respect to k. Then the convergence of (4) implies that lim keM(j k

1/2 )−M(mk 1/2 )

k→+∞

=0

and hence sup keM(j k

1/2 )−M(mk 1/2 )

≤ A,

k∈N

for some A ∈ R+ . Then log k + M(j k 1/2 ) − M(mk 1/2 ) ≤ log A,

∀k ∈ N,

and hence M(j k 1/2 ) − M(mk 1/2 ) ≤ − log k + log A = −2 log(j k 1/2 ) + log(j 2 A) ≤ − log(j k 1/2 ) + log(j 2 A).

(6)

To prove that (6) implies (3) let us first condider t ≥ 1 and choose the smallest k ∈ N such that t ≤ j k 1/2 . Since j (k + 1)1/2 − j k 1/2 = √

j k+1+

√ < j, k

∀k ∈ N,

we have that j k 1/2 ∈ [t, (j + 1)t] and therefore, from (6), M(t) + log t ≤ M(j k 1/2 ) + log(j k 1/2 ) ≤ M(mk 1/2 ) + log(j 2 A)  m 1/2 + log(j 2 A) jk =M j  m (j + 1)t + log(j 2 A), ≤M ∀t ≥ 1, j and hence, for H = max satisfied for all t > 0.



m j (j

 + 1), log(j 2 A) + M(1) , we have that (3) is  

About the Nuclearity of S(Mp ) and Sω

125

So, we automatically obtain Corollary 1 If (Mp ) satisfies (M2) and (1), the space S(Mp ) is nuclear. Proof The spaces S(Mp ) and (Mp ) are isomorphic because (Mp ) satisfies (M2) and (1) by Theorem 3.4 of [7]. Here, without loss of generality, we restrict our proof to the 1-variable case (see [7, §4]). Since (M2) implies (3) (see for instance [6, Prop. 3.4]), the result follows from Theorem 1.   Remark 1 Looking inside the proof of Theorem 3.4 of [7] we can see that in fact Langenbruch needs only (1) and (3), so that in the above corollary we could substitute the assumption (M2) with the condition that the associated function M(t) satisfies (3). Note that (M2) implies (3) by [6, Prop. 3.4], as we already mentioned. On the other hand, if (Mp ) satisfies (M1), then (M2) is equivalent to (3). Indeed, by [6, Prop. 3.2], (Mp ) satisfies (M1) if and only if Mp = M0 sup t≥0

tp , exp M(t)

∀p ≥ 1,

and therefore, if M(t) satisfies (3), then Mp =M0 sup s≥0

(H s)p H p s p eH ≤ M0 sup exp M(H s) s≥0 s exp M(s)

s p−1 =e H M0 sup = eH H p Mp−1 , exp M(s) s≥0 H

(7)

p

i.e. (Mp ) satisfies (M2) .

3 Results for the Space Sω and Examples In this section we give a sufficient condition for the space Sω in the sense of Björck [1] to be nuclear. We consider continuous increasing weight functions ω : [0, +∞) → [0, +∞) satisfying: (α) (β) (γ ) (δ)

∃L > 0 s.t. ω(2t) ≤ L(ω(t) + 1), ∀t ≥ 0; ω(t) = o(t), as t → +∞; ∃a ∈ R, b > 0 s.t. ω(t) ≥ a + b log(1 + t), ϕ : t → ω(et ) is convex.

∀t ≥ 0;

Then we define ω(ζ ) := ω(|ζ |) for ζ ∈ Cd . We denote by ϕ ∗ the Young conjugate of ϕ, defined by ϕ ∗ (s) := sup(ts − ϕ(t)). t≥0

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We recall that ϕ ∗ is increasing and convex, ϕ ∗∗ = ϕ and ϕ ∗ (s)/s is increasing. Moreover, it will be not restrictive, in the following, to assume ω|[0,1] ≡ 0 and hence ϕ ∗ (0) = 0. The space Sω (Rd ) of weighted rapidly decreasing functions is then defined by (see [1]): Definition 1 Sω (Rd ) is the set of all u ∈ L1 (Rd ) such that u, uˆ ∈ C ∞ (Rd ) and (i) ∀λ > 0, α ∈ Nd0 : supx∈Rd eλω(x) |D α u(x)| < +∞, (ii) ∀λ > 0, α ∈ Nd0 : supξ ∈Rd eλω(ξ ) |D α u(ξ ˆ )| < +∞, where D α = (−i)|α| ∂ α . Note that

" ω0 (t) =

0≤t ≤1

0,

(8)

log t, t > 1

is a weight function for which Sω0 (Rd ) coincides with the classical Schwartz class S(Rd ). The space Sω (Rd ) is a Fréchet space with different equivalent systems of seminorms (cf. [2–4]). In particular, we shall use in what follows the family of seminorms pλ (u) = sup sup |x D u(x)|e β

α

−λϕ ∗



|α+β| λ



.

(9)

α,β∈Nd0 x∈Rd

Given a weight function ω we construct the sequence (Mp ) by Mp = e ϕ

∗ (p)

,

∀p ∈ N0 .

(10)

Then the associated function of Mp is equivalent to the given weight ω. Indeed, on one side, since M0 = 1, we have, for t > 0: M(t) = sup log p∈N0

  tp ∗ = sup log t p − log eϕ (p) Mp p∈N0

≤ sup(s log t − ϕ ∗ (s)) = ϕ(log t) = ω(t). s≥0

On the other side, for t > 0: ω(t) = sup(s log t − ϕ ∗ (s)) = sup s≥0

sup

(s log t − ϕ ∗ (s))

p∈N0 p≤s 0,

(11)

and for some A > 0. Moreover, Mp = e ϕ

∗ (p)

  t = exp{sup(pt − ω(et ))} = sup ept e−ω(e ) t≥0

t≥0

    = sup s p e−ω(s) = sup s p e−ω(s) , s≥1

(12)

s≥0

since ω|[0,1] ≡ 0. Let us remark that the sequence (Mp ) satisfies (Mp /M0 )1/p ≥ 1 and the condition of logarithmic convexity (M1)

Mp2 ≤ Mp−1 Mp+1 ,

p ∈ N,

since     2ϕ ∗ (p) = 2 sup(tp − ϕ(t)) ≤ sup t (p − 1) − ϕ(t) + sup t (p + 1) − ϕ(t) t≥0

t≥0 ∗

t≥0 ∗

= ϕ (p − 1) + ϕ (p + 1). If ω satisfies condition (BMM), then also M(t) satisfies condition (BMM) because, by (11), 1 (4ω(t)) ≤ 2 1 ≤ ω(H 2 t) + 2

2M(t) ≤

1 (2ω(H t) + 2H ) 2 3 A 3 H ≤ M(H 2 t) + + H. 2 2 2

(13)

Then, by [6, Prop. 3.6], we obtain that (Mp ) satisfies also the condition of stability under ultradifferential operators: (M2)

∃A, H > 0 s.t.

Mp ≤ AH p min Mq Mp−q . 0≤q≤p

Moreover, the sequence (Mp )p satisfies (1). Indeed, since ω(t) = o(t) as t → +∞, we have that for every ε > 0 there exists Rε > 0 such that ω(t) ≤ εt + Rε for all t ≥ 0. Therefore, for s ≥ ε,     s ϕ ∗ (s) = sup ts − ω(et ) ≥ sup ts − εet − Rε = s log − s − Rε , ε t≥0 t≥0

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and hence

 s s ε

≤ es+ϕ

∗ (s)+R

ε

,

∀s ≥ ε.

Since, for s ≤ ε we have that s s ≤ (εe)s , we finally have that for every s > 0: s s/2 Mp ≤ s s eϕ

∗ (p)

≤ eRε (εe)s eϕ

∗ (s)+ϕ ∗ (p)

≤ eRε (εe)s eϕ

∗ (p+s)

= eRε (εe)s Mp+s .

If ω satisfies (BMM), then (Mp ) coincides with the sequence space   1/2 ω := (ck )k∈N0 : sup |ck |eω(j k ) < +∞ ∀j ∈ N0 ,

(14)

k∈N0

by (11) and (13). Theorem 2 Let ω be a weight function. Then ω is nuclear if and only if ω satisfies condition (3). Proof As in Theorem 1, we use [8, Prop. 28.16] for the sequence space ω .

 

Example 1 Condition (3) for a weight function ω is weaker than (BMM). For instance " 0, 0≤t ≤1 ω(t) = 2 log t, t > 1 satisfies (3) but not (BMM). Corollary 2 Let ω be a weight function satisfying (BMM). Then ω is nuclear. Proposition 1 Let ω be a weight function satisfying (BMM) and (Mp ) the sequence defined by (10). Then Sω (Rd ) is equal (as vector space and as locally convex space) to S(Mp ) and isomorphic to (Mp ) = ω . Proof We endow Sω (Rd ) with the family of seminorms (9). It is isomorphic (and hence equal) to S(Mp ) because, by [5, formulas (5), (6)], the following two conditions hold: ∀j ∈ N ∃λ, c > 0 s.t.

∗ p eλϕ ( λ ) ≤ cj −p Mp ,

∀p ∈ N0 ,

and ∀λ > 0 ∃j ∈ N, C > 0 s.t.

∗ p j −p Mp ≤ Ceλϕ ( λ ) ,

∀p ∈ N0 .

Finally, S(Mp ) is isomorphic to (Mp ) by Theorem 3.4 of [7] (see the proof of Corollary 1), since (Mp ) satisfies (M2) (stronger than (M2) ) and (1). Moreover (Mp ) coincides with ω , as we already remarked in the comment for formula (14).   Condition (3), written in terms of the weight function ω, is equivalent to the nuclearity of ω by Theorem 2, but it is not necessary for the nuclearity of Sω .

About the Nuclearity of S(Mp ) and Sω

129

For example, the weight ω0 (t) defined by (8) does not satisfy (3) and hence ω0 is not nuclear, while S is well known to be nuclear. In particular, ω0 and S are not isomorphic. On the other hand, from the results that we have we do not know if condition (3) is sufficient for the nuclearity of Sω , but we need the stronger condition (BMM), as we state in the following Theorem 3 Let ω be a weight function satisfying (BMM). Then Sω is a nuclear space.  

Proof It follows from Proposition 1 and Corollary 2. (M2)

Example 2 There exist sequences (Mp ) satisfying (M1), but not (M2), for which the space S(Mp ) is nuclear. Let us consider, for example, a weight function ω satisfying (3) but not (BMM) (see Example 1) and construct the sequence (Mp ) as in (10). Then (Mp ) satisfies (1), (M1) and, by (12), with the same computations as in (7) we obtain that (Mp ) satisfies (M2) . Therefore the space S(Mp ) is nuclear by Corollary 1. Comparing with [9] it is then interesting the following Corollary 3 Condition (M2) is not necessary for the nuclearity of S(Mp ) . Acknowledgements We are grateful to Prof. Gerhard Schindl for pointing out that (M2) is equivalent to (3), under (M1). The authors were partially supported by the Projects FAR 2017, FAR 2018 and FIR 2018 (University of Ferrara), FFABR 2017 (MIUR). The research of the second author was partially supported by the project MTM2016-76647-P. The first and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References 1. G. Björck. Linear partial differential operators and generalized distributions. Ark. Mat. 6 (1966), no. 21, 351–407. 2. C. Boiti, D. Jornet, A. Oliaro. Real Paley-Wiener theorems in spaces of ultradifferentiable functions. J. Funct. Anal. 278 (2020), no. 4. https://doi.org/10.1016/j.jfa.2019.108348. 3. C. Boiti, D. Jornet, A. Oliaro. Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms. J. Math. Anal. Appl. 446 (2017), 920–944. 4. C. Boiti, D. Jornet, A. Oliaro. The Gabor wave front set in spaces of ultradifferentiable functions. Monatsh. Math. 188 (2019), no. 2, 199–246. 5. J. Bonet, R. Meise, S.N. Melikhov. A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 3, 425–444. 6. H. Komatsu, Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect IA Math. 20 (1973), 25–105. 7. M. Langenbruch. Hermite functions and weighted spaces of generalized functions. Manuscripta Math. 119 (2006), no.3, 269–285. 8. R. Meise, D. Vogt. Introduction to functional analysis. Clarendon Press, 1997. 9. S. Pilipovi´c, B. Prangoski, J. Vindas. On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution. J. Math. Pures Appl. 116 (2018), 174–210.

Spaces of Ultradifferentiable Functions of Multi-anisotropic Type Chikh Bouzar

Dedicated to Prof. Luigi Rodino on the occasion of his 70th birthday.

Abstract The paper deals first with the relationship between multi-anisotropic Gevrey spaces and Denjoy-Carleman spaces and then it introduces a class of ultradifferentiable functions unifying these both spaces. Keywords Denjoy-Carleman spaces · Gevrey spaces · Ultradifferentiable functions · Linear differential operators · Elliptic iterates

1 Introduction Let A(Ω) denotes the classical space of real analytic functions defined on a non empty open set Ω of the Euclidean space Rn . Different classes of ultradifferentiable functions have been considered and studied as an extension or a generalisation of the space A(Ω). The following scheme illustrates this development : E ω (Ω)

E M (Ω) /

0 A(Ω)

1 E s,λ (Ω)

2 E s,Γ (Ω)

C. Bouzar () Laboratory of Mathematical Analysis and Applications, University of Oran, Oran, Algeria © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_7

131

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C. Bouzar

where the ultradifferentiable function classes are denoted as follows : 1. 2. 3. 4.

E M (Ω) is the Denjoy-Carleman space, see [17, 18, 27]. E ω (Ω) is the Beurling space, see [1, 2, 14]. E s,Γ (Ω) is the multi-anisotropic Gevrey space, see [20, 35]. E s,λ (Ω) is the inhomogeneous Gevrey space, see [25].

Recent works addressed the issue to find the relationship between these different classes : 1. the comparison between E M (Ω) and E ω (Ω) is done in [4]. 2. the formulation of E s,Γ (Ω) as a class E s,λ (Ω) , see [16, 31]. 3. the relationship between E ω (Ω) and E s,λ (Ω) is studied in [15]. This work is aimed to study in first the relationship between the multi-anisotropic Gevrey space E s,Γ (Ω) and the Denjoy-Carleman space E M (Ω) , and then it introduces a class of ultradifferentiable functions unifying these both spaces. Remark 1 We consider in this paper only differential operators with complex constant coefficients. Many results are obtained also in the case of variable coefficients, see [5–10].

2 The Space E M (Ω) The definition and some results of ultradifferentiable functions of Denjoy-Carleman type are given in this section, see [27] and [23] for more details. A sequence of positive numbers Mp p∈Z satisfies the following conditions : + Logarithmic convexity, if Mp2 ≤ Mp−1 Mp+1 , ∀p ∈ N

(H1)

Stability under multiplication, if Mp Mq ≤ M0 Mp+q , ∀p, q ∈ Z+

(H1’)

Stability under ultradifferential operators, if ∃A > 0, ∃H > 0, Mp+q ≤ AH p+q Mp Mq , ∀p, q ∈ Z+

(H2)

Stability under differential operators, if ∃A > 0, ∃H > 0, Mp+1 ≤ AH p Mp , ∀p ∈ Z+

(H2’)

Non-quasi-analyticity, if ∞

Mp−1 < +∞ Mp

p=1

(H3’)

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133

Remark 2 We have (H1)⇒(H1’) and (H2)⇒(H2’). ultradifferential operator is a differential operator of infinite order P =  A M− aγ D γ satisfying the following : for every h > 0 there exists c > 0 such that γ

∀γ ∈ Zn+



aγ M|γ | ≤ ch|γ | We denote by E (Ω) the space of infinitely differentiable functions defined in the open set Ω. Definition 1 The space of M-ultradifferentiable functions, denoted E M (Ω) , is the set of all u ∈ E (Ω) satisfying the following : for every compact K of Ω, ∃C > 0, ∀α ∈ Zn+ , 1 α 1 1∂ u1

L∞ (K)

≤ C |α|+1 M|α|

  Example 1 If Mp p∈Z = (p!s )p∈Z+ , s > 0, we obtain the classical isotropic + Gevrey space E s (Ω) of order s. The space of real analytic functions A (Ω) corresponds to the space E 1 (Ω) , see [3]. The basic properties of the space E M (Ω) are summarized in the following proposition.     Proposition 1 If Mp p∈Z satisfies H 1 the space E M (Ω) is stable under +     product. Moreover if Mp p∈Z satisfies H 2 then E M (Ω) is stable by any linear +   differential operator of finite order with coefficients in E M (Ω) , and if Mp p∈Z + satisfies (H 2) then any ultradifferential operator of class M operates as a sheaf M M homomorphism.  space D (Ω)  E (Ω) ∩ D (Ω) is not trivial if and only if  :=  The  the sequence Mp p∈Z satisfies H 3 . +

Let P be a linear differential operator of order m ∈ N, i.e. P =

aα D α ,

|α|≤m

where aα ∈ C, α = (α1 , . . . .αn ) ∈ Zn+ , |α| = α1 + . . . + αn , D α = D1α1 . . . Dnαn  and Dj = i −1 ∂xj , j = 1, . . . , n. Defining the symbol of P by P (ξ ) := aα ξ α , the ellipticity of the operator P is equivalent to ∃c > 0 such that (1 +

n

m

ξj )  c(1 + |P (ξ )|), ∀ξ ∈ Rn j =1

|α|≤m

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Definition 2 The space of functions u ∈ E (Ω) such that ∀H compact of Ω, ∃C > 0, ∀k ∈ Z+ , we have 1 1 1 k 1 1P u1

L∞ (K)

≤ C k+1 Mmk

,

(1)

denoted E M (Ω, P ) , is called the space of M−ultradifferentiable vectors of the operator P . We recall a result characterising the space E M (Ω) by the space of M−ultradifferentiable vectors due firstly to H. Komatsu [24] in the general case of linear differential operators with constant coefficients and the particular Gevrey case, and then extended by J. L. Lions and E. Magenes in [26] to the general case of linear differential operators with variable coefficients and arbitrary sequence M. Theorem 1 We have P is elliptic ⇒ E M (Ω, P ) = E M (Ω)

(2)

Remark 3 The reverse implication is proved in [28] for the case of the Gevrey sequence (p!σ )p∈Z+ .

3 The Space E s,Γ (Ω) , Let q = (q1 , .., qn ) ∈ Rn+ := ξ ∈ Rn : ξj > 0, j = 1, . . . , n , α (α1 , . . . , αn ) ∈ Zn+ , and α · q := nj=1 αj qj .

=

Definition 3 Let A be a finite subset of Rn+ . The Newton polyhedron of A, denoted Γ (A), is the convex hull of {0} ∪ A. A Newton polyhedron Γ is always characterized by Γ=

  q∈A(Γ )

 α ∈ Rn+ , α · q ≤ 1 ,

where A (Γ ) is a finite subset of Rn  {0} . We say that Γ is regular if qj > 0, ∀j = 1, . . . , n, ∀q = (q1 , . . . , qn ) ∈ A (Γ ) The regularity of a Newton polyhedron means that for every point of Γ all its projections on the coordinate planes of various dimensions belong to Γ and it contains no faces parallel to the coordinate hyperplanes and not belonging to them. As a consequence, Γ has vertices on the coordinates axes.

Multi-anisotropic Ultradifferentiable Functions

135

The following elements are associated to every regular Newton polyhedron Γ ;   V (Γ ) = s 0 = 0, s 1 , . . . , s m(Γ ) the set of vertices of Γ   k (α, Γ ) = inf t > 0, t −1 α ∈ Γ the jauge of Γ μ (Γ ) = max μj (Γ ) the formal order of Γ 1≤ j ≤n

μj (Γ ) = max qj−1 q∈A(Γ )

q (Γ ) = (

μn (Γ ) μ1 (Γ ) ,..., ) μ (Γ ) μ (Γ )

Definition 4 Let Γ be a regular Newton polyhedron and s > 0. The space of u ∈ E(Ω) such that ∀K compact of Ω, ∃C > 0, ∀α ∈ Zn+ , 1 α 1 1∂ u1

L∞ (K)

 C |α|+1 |α|sμ(Γ )k(α,Γ ) ,

(3)

denoted E s,Γ (Ω), is called the multi-anisotropic Gevrey space of order s. Remark 4 The value |α|sμ(Γ )k(α,Γ ) can be replaced by k(α, Γ )sμ(Γ )k(α,Γ ) . The multi-anisotropic Gevrey spaces E s,Γ (Ω), introduced explicitly in the mathematical literature in the paper [35] of L. Zanghirati to study the multi-anistropic Gevrey regularity of multi-quasi-elliptic linear partial differential operators by the method of elliptic iterates, were well-known in first as the anisotropic Gevrey spaces E s,l (Ω) in the study of parabolic and quasi-elliptic linear partial differential operators, see [20] and [33]. Example 2 Let l = (l1 , . . . , ln ) ∈ Rn+ with minlj = 1. The anisotropic Gevrey j

space is defined classically as  E

s,l

(Ω) :=

u ∈ E (Ω) : ∀K compact of Ω, ∃C > 0, ∀α ∈ Zn+ , ∂ α u L∞ (K) ≤ C |α|+1 α1 !sl1 . . . αn !sln



We have E s,l (Ω) = E s,Iq (Ω), where Iq is the regular Newton polyhedron defined by the simplex   Iq = α ∈ Rn+ : α · q ≤ 1 , 1 1 m ,..., ), mj = , and m = maxlj . The simplex Iq has its set of m1   mn, lj - j , vertices V Iq = 0, mj ej , j = 1, .., n , where ej , j = 1, . . . , n represents  the canonical basis of Rn , in this case we have k(α, Iq ) = α · q and μ Iq = m. q = (

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If l1 = . . . = ln = 1 we obtain Gevrey space E s (Ω) , i.e.  the classical isotropic  E s (Ω) = E s,I (Ω), where I : = α ∈ Rn+ : |α| ≤ 1 is the unitary simplex. Let’s give the following result. Proposition 2 Let Γ be a regular Newton polyhedron, then ∀h > 0, E s,hΓ (Ω) = E s,Γ (Ω)

(4)

Proof A direct computation shows that we have μ (hΓ ) = hμ (Γ ) and k (α, hΓ ) = 1   h k (α, Γ ) , so μ (hΓ ) k (α, hΓ ) = μ (Γ ) k (α, Γ ) which gives the result. Remark 5 Due to this Proposition, the study of the space E s,Γ (Ω) with a regular Newton polyhedron Γ having vertices in Qn+ is reduced to the case Γ with vertices in Zn+ . Indeed, let h ∈ N be such that the regular Newton polyhedron hΓ has vertices in Zn+ , then E s,Γ (Ω) = E s,hΓ (Ω) . The so-called multi-quasi-elliptic linear differential operators appeared in the paper [30] of S. N. Nikolsky, their characterization was done by V. P. Mikhaïlov in [29], then J. Friberg in [19] and S. G. Gindikin and L. R. Volevich in [34] extended their study. Since then they became a subject of study in the mathematical literature, the book [21] of S. G. Gindikin and L. R. Volevich gives the state of the art of a general theory developed from multi-quasi-elliptic linear differential operators, see also the recent book of L. Rodino and F. Nicola [32]. Definition 5 The Newton polyhedron of a linear differential operator P , denoted Γ (P ), is the convex hull of the set , {0} ∪ α ∈ Zn+ : aα = 0 In the same way the elements A (P ) , V (P ) , μj (P ) , μ (P ) , k (α, P ) are associated to the operator P and also the weight function |ξ |P :=

|ξ |α , ξ ∈ Rn ,

α∈V (P )

where |ξ |α = |ξ1 |α1 . . . |ξn |αn . Definition 6 An operator P is said multi-quasi-elliptic, if i) its Newton’s polyhedron Γ (P ) is regular. ii) ∃c > 0 such that ∀ξ ∈ Rn we have |ξ |P  c(1 + |P (ξ )|) An important property of multi-quasi-elliptic operators is given by the following result, see [19] and [34].

Multi-anisotropic Ultradifferentiable Functions

137

Proposition 3 Every multi-quasi-elliptic operator P is hypoelliptic. Remark 6 The converse is not true, see [12]. Remark 7 A well-known result of L. Hörmander, see [22], says that every hypoelliptic linear differential operator P is characterised by an anisotropic Gevrey regularity. It is proved in [12] and [13] that for every such an operator there exists a Newton polyhedron H ⊂ Γ (P ) and σ > 0 such that every solution u of P u = 0 satisfies u ∈ E σ,H (Ω) , and this result is optimal and more precise. Definition 7 Let s > 0, the space of u ∈ E (Ω) such that ∀H compact of Ω, ∃C > 0, ∀k ∈ Z+ , 1 1 1 k 1 1P u1

L∞ (H )

≤ C k+1 k sμ(P )k ,

(5)

denoted E s (Ω, P ) , is called the space of multi-anisotropic Gevrey vectors of the operator P . The following result is first due to L. Zanghirati in [35] and completed in [9]. Theorem 2 We have P is multi-quasi-elliptic ⇔ E s (Ω, P ) = E s,Γ (P ) (Ω)

(6)

We give an interesting result on the spaces E s (Ω, P ) . Proposition 4 Let P be a multi-quasi-elliptic operator, then ∀k ∈ N,   E s Ω, P k = E s (Ω, P ) Proof It is known that Γ (P k ) = kΓ (P ) see [21]. Applying Theorem 2 and Proposition 2 we obtain   k E s Ω, P k = E s,Γ (P ) (Ω) = E s,kΓ (P ) (Ω) = E s,Γ (P ) (Ω) = E s (Ω, P )  

4 The Space of Multi-Anisotropic Gevrey Vectors A generalization of classical Gevrey spaces is the space of Gevrey vectors of systems of linear differential operators, as they are defined by replacing in the definition of the Gevrey space E s (Ω) the elementary system of the first partial derivatives D1 , . . . , Dn by a general system of differential operators P1 , . . . , PL .

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Let Pj , j = 1, .., L, be linear differential operators, i.e.

Pj =

aj α D α

f inite

and P : = (P1 , P2 , .., PL ) denotes the associated system. Definition 8 The Newton polyhedron of the system P, denoted by Γ (P), is the convex hull of the set , {0} ∪ α ∈ Zn+ : ∃j ∈ {1, .., L} , aj α = 0 We denote by A (P) , V (P) , μj (P) , μ (P) , k (α, P) and the weight function |ξ |P the elements associated to the system P.  L Definition 9 We say that the system P = Pj j =1 is multi-quasi-elliptic, if i) its Newton polyhedron P is regular. ii) it satisfies the estimate |ξ |P ≤ c(1 +

L



Pj (ξ ) ), ∀ξ ∈ Rn ,

(7)

j =1

for a positive constant c independent of ξ. Example 3 A linear differential operators P =



aα D α of order m ∈ N, where

α·l≤m

l = (l1 , . . . , ln ) ∈ Qn+ with minlj = 1, and all lj ≥ 1, is said l-quasi-elliptic if it j

satisfies the following estimate for a positive constant c independent of ξ, (1 +

n

|ξ |mj ) ≤ c(1 + |P (ξ )|), ∀ξ ∈ Rn ,

j =1

m ∈ N. The l-quasi-ellipticity gives the multi-quasi-ellipticity, and in lj , this case Γ (P ) has its vertices 0, mj ej , j = 1, .., n . It follows from the l-quasim m ellipticity of P that l = ( , . . . , ) and m = maxmj . j m1 mn where mj =

Remark 8 A general definition of multi-quasi-ellipticity for systems in the sense of Douglis-Nirenberg is studied in [11]. Definition 10 A system of linear differential operators is said regular if its Newton polyhedron is regular.

Multi-anisotropic Ultradifferentiable Functions

139

Let β = (β1 , β2 , .., βL ) ∈ ZL + and define β

β

β

Pβ := P1 1 ◦ P2 2 ◦ · · ◦PL L Definition 11 The space of functions u ∈ E (Ω) satisfying ∀H compact of L, Ω, ∃C > 0, ∀β ∈ Z+ 1 β 1 1P u1 ∞ ≤ C |β|+1 |β|sμ(P)|β| , L (H )

(8)

 L denoted E s (Ω, Pj j =1 ), is called the space of multi-anisotropic Gevrey vectors of  L the system Pj j =1 .  n ∂ Example 4 The space E s (Ω, ∂xjj ) coincides with E s (Ω) the classical j =1

isotropic Gevrey space of order s.

Example 5 If P1 (D) = · · · = PL (D) = P (D) we obtain E s (Ω, P ) . The extension of the classical problem of elliptic iterates to systems is done in [9] where the following general result is obtained generalizing Theorem 2. Theorem 3 We have  L  L Pj j =1 is multi-quasi-elliptic ⇔ E s (Ω, Pj j =1 ) = E s,Γ (P) (Ω)

(9)

5 The Characterization of the Space E s,Γ (Ω) The study of the space of multi-anisotropic Gevrey vectors of systems of linear differential operators reveals a characterization of multi-anisotropic Gevrey spaces.  σ Lemma 1 Let vj j =1 ⊂ Nn be the vertices of a regular Newton polyhedron Γ and  σ h ∈ N, then the system D hvj j =1 is multi-quasi-elliptic. σ  Proof It is easy to see that the multi-quasi-ellipticity of the system D hvj j =1 is equivalent to the existence of c > 0 such that ∀ξ ∈ Rn , the following estimate holds (1 +

σ

j =1

|ξ |

hvj

σ



Pj (ξ ) ), ) ≤ c(1 + j =1

where Pj (ξ ) = ξ hvj , which is obviously satisfied.

 

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We give the main result of this section.  σ Proposition 5 Let vj j =1 ⊂ Nn be the set of vertices of a regular Newton polyhedron Γ, h ∈ N and s ≥ 1 , then the following statements are equivalent for every u ∈ E (Ω) : i) u ∈ E s,Γ (Ω) . ii) ∀H compact of Ω, ∃C > 0, ∀k = (k1 , . . . , kσ ) ∈ Zσ+ , 1 1 1 k1 hv1 1 . . . ∂ kσ hvσ u1 1∂

L∞ (H )

≤ C |k|+1 |k|shμ(Γ )|k|

.

(10)

iii) ∀H compact of Ω, ∃C > 0, ∀k = (k1 , . . . , kσ ) ∈ Zσ+ , 1 1 1 k1 v1 1 . . . ∂ kσ vσ u1 1∂

L∞ (H )

≤ C |k|+1 |k|sμ(Γ )|k|

.

(11)

σ  Proof It is clear that hvj j =1 ⊂ Nn are the vertices of the regular Newton polyhe-

dron hΓ. We associate to hΓ the system of linear differential operators (D hvj )σj=1 which ismulti-quasi-elliptic by Lemma 1, then by the result of Theorem 3 we  σ s hv j have E Ω, (D )j =1 = E s,hΓ (Ω) and due to Proposition 2 E s,hΓ (Ω) =   E s,Γ (Ω) = E s Ω, (D vj )σj=1 , so i)⇒ii)⇒iii) are proved. Then, from the multiquasi-ellipticity of the system (D vj )σj=1 due to Lemma 1, by Theorem 3 we have u ∈ E s,Γ (Ω) , i.e. iii)⇒i).   As a consequence we obtain a characterisation of anisotropic Gevrey spaces. Corollary 1 Let l = (l1 , .., ln ) ∈ Qn+ , then the anisotropic Gevrey space E s,l (Ω) coincides with the space of functions u ∈ E (Ω) such that ∀H compact of Ω, ∃C > 0, ∀k ∈ Zn+ , we have 1 1 1 1 k1 v1 1∂1 . . . ∂nkn vn u1

L∞ (H )

≤ C |k|+1 |k|smh|k| ,

where m = max lj and h ∈ N is such that vj := h

m ∈ N, j = 1, . . . , n. lj

Example 6 A function u ∈ E s,(2,1) (Ω) if and only if ∀H compact of Ω, ∃C > 0, ∀k = (k1 , k2 ) ∈ Z2+ , 1 1 1 k1 2k2 1 1∂t ∂x u1

L∞ (H )

≤ C |k|+1 |2k|!s

We obtain a classical result on the regularity of solutions of the heat equation. 

Corollary 2 let u ∈ D (Ω) be a distributional solution of the heat equation ∂t u − ∂x2 u = 0, (t, x) ∈ R2 . Then u ∈ E 1,(2,1) (Ω).

Multi-anisotropic Ultradifferentiable Functions

141 

Proof As the heat operator is hypoelliptic, if u ∈ D (Ω) satisfies the equation ∂t u − ∂x2 u = 0 then u ∈ E(Ω). We have ∂tk1 ∂x2k2 u = ∂x2(k1 +k2 ) u, ∀(k1 , k2 ) ∈ Z2+ , by a classical result the function u is real analytic with respect to the variable x, consequently ∀H compact of Ω, ∃C > 0, ∀(k1 , k2 ) ∈ Z2+ , 1 1 1 k1 2k2 1 1∂t ∂x u1

L∞ (H )

≤ C |k|+1 |2k|!,

the Corollary 1 gives that u ∈ E 1,(2,1) (Ω).

 

6 The Space E M,Γ (Ω)  σ Let vj j =1 ⊂ Nn be the set of vertices of a regular Newton polyhedron Γ, the differential operator ∂Γ := ∂ v1 · · · ∂ vσ is said the Γ -derivation. A Γ -derivation of order k = (k1 , . . . , kσ ) ∈ Zσ+ is defined by ∂Γk := ∂ k1 v1 · · · ∂ kσ vσ  ∞ Definition 12 Let M = Mj j =0 be a sequence of real positive numbers and Γ a regular Newton polyhedron. We define the space of multi-anisotropic ultradifferentiable functions, denoted E M,Γ (Ω) , as the space of functions u ∈ E (Ω) such that ∀H compact of Ω, ∃C > 0, ∀k ∈ Zσ+ , 1 1 1 k 1 1∂Γ u1

L∞ (H )

≤ C |k|+1 M|k|

(12)

The spaces E M,Γ (Ω) gives as examples the classical Gevrey spaces. Example 7 By the result of the last section we have that the multi-anisotropic s,Γ M,Γ (Ω) with M equals the Gevrey Gevrey space  sμ(ΓE)j ∞(Ω) corresponds to E , i.e. sequence j j =0 " E

s,Γ

(Ω) =

u ∈ E (Ω) Ω, ∃C > 0, ∀k ∈ Zσ+ , 1 compact of 1 k: ∀H |k|+1 1∂ u1 ∞ |k|sμ(Γ )|k| . ≤C Γ L (H )

2 (13)

s M,Γ (Ω) with M the Gevrey Example 8 The ∞ classical Gevrey space E (Ω) is E sj sequence j j =0 and Γ the identity simplex I.

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Example 9 The classical Denjoy-Carleman space E M (Ω) corresponds to the space E M,Γ (Ω) with Γ the identity simplex I. Definition 13 A linear Γ -differential operator P of order m ∈ N is defined by P =

ak ∂Γk ,

|k|≤m

where ak ∈ C.

  We will always suppose that the sequence Mp p∈Z satisfies the conditions + (H 1), (H 2) and (H 3 ).

Proposition 6 The space E M,Γ (Ω) is a vector space stable under action of linear Γ -differential operators of finite order. Proof That E M,Γ (Ω) is vector space is obvious. For the stability under linear Γ differential operators it suffices to show that ∂Γ u ∈ E M,Γ (Ω) if u ∈ E M,Γ (Ω) . Indeed, we have the following : for every H compact of Ω, ∃C > 0, ∀k ∈ Zσ+ , 1 1 1 k 1 1∂Γ ∂Γ u1

L∞ (H )

≤ C |k+σ |+1 M|k+σ | ≤ AH |k+σ | M|σ | C |k+σ |+1 M|k| ≤ B |k|+1 M|k| ,

for a constant B > 0 independent of k ∈ Zσ+ . A Γ -differential operator of infinite order P =

 γ

  γ aγ ∂Γ

is said a M-Γ -

ultradifferential operator if for every h > 0 there exists c > 0 such that ∀γ ∈ Zσ+ we have



aγ M|γ | ≤ ch|γ | Proposition 7 The space E M,Γ (Ω) is stable under the action of M-Γ ultradifferential operators. Proof Let u ∈ E M,Γ (Ω) then ∀H compact of Ω, ∃C > 0, ∀γ ∈ Zσ+ , ∃c > 0, ∀k ∈ Zσ+ , we have 1 1 1 k γ 1 1∂Γ (aγ ∂Γ u)1

L∞ (H )



≤ C |γ +k|+1 aγ M|γ +k|



≤ AH |k| C |γ +k|+1 M|k| aγ H |γ | M|γ | ≤ AH |k| C |k|+1 M|k| c(CH h)|γ |

Choose h > 0 such that

 γ

(CH h)|γ | converges we then obtain the result.

 

Multi-anisotropic Ultradifferentiable Functions

143

The following diagram summarizes the development of Gevrey spaces, E M,I (Ω) 0

2

A(Ω) → E s,I (Ω)

E M,Γ (Ω) 2

0 E s,Γ (Ω)

Remark 9 The subject of a forthcoming work is a thorough study of the space E M,Γ (Ω), as well as an application tackling the following general result on elliptic  L iterates : if P = Pj j =1 is a regular system of linear differential operator, then we have      L L Pj j =1 is multi-quasi-elliptic ⇔ E M Ω, Pj j =1 = E M,Γ (P) (Ω) , where     L E M Ω, Pj j =1 :=

"

u ∈ E (Ω) :1∀H compact of Ω, ∃C > 0, 1 L , 1Pβ u1 ∀β ∈ Z+ ≤ C |k|+1 M|β| . L∞ (H )

2

Such a result contains all the above cited results on elliptic iterates. Acknowledgements The author thanks the anonymous referee whose remarks helped to improve the quality of the text.

References 1. A. Beurling. Quasi-analyticity and general distributions. Lecture 4 and 5, AMS Summer Institute, Standford, (1961). 2. G. Björck. Linear partial differential operators and generalized distributions. Ark. Mat., Vol. 6, p. 351–407, (1966). 3. R. P. Boas. A theorem on analytic functions of a real variable. Bull. A. M. S., Vol. 41, p. 233–236, (1935). 4. J. Bonet, R. Meise, S. N. Melikhov. A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin, Vol. 14–3, p. 425–444, (2007). 5. C. Bouzar, R. Chaïli. Vecteurs Gevrey de systèmes quasielliptiques. Annales de Mathématiques de l’Université de Sidi Bel Abbés, T. 5, p. 33–43, (1998). 6. C. Bouzar, R. Chaïli. Vecteurs Gevrey d’opérateurs différentiels quasi-homogènes. Bull. of the Belgian Math. Society, Vol. 9, NÂ◦ 2, p. 299–310, (2002). 7. C. Bouzar, R. Chaïli. Régularité des vecteurs de Beurling de systèmes elliptiques. Maghreb Math. Rev., T. 9, NÂ◦ 1–2, p. 43–53, (2000). 8. C. Bouzar, R. Chaïli. Une généralisation de la propriété des itérés. Arch. Math., Vol. 76, NÂ◦ 1, p. 57–66, (2001).

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9. C. Bouzar, R. Chaïli. Gevrey vectors of multi-quasi-elliptic systems. Proc. Amer. Math. Soc., Vol. 131–5, p. 1565–1572, (2003). 10. C. Bouzar, R. Chaïli. Iterates of differential operators. Progress in Analysis, Vol. I, (Berlin 2001), p. 135–141, World Sci. Publ., (2003). 11. C. Bouzar, L. R. Volevich. Hypoelliptic systems connected with Newton’s polyhedron. Math. Nachr., Vol. 273, p. 14–27, (2004). 12. C. Bouzar, A. Dali. Multi-anisotropic Gevrey regularity of hypoelliptic operators. Operator Theory : Advances and Applications, Vol. 189, p. 265–273, (2008). 13. C. Bouzar, A. Dali. The Gevrey regularity of multi-quasielliptic operators. Annali dell Universita di Ferrara, Vol. 57, p. 201–209, (2011). 14. R. W. Braun, R. Meise, B. A. Taylor. Ultradifferentiable functions and Fourier analysis. Results Math., T. 17, N 3–4, p. 206–237, (1990). 15. D. Calvo, M. C. Gomez-Collado. On some generalizations of Gevrey classes. Math. Nach., Vol. 284, NÂ◦ 7, p. 856–874, (2011). 16. D. Calvo, A. Morando, L. Rodino. Inhomogeneous Gevrey classes and ultradistributions. J. Math. Anal. Appl., 297, p. 720–739, (2004). 17. T. Carleman. Les fonctions quasi-analytiques. Gauthier-Villars, (1926). 18. A. Denjoy. Sur les fonctions quasi-analytiques de variable réelle. C. R. Acad. Sci. Paris, T. 173, p. 1320–1322, (1921). 19. J. Friberg. Multi-quasielliptic polynomials. Ann. Sc. Norm. Sup. Pisa, Cl. di Sc., Vol. 21, p. 239–260, (1967). 20. M. Gevrey. Sur la nature analytique des solutions des équations aux dérivées partielles. Ann. Ec. Norm. Sup. Paris, T. 35, p. 129–190, (1918). 21. S. G. Gindikin, L. R. Volevich. The method of Newton polyhedron in the theory of partial differential equations. Kluwer, (1992). 22. L. Hörmander, Linear partial differential operators. Springer, (1963). 23. L. Hörmander. Distribution theory and Fourier analysis. Springer, (2000). 24. H. Komatsu. A characterization of real analytic functions. Proc. Japan Acad., Vol. 36, p. 90–93, (1960). 25. O. Liess, L. Rodino. Inhomogeneous Gevrey classes and related pseudodifferential operators. Suppl. Boll. Un. Mat. It., Vol. 3, 1-C, p. 233–323, (1984). 26. J. L. Lions, E. Magenes. Non homogenous boundary value problems and applications, Vol. 3. Springer, (1973). 27. S. Mandelbrojt. Séries adhérentes, régularisations des suites, Applications. Gauthier-Villars, (1952). 28. G. Métivier. Propriété des itérés et ellipticité. Comm. P.D.E., Vol. 3, N◦ 9, p. 827–876, (1978). 29. V. P. Mikhailov. On the behavior at infinity of a class of polynomials. Trudy Math. Inst. Steklov, T. 91, p. 59–81, (1967). 30. S. N. Nikolsky. The first boundary-value problem for a general linear equation. Dokl. Akad. Nauk SSSR, T. 146, p. 767–769, (1962). 31. L. Rodino, Linear partial differential operators in Gevrey spaces. World Scientific, (1993). 32. L. Rodino, F. Nicola. Global pseudo-differential calculus on euclidian spaces. Springer, (2010). 33. L. R. Volevich. Local properties of solutions of quasi-elliptic systems. Math. Sbornik (N. S.), T. 59 (101), p. 3–52, (1962). 34. L. R. Volevich, S. Gindikin, On a class of hypoelliptic operators. Mat. Sb., 75 (117) (3), p. 400–416, (1968). 35. L. Zanghirati. Iterati di una classe di operatori ipoellittici e classi generalizzate di Gevrey. Boll. U.M.I., Vol. 1, suppl., p. 177–195, (1980).

Comparison Principle for Non-cooperative Elliptic Systems and Applications Georgi Boyadzhiev and Nikolay Kutev

Abstract In this paper are given some sufficient conditions for validity of the comparison principle for linear and quasi-linear non-cooperative elliptic systems. Existence of classical solutions is proved as well. Keywords Comparison principle · Elliptic systems · Non-cooperative

1 Introduction The main object of interest in this paper is the comparison principle (CP) for noncooperative elliptic systems. General definition of comparison principle follows: Definition 1 Let O˜ be an operator in some bounded domain D and let u and u ˜ ≤ Ou ˜ in D. Comparison principle holds for O˜ if u ≤ u satisfy the inequality Ou on ∂D yields u ≤ u in D Let Ω ∈ R n be a bounded domain with smooth boundary ∂Ω. In this paper are considered weakly coupled linear elliptic systems in Ω of the form LM u = f (x) ,

(1)

u(x) = g(x) ,

(2)

and boundary data on ∂Ω

G. Boyadzhiev () · N. Kutev Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_8

145

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G. Boyadzhiev and N. Kutev

where LM = L + M, L is a matrix operator with null off-diagonal elements L = diag (L1 , L2 , . . . LN ), and matrix M = {mki (x)}N k,i=1 . Without loss of generality we assume mkk ≡ 0, because they can be included in Lk uk . Scalar operators Lk uk = −

n

n   Dj aijk (x)Di uk + bik (x)Di uk + ck (x)uk ,

i,j =1

i=1

are supposed uniformly elliptic ones in Ω for k = 1, 2, . . . N, i.e. there are constants λ, Λ > 0 such that λ |ξ |2 ≤

n

aijk (x)ξi ξj ≤ Λ |ξ |2 ,

i,j =1

for every k and any ξ = (ξ1 , . . . ξn ) ∈ R n . The right-hand side f (x) is supposed a bounded vector-function, that is |f l (x)| ≤ C ,

(3)

in Ω for every l=1,. . . N, where C is a positive constant. Coefficients ck and mik in (1) are supposed continuous ones in Ω, aijk (x) ∈ C 1 (Ω) and

k ∂aij ∂xj

, bik (x) are Holder continuous in Ω with Holder constant α < 1.

Hereafter by f − (x) = min(f (x), 0) and f + (x) = max(f (x), 0) are denoted the non-negative and, respectively, the non-positive part of the function f. The same convention is valid for matrices as well. We recall that system (1) is called cooperative one if mj k ≤ 0 for j = k, and competitive one if mj k ≥ 0 for j = k. Analogously we call L+ M u the cooperative part of the operator, and L− u the competitive one. M The theory of the positive operators in positive cone, which proved to be so useful in the cooperative case, is not applicable for non-cooperative elliptic system. So far there are no general criteria for validity of CP for non-cooperative elliptic operators. Some results are obtained by G.Sweers [14] and G.Caristi, E.Mitidieri [5] for noncooperative elliptic systems, obtained by small perturbations of cooperative ones. In this case is used the theory of the positive operators in positive cone and the fact that small perturbations preserve some key features of the operator, see [12]. In this paper is presented a different, spectral approach. Analogously to the scalar case, we are interested in the relation between the sign of the first eigenfunction and validity of the CP. Unfortunately, not any non-cooperative system has first eigenfunction, an example is given by Hess in [9]). Furthermore, the spectrum of non-cooperative systems is still not studied well. On the other hand, the spectral properties of cooperative systems are well studied (see [15]. Theorem 3.1). The main

Comparison Principle for Non-cooperative Elliptic Systems

147

idea in Theorem 1 below is to divide the non-cooperative system into two parts— cooperative and competitive one. By means of the well known spectral properties of the cooperative part we obtain conditions for validity of CP for the original noncooperative system. The full proof is given in [1]. In [8] are considered existence and local stability of the positive solutions of systems with Lk = −dk Δ, linear cooperative and non-linear competitive part, and Neumann boundary conditions. Theorem 3.4 in [8] is similar to Theorem 1 for Lk = −dk Δ. One of the basic applications of CP is the method of sub- and super-solutions. It is applied for cooperative systems in general case in [2], and many authors build solutions for particular systems, for instance in [13]. In non-cooperative case existence theorem is proved firstly for competitive systems (Theorem 3 below) and then for general non-cooperative ones (Theorem 4 below). In Theorem 3 and Theorem 4 is given a new, improved proof of existence theorem in [3].

2 Comparison Principle for Non-Cooperative Elliptic Systems Comparison principle for cooperative elliptic systems is well-studied. Since the cooperative elliptic system is a positive operator, a key instrument in this case is the theory of the positive operators in the positive cone. This approach is not applicable for the non-cooperative systems. The main idea of the following theorem is to find a relation between the first eigenvalue of the cooperative part of the system and the non-cooperative part, so we derive some conditions for validity of CP. Let us denote by L∗M the L2 -adjoint operator of LM . In the formulation of Theorem 1 below is used a definition for “irreducible” matrix. Irreducible one is a matrix that can not be decomposed to matrices of lower rank, and respectively, the reducible matrix can be decomposed. Theorem 1 Let (1), (2) be a weakly coupled uniformly elliptic system and the cooperative part of the operator L∗M − is irreducible . Then the comparison principle holds for the classical solutions of system (1), (2) if there is x0 ∈ Ω such that λ+

N

m+ kj (x0 ) > 0 ,

(4)

k=1

for j=1. . . N and λ + m+ jj (x) ≥ 0 ,

(5)

for every x ∈ Ω and j = 1 . . . N, where λ is the principal eigenvalue of the operator LM − in Ω.

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If the the cooperative part of L∗M − is reducible one can refer to Theorems 4 and 5 in [1]. This result can be transferred to quasi-linear systems. Let us consider the problem for weakly coupled elliptic systems of the type − diva l (x, ul , Dul ) + F l (x, u1 , . . . uN , Dul ) = f l (x) ,

(6)

in Ω with boundary conditions (2), D = (D1 , . . . Dn ), Di = ∂x∂ i . For convenience we assume a l (x, ul , 0) = 0, F l (x, 0, 0) = 0, l = 1, . . . N for every x ∈ Ω, ul ∈ R. li ∂a li ∂F l The coefficients of system (6), (2) are smooth enough in Ω, i.e. ∂a ∂pj , ∂ul , ∂ui , ∂F l ∂pi

∈ L1 (Ω), i, j = 1, . . . n, l = 1, . . . N. As for the right hand side of (1), we suppose f l (x) ∈ L2 (Ω).  N Let u(x), u(x) ∈ C 2 (Ω) ∩ C(Ω) be sub- and super-solution of (6), (2). Then w 3(x) = u(x) − u(x) is weak sub-solution of the following problem −

n

N n  

Di Bjli Dj w 3l + B0li w 3l + Ekl w 3k + Hil Di w 3l = 0

i,j =1

k=1

i=1

in Ω with non-positive boundary data on ∂Ω, i.e. for every test-function ηl ∈ H01 (Ω) and l = 1, . . . N holds ⎛ ⎞  n  N n 

⎝ Bjli Dj w 3l + B0li w 3l ηxl i + Ekl w 3k ηl + Hil Di w 3l ηl ⎠ dx ≤ 0 Ω

i,j =1

k=1

i=1

Here  Bjli

=

∂a li (x, P l )ds, ∂pj

1

∂a li (x, P l )ds, ∂ul

0

 B0li =

1

0

  P l = v l + s(ul − v l ), Dv l + sD(ul − v l )  Ekl =

∂F l (x, S l )ds, ∂uk

1

∂F l (x, S l )ds, ∂pi

0

 Hil =

1

0

  S l = v + s(u − v), Dv l + sD(ul − v l ) .

Comparison Principle for Non-cooperative Elliptic Systems

149

Therefore w 3+ (x) = max (3 w (x), 0) is a weak sub-solution in Ω of −

n

i,j =1

N n  

l l k l + Di Bjli Dj w 3+ + B0li w 3+ Ekl w 3+ + Hil Di w 3+ =0, k=1

(7)

i=1

with null boundary data on ∂Ω. Equation (7) is equivalent to BE w 3+ = (B + E)3 w+ = 0 ,    l + B li w l 3+ in Ω, where B = diag(B1 , B2 , . . . BN ), Bl = − ni,j =1 Di Bjli Dj w 0 3+  l and E = {E l }N + ni=1 Hil Di w 3+ k l,k=1 . Then the following theorem (Theorem (8) in [1]) holds: Theorem 2 Let (6), (2) be quasi-linear uniformly elliptic system. Then comparison principle holds for it if either BE − is irreducible one and there is x0 ∈ Ω such that for every j = 1 . . . n hold λ+

N

∂F k k=1

∂pj

(x, p, q ) + l

N

i=1

∂a j i Di j (x, pj , q j ) ∂p

+ >0,

and for every x ∈ Ω λ+

 n

i=1

∂a j i ∂F j Di j (x, pj , q j ) + (x, p, q j ) ∂p ∂pj

+ ≥0,

where p, q ∈ R n and λ is the first eigenvalue of operator BE − in Ω; or if BE − is reducible one and there is x0 ∈ Ω such that for every j = 1 . . . n hold λj +

N

∂F k k=1

∂pj

(x, p, q ) + j

N

i=1

∂a j i Di j (x, pj , q j ) ∂p

+ >0,

and for every x ∈ Ω  λj +

n

i=1

+ ∂a j i ∂F j j j j Di j (x, p , q ) + (x, p, q ) ≥0, ∂p ∂pj

where p, q ∈ R n and λl is the first eigenvalue of operator Bl in Ω.

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3 Existence of Classical Solution for Linear Non-Cooperative Elliptic System In this section is briefly given one of the most useful applications of CP—the method of sub- and super-solutions. All proofs are given in details in [2]. Solvability of system (1), (2) could be studied using the theorem of Leray– Schauder and the classical method of continuation of solutions along a parameter, since the coefficients of (1), (2) are smooth. In order to apply the theorem of Leray–Schauder one have to find a priori estimates for maxΩ |u(x)|, maxΩ |∇u(x)|, max∂Ω |∇u(x)| and of Holder norms |u(x)|α,Ω . This approach is well described in [7, 10] for systems with the same principal symbol for all equations of the system, which is significant constrain for many applications. Transfer of the results of O.A.Ladyzhenskaya and N.Ural’tseva [10] for elliptic systems with arbitrary principal symbol is not trivial and faces a number of technical obstacles. Another approach to the solvability problem is the method of sub- and supersolutions. It is widely used for scalar equations and reads that if comparison principle holds for system (1), (2), and there are super- and sub-solutions of the same system, then exists a solution of system (1), (2). In order to use the method of sub- and super-solutions we need some additional conditions on the growth of the coefficients. Assume that for every k = 1, . . . N ⎧ ⎫ ⎞2 ⎛ ⎪ ⎪ n n ⎨ ⎬

⎝ max Dj aijk (x) + bik (x)⎠ , |ck | ≤ b , ⎪ ⎪ ⎩ i=1 j =1 ⎭ holds for x ∈ Ω, where b is a positive constant, 8 7 n n

bik (x).pi .uk + ck uk + mk,i (x).ui (x) uk ≥ c1 |u|2 − c2 , i=1

(8)

(9)

i=1

for every x ∈ Ω, l = 1, . . . N and arbitrary vectors u and p, where c1 = const > 0 and c2 = const ≥ 0,

n

n





k k k k bi (x).pi .u + c u + mk,i (x).ui (x) ≤



i=1

i=1

≤ ε(CM ) + P (p, CM )(1 + |p|2 ) ,

(10)

where P (p, CM ) → 0 for |p| → ∞ and ε(CM ) is sufficiently small and depends only on n, N, CM , λ and Λ. λ and Λ are the constants from elliptic condition and "

CM

2max|f (x)| , = max max∂Ω |u|, c1 n

9 2c2 c1 n

2 .

(11)

Comparison Principle for Non-cooperative Elliptic Systems

151

Furthermore, in the case of competitive systems we need the following additional condition:



n m+ (x)

i=1 ki

(12) max

≤K 0.  Then system (1), (2) is solvable in C 2 (Ω) C(Ω). The proof is accomplished in two steps. In the first one is proved the existence theorem for competitive elliptic systems with, roughly speaking, small w.r.t. ck coupling coefficients mki (x) for all k = 1, . . . N. In this step no CP is required. In the second step the result about competitive systems and CP are employed in order to prove existence theorem for general non-cooperative system. Suppose system (1) is competitive one, i.e. mij (x) ≥ 0. Then the following theorem holds: Theorem 3 Suppose conditions (3)–(12) hold for the competitive and uniformly elliptic system (1), (2). Then the vector function m = (CM , CM , . . . CM ) is a super - solution  of (1), (2), where CM is the constant from (11) and there exists a classical C 2 (Ω) C(Ω) solution v(x) of the problem (1), (2). Since the system (1), (2) is a linear one, we assume without loss of generality in the following proof that g(x) = 0. Proof Consider the sequence of vector-functions v0 , v1 , . . . vl , . . ., where v0 is a super-solution and vl ∈ H01 (Ω) defines vl+1 by induction as a solution of the problem with null boundary conditions Lvl+1 + σ vl+1 = f (x) − M + vl + σ vl ,

(13)

or in details −

N

k Di (aijk (x)Dj vl+1 )+

i,j =1

N

k k k bik (x)Dvl+1 + ck vl+1 + σ vl+1 =

i=1

= f k (x) −

n

i k m+ ki (x)vl + σ vl

i=1

in Ω for every k = 1, . . . N, σ is a positive constant. Constant σ is a technical one and allows application of Theorem 1 in [11] for ck + σ > 0 for all k.

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1. By Theorem 1 in [11] conditions (3) to (10) are sufficient for solvability of the corresponding PDEs, while by Theorem 4 in [11], p. 120, see also [7], conditions (14)–(16) below are derived in the whole domain Ω. Therefore for k (x) ∈ C 2 (Ω) there is constant β = β(l + 1) ∈ (0, 1) such that the solution vl+1 k C β (Ω) < c3 , vl+1

1 1 1 ∂v k 1 1 l+1 1 1 1 1 ∂xi 1

(14)

< c4 ,

(15)

< c5 (ρ) ,

(16)

C β (Ω)

for every i = 1, . . . n and k = 1, . . . N. For every compact set K0 ⊂ Ω holds 1 1 1 ∂ 2vk 1 1 l+1 1 1 1 1 ∂xi ∂xj 1

C β (K0 )

for every i, j = 1, . . . n, ρ = dist (K0 , ∂Ω), and constants c3 − c5 are independent on k. k − v k is a solution as well. Furthermore, 2. Since system (13) is linear one then vl+1 l ck + σ > 0 and by inequality (1.5), page 145 in [10] we have

n + i − vi )

(m (x) − δ σ )(v ik



i=1 l ki l−1 k max |vl+1 − vlk | ≤ max



x∈Ω x∈Ω

ck + σ k − since we consider the problem with null boundary data and then max∂Ω |vl+1 vlk | = 0. Therefore, by (12) we have

k max |vl+1 x∈Ω

− vlk |



N (m+ (x) − δ σ )

ik

i=1 ki

i ≤ max

)| ≤

. max |(vli − vl−1

i,x∈Ω x∈Ω

ck + σ



N (m+ (x))

i=1 ki

i i )| ≤ K. max |vli − vl−1 | ≤ max

max |(vli − vl−1 k

i,x∈Ω x∈Ω

i,x∈Ω c Hence the operator used for construction of sequence v0 , v1 , . . . vl , . . . is contracting one since for all k − 1, . . . N we have k i − vlk | ≤ K. max |vli − vl−1 |, max |vl+1 x∈Ω

i,Ω

(17)

3. The sequence of vector-functions {vl } (vl = (vl1 , . . . vlN ) is contracting in Ω by (17). Therefore there is a function v such that vl (x) → v(x) point-wise in Ω. Furthermore, (14) yields {vl } is uniformly bounded and equicontinuous in

Comparison Principle for Non-cooperative Elliptic Systems

153

Ω and {vl } < const, since vlk (x) is Holder continuous and therefore |vlk (x) − vlk (x0 )| ≤ c(|x − x0 |β ) for every k = 1, . . . N. By Arzela–Ascoli compactness criterion there is a sub-sequence {vlj } that converges uniformly to v ∈ C(Ω). For convenience we denote {vlj } by {vl } (or component-wise {vlkj } by {vlk } for all k). Since v ∈ C(Ω) and all functions {vlk } satisfy the null boundary conditions, then v satisfies the boundary conditions as well. The functions vl are Holder continuous with the same Holder constant, therefore v is Holder continuous as well with the same Holder constant, i.e. v ∈ C β (Ω). Since the sequence vl (x) is contracting and v(x) is continuous, then {(vl )2 } → v 2 in Ω. Then the Dominated Convergence Theorem (Theorem 5 at p.648 in [6]) yields vl → v(x) in (L2 (Ω))N . 4. Analogously to the previous step, (15) yields {Di vl } is uniformly bounded and equicontinuous in Ω and {Di vl } < const. According to Arzela–Ascoli compactness criterion there is sub-sequence {Di vlj } that converges uniformly to Di v ∈ C(Ω). For convenience we denote {vlj } by {vl } as well. 5. For every 0 < η(x) = (η1 (x), . . . ηN (x)) ∈ (H01 (Ω))N we have ⎛



⎝ Ω

N

k aijk (x)Dj vl+1 Di ηk (x) +

i,j =1

N

⎞ k k bik (x)Dvl+1 ηk (x) + (ck + σ )vl+1 ηk (x)⎠ dx =

i=1

 (f k (x) −

= Ω

n

i k k m+ ki (x)vl + σ vl )η (x)dx

i=1

holds and for l → ∞ we obtain ⎞ ⎛  N N

⎝ aijk (x)Dj v k Di ηk (x) + bik (x)Dv k ηk (x) + ck .v k ηk (x)⎠ dx = Ω

i,j =1

i=1

 (f k (x) −

= Ω

n

i k m+ ki (x)v )η (x)dx

i=1

that is v(x) is a weak solution of (1), (2). 6. Since the coefficients aijk (x) of the principal symbol in (1) are C 1+α (Ω) smooth and Dx2 v k (x) are locally bounded, then Dx2 v(x) ∈ C(Ω). In sets κr , κr ⊂ κr+1 ⊂ Ω and  fact by the exhaustion of Ω by compact κr = Ω, and by (16) we have Dx2 vl ∈ C β (Kr ) are uniformly bounded and equicontinuous in κr . Applying Arzela–Ascoli theorem and Cantor diagonal process (for sub-sequence and compact) yields C 2 smoothness in Ω of the limit v(x). Therefore v(x) ∈ C 2 (Ω))N is classical solution of (1), (2).

 

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For general non-cooperative elliptic system the following theorem holds: Theorem 4 Suppose conditions (3)–(15) hold for the uniformly elliptic system (1), (2). Then the vector function m = (−CM , −CM , . . . − CM ) is a subsolution of (1),  (2), where CM is the constant from (11) and there exists a classical C 2 (Ω) C(Ω) solution u(x) of the problem (1), (2) with null boundary data. Theorem 4 is proved by the method of sub- and super-solutions. A key-point of this method is the comparison principle. Since the system (1), (2) is a linear one, we assume in the following proof without loss of generality that g(x) = 0. Proof Let us consider the sequence of vector-functions u0 , u1 , . . . ul , . . ., where u0 = m is the subsolution and ul ∈ H01 (Ω) defines ul+1 by induction as a solution of the problem LM + u + σ u = f (x) − M − u + σ u ,

(18)

with null boundary conditions ukl+1 (x) = 0 on ∂Ω for every k = 1, . . . N, σ > 0 is a constant. The problem (18) is competitive system and by Theorem 3 it is solvable. Furthermore ul0 ≤ ul1 ≤ . . . ≤ ukl+1 ≥ . . . by the comparison principle and the fact that f k (x) −

n

i k k m− ki (x)ul + σ ul − f (x) +

i=1

=−

n

i k m− ki (x)ul−1 − σ ul−1 =

i=1 n

i i k k m− ki (x)(ul − ul−1 ) + σ (ul − ul−1 ) ≥ 0

i=1 l l l since ukl ≥ ukl−1 and −m− ki (x) ≥ 0. u0 ≤ u1 is trivial inequality since u0 is the subsolution of (1), (2). 4. The sequence of vector-functions {ul } (ul = (uk1 , . . . ukN )) is monotonously increasing and bounded from above in Ω. Therefore there is a function u such that ul (x) → u(x) point-wise in Ω. The rest of the proof follows the proof of Theorem 3.  

The complete proof of Theorem 4 is given in [4].

Comparison Principle for Non-cooperative Elliptic Systems

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References 1. G. Boyadzhiev. Comparison principle for non-cooperative elliptic systems. Nonlinear Analysis, Theory, Methods and Applications, 69, (2008), no. 11, 3838–3848. 2. G. Boyadzhiev. Existence theorem for cooperative quasi-linear elliptic systems. C.R. Acad. Bulg. Sci., 63 (2010), no 9, 665–672. 3. G. Boyadzhiev. Existence of Classical Solutions of Linear Non-cooperative Elliptic Systems. C.R. Acad. Bulg. Sci., 68 (2015), no. (2), 159–164. 4. G. Boyadzhiev and N. Kutev. Existence of classical solutions of linear non - cooperative elliptic systems. Pliska Stud. Math. 30 (2019), 45–54. 5. G. Caristi and E. Mitidieri. Further results on maximum principle for non-cooperative elliptic systems. Nonl.Anal.T.M.A., 17 (1991), 547–228. 6. L.C. Evans. (P)artial (D)ifferential (E)quations, series Graduate Studies in Mathematics, AMS, 1998. 7. D. Gilbarg and N. Trudinger. (E)lliptic (P)artial (D)ifferential (E)quations of (S)econd (O)rder. 2nd ed., Springer - Verlag, New York. 8. Li Jun Hei, Juan Hua Wu : Existence and Stability of Positive Solutions for an Elliptic Cooperative System. Acta Math. Sinica 21 (2005), No 5, 1113–1130. 9. P. Hess. On the Eigenvalue Problem for Weakly Coupled Elliptic Systems, Arch. Ration. Mech. Anal. 81 (1983), 151–159. 10. O.A. Ladyzhenskaya and N.Ural’tseva. (L)inear and (Q)uasilinear (E)quations of (E)lliptic (T)ype. Nauka, Moskwa, 1964. 11. O.A.Ladyzhenskaya, V. Rivkind and N. Ural’tseva. Classical solvability of diffraction problem for elliptic and parabolic equations with discontinuous coefficients. trudy Mat. Innst. Steklov, 92 (1996), 116–146. ( In Russian) 12. E. Mitidieri and G. Sweers. Weakly coupled elliptic systems and positivity. Math.Nachr. 173 (1995), 259–286. 13. P. Popivanov. Explicit formulaes to the solutions of Dirichet problem for equations arising in geometry and physics. C.R.Acad.Bulg. Sci., 68 (2015), no 1, 19–24. 14. G. Sweers. A strong maximum principle for a noncooperative elliptic systems. SIAM J. Math. Anal., 20 (1989), 367–371. 15. G. Sweers. Strong positivity in C(Ω) for elliptic systems. Math.Z. 209 (1992), 251–271.

On the Simple Layer Potential Ansatz for the n-Dimensional Helmholtz Equation Alberto Cialdea, Vita Leonessa, and Angelica Malaspina

To Prof. Luigi Rodino on occasion of his 70th birthday

Abstract In the present paper we consider the Dirichlet problem for the ndimensional Helmholtz equation. In particular we deal with the problem of representability of the solutions by means of simple layer potentials. The main result concerns the solvability of a boundary integral equation of the first kind. Such a result is here obtained by using the theories of differential forms and reducible operators. Keywords Helmholtz equation · Potential theory · Integral representations

1 Introduction As well known, the classical indirect method of Fredholm gives the solution of the Dirichlet problem for the n-dimensional Laplacian in terms of a double layer potential. Another indirect approach consists in looking for the solution in the form of a simple layer potential. In this case, imposing the boundary condition leads to a boundary integral equation of the first kind. Muskhelishvili [18] gave a method for solving this equation for n = 2. His approach heavily hinges on the theory of holomorphic functions of one complex variable. In [2] Muskhelishvili’s method was generalized to the case of n real variables by the first author (see also [4]). Such a generalization was obtained by replacing holomorphic functions by conjugate differential forms and applying the theory of reducible operators. Neither pseudodifferential operators nor hypersingular integrals have been used. This approach has

A. Cialdea · V. Leonessa · A. Malaspina () Department of Mathematics, Computer Science and Economics, University of Basilicata, Potenza, Italy e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_9

157

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A. Cialdea et al.

been applied to different BVPs for other PDEs in simply and multiply connected domains (see [3, 5–12, 15, 16]). The aim of the present paper is to apply this method to the Dirichlet problem for the n-dimensional Helmholtz equation. The classical reference text on integral methods for Helmholtz equation is the well known monograph [13] by Colton and Kress. In this book several questions are investigated. In particular the solutions of the Dirichlet problem are obtained in the form of double layer potentials. Their results are considered in spaces of continuous functions on a C 2 boundary, but nowadays it is not difficult to see by standard arguments that they are valid under more general assumptions. In particular they hold in Lp spaces on a Lyapunov boundary. When we consider their results, we shall always refer to them under these more general hypotheses. In [13, Section 3.5] Colton and Kress study also the integral equation of the first kind related to the rappresentability of the solution of the Dirichlet problem by means of a simple layer potential. The present paper has to be considered as an alternative approach to their one. The paper is structured as follows. Section 2 is devoted to some notations and definitions. In Sect. 3 we construct a reducing operator that we use in the study of the integral equation of the first kind arising when we impose the Dirichlet boundary condition to a simple layer potential. In Sect. 4 we find a solution of the Dirichlet problem for the Helmholtz equation in terms of a simple layer potential.

2 Definitions Throughout this paper,  is a bounded domain (open connected set) of Rn (n ≥ 3) such that its boundary is a Lyapunov hypersurface  (i.e.  has a uniformly Hölder continuous normal field of some exponent λ ∈ (0, 1]), and such that Rn \  is connected; ν(x) = (ν1 (x), . . . , νn (x)) denotes the outwards unit normal vector at the point x = (x1 , . . . , xn ) ∈ . The symbol | · | denotes the Euclidean norm for elements of Rn . Given k ∈ C \ {0} with I m(k) ≥ 0, we consider the n-dimensional Helmholtz equation u + k 2 u = 0

(1)

where u :  → C. By (·) we denote the fundamental solution of (1) (see, e.g., [1, p. 42]) i (x) = 4 (1)



k 2π |x|

(n−2)/2

(1)

H(n−2)/2 (k|x|)

where Hμ is the Hankel function of the first kind of order μ.

On the Simple Layer Potential Ansatz for the n-Dimensional Helmholtz Equation

159

For n ≥ 3 let s(x) =

1 |x|2−n (n − 2)cn

(cn =

2π n/2 ) (n/2)

be a fundamental solution of the Laplace operator −. Setting h(x) = (x) − s(x),

(2)

we have that |∇h(x)| ≤ c|x|3−n ,

∀ x ∈ Rn \ {0}

(3)

(see [19, Lemma A.5]). Then, recalling that |∇s(x)| ≤ c1 |x|1−n , thanks to (3) we get |∇(x)| ≤ c2 |x|1−n . From now on we consider p ∈ (1, +∞). By W 1,p () we denote the usual Sobolev p 1,p space. By Lh () (Wh ()) we mean the space of the differential forms of degree h ≥ 1 whose components are Lp (W 1,p ) complex-valued functions in a coordinate system of class C 1 (and then in every coordinate system of class C 1 ). We recall that if u is a h-form in , the symbol du denotes the differential of u, while ∗u denotes the star Hodge operator. Finally, we write ∗ w = w0 if w is an (n − 1)-form on  

and w = w0 dσ . If B and B  are two Banach spaces and S : B → B  is a continuous linear operator, we say that S can be reduced on the left if there exists a continuous linear operator S  : B  → B such that S  S = I + T , where I stands for the identity operator on B and T : B → B is compact. Analogously, one can define an operator S  reducible on the right. If S is a reducible operator, then it is known that its range is closed and then the equation Sα = β has a solution if and only if γ , β = 0, for any γ ∈ B ∗ such that S ∗ γ = 0, S ∗ being the adjoint of S (see, e.g., [14] or [17]).

3 Reduction of a Certain Integral Equation Let us consider the BVP ⎧ 1,λ 2 ⎪ ⎪ ⎨
∈ C () ∩ C () 
+ k 2
= 0 ⎪ ⎪ ⎩
= 0

in 

(4)

on .

Denote by V0 the space of solutions of (4). Note that, if k 2 is not a Dirichlet eigenvalue, than V0 = {0}.

160

A. Cialdea et al.

As proved in [13, Theorem 3.22], the space 

∂


:
∈ V0 ∂ν 



coincides with the kernel of the boundary integral operator I − K  , where K  : Lp () → Lp () is given by K  ψ(x) = 2

 ψ(y) 

∂ (y − x) dσy . ∂νx

(5)

Given f ∈ W 1,p (), 1 < p < +∞, such that  f 

∂
dσ = 0, ∂ν

∀
∈ V0 ,

(6)

we want to determine a solution of the Dirichlet problem "

u + k 2 u = 0 in  u=f

(7)

on 

in the form of a simple layer potential  u(x) =

ϕ(y) (y − x) dσy ,

x∈

(8)



with density ϕ ∈ Lp (). Observe that conditions (6) are necessary for the solvability of the problem (7) because of Green’s formulas. By imposing the boundary condition to (8), an integral equation of the first kind  ϕ(y)(y − x) dσy = f (x),

x∈

(9)



arises. Following [2], we take the exterior differential d of both sides of equation (9) and the singular integral equation  ϕ(y)dx [(y − x)]dσy = df (x),

x∈

(10)



comes out. Note that in (10) the unknown is a function ϕ ∈ Lp (), while the data p is a differential form of degree 1 belonging to L1 (). We are going to show that p the operator on the left-hand side of (10), acting from Lp () into L1 (), can be reduced on the left. First, we recall the next result proved in [2, p. 186].

On the Simple Layer Potential Ansatz for the n-Dimensional Helmholtz Equation

161

p

Theorem 1 Let J : Lp () −→ L1 () be the singular integral operator  ϕ(y)dx [s(y − x)]dσy ,

J ϕ(x) =

x ∈ .

 

p

Then J can be reduced on the left by the operator J : L1 () −→ Lp () defined by 



J ψ(z) = ∗

 

ψ(x) ∧ dz [sn−2 (z, x)],

z ∈ ,

(11)

where sn−2 is the Hodge double (n − 2)-form:

sn−2 (y, x) =

s(y − x)dx j1 . . . dx jn−2 dy j1 . . . dy jn−2 .

j1 1, we split the integral into two parts: ˆ ·)uˆ 1 2 2 = K(t, L (|ξ |≤1)

 

=

e−t|ξ | ω−2 sin2 (tω) dξ 2

|ξ |≤1



|ξ |≤t −1

. . . dξ +

t −1 ≤|ξ |≤1

. . . dξ,

and in the first integral we now use the estimate ω−1 sin(tω) ≤ t. Then, we derive ˆ ·)uˆ 1 2 2  t2 K(t, L (|ξ |≤1)

 |ξ |≤t −1

"

 t 2−n + that is, we proved (10).

e

−t|ξ |2

 dξ +

t −1 ≤|ξ |≤1

t

if n = 1,

log t

if n = 2,

|ξ |−2 dξ

 

Decay Estimates and Gevrey Smoothing for a Strongly Damped Plate Equation

179

References 1. H. Chen, L. Rodino, Micro-elliptic Gevrey regularity for nonlinear partial differential equations, Boll. Un. Mat. Ital. 10-B (1996), 199–232. 2. H. Chen, L. Rodino, General theory of PDE and Gevrey classes, in: “General theory of PDE and microlocal analysis”, M. Y. Qi and L. Rodino, editors, Pitman Res. Notes Math. Ser. 349 (1996), 6–81. 3. H. Chen, L. Rodino, Nonlinear microlocal analysis and applications in Gevrey classes, in: “Differential Equations, Asymptotic Analysis and Mathematical Physics”, Math. Res., Akademie Verlag 100 (1997), 47–53. 4. M. D’Abbicco, A benefit from the L1 smallness of initial data for the semilinear wave equation with structural damping, in Current Trends in Analysis and its Applications, 2015, 209–216. Proceedings of the 9th ISAAC Congress, Krakow. Eds V. Mityushev and M. Ruzhansky, http:// www.springer.com/br/book/9783319125763. 5. M. D’Abbicco, R. Chãrao, C. da Luz, Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient, Discrete and Continuous Dynamical Systems A 36 (2016) 5, 2419–2447, http://dx.doi.org/10.3934/dcds.2016.36.2419 6. M. D’Abbicco, M.R. Ebert, Diffusion phenomena for the wave equation with structural damping in the Lp − Lq framework, J. Differential Equations, 256 (2014), 2307–2336, http:// dx.doi.org/10.1016/j.jde.2014.01.002. 7. M. D’Abbicco, M.R. Ebert, An application of Lp − Lq decay estimates to the semilinear wave equation with parabolic-like structural damping, Nonlinear Analysis 99 (2014), 16–34, http:// dx.doi.org/10.1016/j.na.2013.12.021. 8. M. D’Abbicco, M.R. Ebert, A classification of structural dissipations for evolution operators, Math. Meth. Appl. Sci. 39 (2016), 2558–2582, http://dx.doi.org/10.1002/mma.3713. 9. M. D’Abbicco, M.R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Analysis 149 (2017), 1–40; http://dx.doi.org/10. 1016/j.na.2016.10.010. 10. M. D’Abbicco, G. Girardi, J. Liang, L1 − L1 estimates for the strongly damped plate equation, Journal of Mathematical Analysis and Applications 478 (2019), 476–498, https://doi.org/10. 1016/j.jmaa.2019.05.039 11. M. D’Abbicco, E. Jannelli, A damping term for higher-order hyperbolic equations, Ann. Mat. Pura ed Appl. 195, 2, 2016, 557–570, http://dx.doi.org/10.1007/s10231-015-0477-z. 12. M. D’Abbicco, E. Jannelli, Dissipative Higher Order Equations, Communications in Partial Differential Equations, 42 (2017) 11, 1682–1706, http://dx.doi.org/10.1080/03605302.2017. 1390674. 13. M. D’Abbicco, M. Reissig, Semilinear structural damped waves, Math. Methods in Appl. Sc., 37 (2014), 1570–1592, http://dx.doi.org/10.1002/mma.2913. 14. Han Yang, A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. math. 124, 5 (2000) 415–433. 15. C. Hua, L. Rodino, Paradifferential Calculus in Gevrey Classes, J. Math. Kyoto Univ. (JMKYAZ) 41 (2001), 1–31. 16. R. Ikehata, Asymptotic Profiles for Wave Equations with Strong Damping, J. Differential Equations 257 (2014), 2159–2177, http://dx.doi.org/10.1016/j.jde.2014.05.031. 17. M. Kainane, M. Reissig, Qualitative properties of solution to structurally damped σ -evolution models with time decreasing coefficient in the dissipation, Complex Analysis and Dynamical Systems VI, Contemporary Mathematics, Amer. Math. Soc., 2015, 191–218 18. M. Kainane, M. Reissig, Qualitative properties of solution to structurally damped σ -evolution models with time increasing coefficient in the dissipation, Advances in Differential Equations 20 (2015) 5–6, 433–462. 19. G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Mathematica 143 (2000), 2, 175–197.

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20. T. Krauthammer, E. Ventsel. Thin Plates and Shells Theory: Analysis, and Applications. Marcel Dekker, Inc., New York, 2001. 21. P. Marcati, K. Nishihara, The Lp -Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Eq. 191 (2003), 445–469. 22. A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS. 12 (1976), 169–189. 23. S. Matthes, M. Reissig, Qualitative properties of structurally damped wave models, Eurasian Math. J. 3 (2013) 4, 84–106. 24. T. Narazaki, Lp −Lq estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004), 586–626. 25. T. Narazaki, M. Reissig, L1 estimates for oscillating integrals related to structural damped wave models, in Studies in Phase Space Analysis with Applications to PDEs, Cicognani M, Colombini F, Del Santo D (eds), Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, 2013; 215–258. 26. K. Nishihara, Lp − Lq estimates for solutions to the damped wave equations in 3-dimensional space and their applications, Math. Z. 244 (2003), 631–649. 27. J. C. Peral, Lp estimates for the wave equation, Journal of functional analysis, 36 (1980), 114–145. 28. D.T. Pham, M. Kainane, M. Reissig, Global existence for semi-linear structurally damped σ evolution models, Journal of Mathematical Analysis and Applications 431 (2015) 1, 569–596. 29. Y. Shibata, On the Rate of Decay of Solutions to Linear Viscoelastic Equation, Math. Meth. Appl. Sci., 23 (2000), 203–226. 30. G. Todorova, B. Yordanov, Critical Exponent for a Nonlinear Wave Equation with Damping, Journal of Differential Equations 174 (2001), 464–489.

Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation Marcello D’Abbicco and Alessandra De Luca

Abstract We study the Cauchy problem for the linear generalized double dispersion equation and derive long time decay estimates for the solution in Lp spaces and in real Hardy spaces. Keywords Double dispersion equation · Decay estimates · Hardy spaces · Fourier multipliers

1 Introduction In this note, we extend the results recently obtained by the authors [2] for the Cauchy problem for the linear generalized double dispersion equation " utt − Δu − aΔutt + bΔ2 u − dΔut = 0,

t ≥ 0, x ∈ Rn ,

u(0, x) = u0 (x), ut (0, x) = u1 (x),

(1)

with a = b = d = 1, to the general case of parameters a > 0, b > 0 and d > 0. By using a Mikhlin-Hörmander type multiplier theorem, which provides Hp boundedness of parameter-dependent operators, we are able to estimate the solution in real Hardy spaces Hp with p ≤ 2 (we recall that Hp = Lp for p > 1). Our main result is the following. Theorem 1 Let n ≥ 1, p ∈ (0, 2], q0 , q1 ∈ (0, p], k ∈ N and α ∈ Nn . Let θ = θ+k+|α| θ (n, p) = n(1/p − 1/2). Assume that u0 ∈ Hq0 with (1 − Δ) 2 u0 ∈ Hp ,

M. D’Abbicco University of Bari, Bari, Italy A. De Luca () University of Milano-Bicocca, Milano, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_11

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and u1 ∈ Hq1 with (1 − Δ)

θ+k+|α|−1 2

u1 ∈ Hp . Moreover, assume that

 1 1 ≥ 1, n − q1 p if k = |α| = 0. Then the solution to (1) verifies the estimate ∂tk ∂xα u(t, ·) Hp

 1

− 21 n

≤ C(1 + t)

1 q0 − p



−θ+k+|α|

 u0 Hq0

θ+k+|α| 2

+ Ce−ct (1 − Δ) u0 Hp     − 21 n q1 − p1 −θ−1+k+|α| 1 u1 Hq1 + C(1 + t) + Ce−ct (1 − Δ)

θ+k+|α|−1 2

u1 Hp ,

(2) (3) (4) (5)

for any t ≥ 0 and for some C, c > 0, independent of the initial data. The statement of Theorem 1 is the same of [2, Theorem 1.2], but its proof need suitable modifications when the parameters a, b, d fail to fullfill a condition which is always verified when a = b = d = 1. Namely, when a zone of the phase space appears, where the two characteristic roots of the full symbol of (1) are real-valued, and not complex valued. Since this zone only appears at intermediate frequencies, the dissipation remains noneffective and the decay estimates are independent on the specific values assigned to the constants a, b, d > 0. We address the interested reader to [7] for a classification of effective and noneffective structural dissipation for damped evolution equations. Decay estimates for evolution equations with effective structural dissipation are obtained in Lp spaces in [3–6, 8, 10] and in real Hardy spaces in [9]. Even if problem (1) is interesting by itself from a theoretical mathematical point of view, it is originated by a real world physical problem. A presentation of the model is provided in [19]: in some problems of nonlinear wave propagation in waveguides, in case of energy exchange between the surface of nonlinear elastic rod in material (e.g., the Murnaghan material) and an external medium, the following double dispersion equation (DDE) utt − Δu =

1 (6Δu2 + aΔutt − bΔ2 u) 4

(6)

and the general cubic DDE (CDDE) utt − Δu =

1 (cΔu3 + 6Δu2 + aΔutt − bΔ2 u + dΔut ) 4

(7)

can be derived from Hamilton Principle. Here u(t, x) is proportional to strain ∂U ∂x , where U (t, x) is the longitudinal displacement, a > 0, b > 0, and d = 0 are

Decay Estimates for the DDE

183

some constants depending on the Young modulus, the shear modulus μ, density of waveguide ρ and the Poisson coefficient ν. Equations (6) and (7) were studied in literature: the travelling wave solutions, depending upon the phase variable z = x ± ct were studied by Samsonov in [16, 17], the strain solutions of equations (6) and (7) were analyzed in [12, 18]. Equation (7) is a special case of the following generalized double dispersion equation utt − Δu − aΔutt + bΔ2 u − dΔut = Δf (u).

(8)

The double dispersion equation and its generalized form have attracted lots of researchers’ interests and many interesting results have been established: the global existence and asymptotic decay of solution to the problem (8) are proved in [2] for a = b = d = 1 and nonsmooth f (u). As customary, the proof is based on the contraction mapping principle and makes use of the sharp decay estimates for the linearized problem. However, in this case the oscillations coming from the wave part of the equation produces two issues when one works in Lp spaces with p ∈ (1, 2): a loss of regularity which is known from the theory of damped wave equations, and a loss of decay rate, which is known from the theory of strongly damped wave equations. The double dispersion equation has been well investigated in recent times, in particular see [1, 15, 19, 20].

2 Notation We denote by F the Fourier transform with respect to the space variable x,  F ϕ(ξ ) =

Rn

ϕ(x)e−ixξ dx,

and we write ϕ(ξ ˆ ) = F f (ξ ), and ϕ(t, ˆ ξ ) = (F ϕ(t, ·))(ξ ). Differential operators are denoted by ∂xα = ∂xα11 · · · ∂xαnn , where α = (α1 , . . . , αn ) ∈ Nn and |α| = α1 + · · · + αn is the length of α.  With the symbol Δ we denote the Laplace operator as Δ = ni=1 ∂x2i . Fractional powers s > 0 of −Δ and 1 − Δ are intended as defined by their action ˆ (−Δ)s ϕ = F −1 (|ξ |2s ϕ),

(1 − Δ)s ϕ = F −1 (ξ 2s ϕ), ˆ

where 1

ξ  = (1 + |ξ |2 ) 2 . Similarly, we define the Riesz potentials (see also the Appendix) for s > 0: ˆ Is ϕ = F −1 (|ξ |−s ϕ),

(1 − Δ)−s ϕ = F −1 (ξ −2s ϕ). ˆ

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By W m,p , p ∈ [1, ∞] we denote the usual Sobolev space of Lp functions with m derivatives up to the order m in Lp , recalling that W m,p = (1 − Δ)− 2 Lp if p > 1. Moreover we use the following. Definition 1 Let f, g : Ω → R be two functions. We use the notation f  g (resp. f  g) if there exists a constant C > 0 such that f (y) ≤ Cg(y) (resp. f (y) ≥ Cg(y)) for all y ∈ Ω. The definition of real Hardy spaces Hp and some of their properties are collected in the Appendix.

3 Fundamental Solution and Decay Estimates Applying to (1) Fourier transform w.r.t. x, we get 

uˆ tt + |ξ |2 uˆ + a|ξ |2 uˆ tt + b|ξ |4 uˆ + d|ξ |2 uˆ t = 0, u(0, ˆ ξ ) = uˆ 0 (ξ ), uˆ t (0, ξ ) = uˆ 1 (ξ ).

t ≥ 0, ξ ∈ Rn ,

(9)

Solving the characteristic equation (1 + a|ξ |2 )λ2 + d|ξ |2 λ + (|ξ |2 + b|ξ |4 ) = 0, we have the characteristic roots: + −d|ξ |2 ± |ξ | −4ab|ξ |4 + (d 2 − 4a − 4b)|ξ |2 − 4 . λ± = 2(1 + a|ξ |2 )

(10)

(11)

If we consider ξ− < |ξ | < ξ+ and d > d, where explicity 9

+

(d 2 − 4a − 4b)2 − 64ab , 8ab : √ √ √ d = 4a + 4b + 8 ab = 2 a + 2 b,

ξ± =

(d 2 − 4a − 4b) ±

(12) (13)

then the characteristic roots are real and distinct. In this zone, it holds u(t, ˆ ξ) =

λ+ eλ− t − λ− eλ+ t eλ+ t − eλ− t uˆ 0 + uˆ 1 . λ+ − λ− λ+ − λ−

(14)

The presence of this zone is neglected in [2], due to the choice a = b = d = 1, which implies that d¯ = 4 > 1 = d. However, out of this zone, namely at low frequencies |ξ | < ξ− and at high frequencies |ξ | > ξ+ , the analysis is qualitatively equivalent to the study carried on in [2]. For this reason, we omit the study of these two zones and only study the “new” intermediate zone |ξ | ∈ (ξ− , ξ+ ).

Decay Estimates for the DDE

185

More precisely, we will fix ε > 0 in the proof, sufficiently small, and we will study the region ξ− + ε ≤ |ξ | ≤ ξ+ − ε. For the sake of brevity, we also omit the study of the two transition regions |ξ | ∈ (ξ− − ε, ξ− + ε) and |ξ | ∈ (ξ+ − ε, ξ+ + ε) (near the surfaces |ξ | = ξ± , at which λ− = λ+ ). For the sake of brevity, in the following we deal with u0 = 0. We denote ˆ ξ )uˆ 1 ). u(t, ·) = G(t, ·) ∗ u1 = F −1 (G(t,

(15)

ˆ ξ ) come into In order to prove Hp estimates with p ∈ (0, 2), the derivatives of G(t, play. Theorem 2 Let n ≥ 1, p ∈ (0, 2), q ∈ (0, p], k ∈ N and α ∈ Nn . Assume that ϕ ∈ Hq with ϕ supported in {ξ ∈ Rn : |ξ | ∈ [ξ− + ε, ξ+ − ε]}. Then we have the estimate ∂tk ∂xα G(t, ·) ∗ ϕ Hp  e−ct ϕ Hp ,

(16)

for any t ≥ 0 and for some c > 0. Proof We consider the Fourier multiplier (see Definition 3) ˆ ξ ), m(t, ξ ) = ξ −θ−k−|α| (iξ )α ∂tk G(t, and we prove that the operator Tm is Hp -bounded, with m(t, ·) M(Hp )  e−ct ,

(17)

for some c > 0. Let us fix ε > 0, sufficiently small. For any ξ− + ε ≤ |ξ | ≤ ξ+ − ε, it holds λ+ − λ−  cε > 0. We notice that we may estimate |∂ξ (λ+ − λ− )(ξ )|  |ξ |−|γ | . γ

Taking into account of (18), writing eλ+ t − eλ− t eλ+ t = (1 − e(λ− −λ+ )t ), λ+ − λ− λ+ − λ−

(18)

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together with ∂ξj eλ+ t = teλ+ t ∂ξj λ+ , ∂ξj e(λ− −λ+ )t = te(λ− −λ+ )t ∂ξj (λ− − λ+ ), we may estimate γ ˆ ξ )|  |ξ |k−|γ | (1 + t)|γ | eλ+ t , |∂ξ ∂tk G(t,

where we used that (1 + t |γ | )  (1 + t)|γ | . Therefore, |∂ξ m(t, ξ )|  |ξ |−|γ | (1 + t)|γ | e−ct , γ

where c=

min

(−λ+ ) > 0.

|ξ |∈[ξ− ,ξ+ ]

The minimum is nonnegative, since λ+ is nonpositive. We remark that λ+ → −

d|ξ |2± 2(1 + a|ξ |2± )

as |ξ | → ξ± .

By applying Theorem 3 in the Appendix, with a = 0 and A = 1 + t, we obtain m(t, ·) M(Hp )  (1 + t)θ e−ct . Therefore, we obtain (17) with a different c. This completes the proof.

 

Remark 1 The polynomial decay rate of formula (4) comes from the multiplier estimate at low frequencies (as it happens for damped waves in the whole space Rn , θ+k+|α|−j 2 uj ∈ Hp , in general), whereas the regularity of the initial data (1 − Δ) j = 0, 1, comes from the multiplier estimate at high frequencies (see [2] for the proof). In the intermediate frequencies, on the one hand we derive an exponential decay, on the other hand, no regularity issue comes into play. Acknowledgements The results for the linear problem in this contribution are a variant of the one contained in the master thesis of the second author, who has been a student at University of Bari.

Appendix We recall how the Hardy spaces Hp (Rn ) are presented by Fefferman and Stein [11]. We use the notation Hp instead of the classical notation H p to avoid possible confusion with the Sobolev space W p,2 .

Decay Estimates for the DDE

187

Fix, once for all, a radial nonnegative function φ ∈ Cc∞ (Rn ) supported in the unit ball with integral equal to 1. For u ∈ S  (Rn ) we define the maximal function Mφ u by Mφ u(x) = sup |(u ∗ φt )(x)|, 0 0 is a constant independent of A. Theorem 4 Let p ∈ (0, 2), and θ = θ (n, p) = n(1/p − 1/2). Assume that m ∈ C k (Rn \ {0}), with m(ξ ) = 0 for |ξ | ≥ 1, and k = max{[θ ], [ n2 ]} + 1. If |∂ξ m(ξ )| ≤ |ξ |aθ (A|ξ |−a−1 )|γ | , |γ | ≤ k, γ

for some constant a ≥ 0 and A ≥ 1, then m ∈ M(Hp (Rn )) and m M(Hp (Rn )) ≤ CAθ , where C is a constant independent of A.

Decay Estimates for the DDE

189

Let Ir be the Riesz potential with order r > 0, defined by means of Ir f (ξ ) = F −1 (|ξ |−r fˆ(ξ )). If r ∈ (0, n), then there exists cn,r such that  Ir f (x) = cn,r

Rn

f (y) dy |x − y|n−r

and sufficiently smooth f . Real Hardy spaces have the property that the HardyLittlewood-Sobolev theorem for Riesz potential, valid in Lp spaces, with p > 1, extends to Hp , with p ∈ (0, ∞), see [14, Theorem F]. Theorem 5 Consider r > 0 and 0 < p < n/r. Then, there exists C = C(r, p) > 0 such that Ir f Hq (Rn ) ≤ C f Hp (Rn ) ,

1 1 r = − . q p n

References 1. G. Chen, Y. Wang, S. Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl. 299 (2004) 563–577. 2. M. D’Abbicco, A. De Luca, Long time decay estimates in real Hardy spaces for the double dispersion equation, J. Pseudo-Differ. Oper. Appl. (2019). https://doi.org/10.1007/s11868019-00287-1. 3. M. D’Abbicco, A benefit from the L∞ smallness of initial data for the semilinear wave equation with structural damping, in Current Trends in Analysis and its Applications, 2015, 209–216. Proceedings of the 9th ISAAC Congress, Krakow. Eds V. Mityushev and M. Ruzhansky, http:// www.springer.com/br/book/9783319125763. 4. M. D’Abbicco, L1 −L1 estimates for a doubly dissipative semilinear wave equation, Nonlinear Differential Equations and Applications NoDEA 24 (2017), 1–23, http://dx.doi.org/10.1007/ s00030-016-0428-4 5. M. D’Abbicco, M.R. Ebert, Diffusion phenomena for the wave equation with structural damping in the Lp − Lq framework, J. Differential Equations, 256 (2014), 2307–2336, http:// dx.doi.org/10.1016/j.jde.2014.01.002. 6. M. D’Abbicco, M.R. Ebert, An application of Lp − Lq decay estimates to the semilinear wave equation with parabolic-like structural damping, Nonlinear Analysis 99 (2014), 16–34, http:// dx.doi.org/10.1016/j.na.2013.12.021. 7. M. D’Abbicco, M.R. Ebert, A classification of structural dissipations for evolution operators, Math. Meth. Appl. Sci. 39 (2016), 2558–2582, http://dx.doi.org/10.1002/mma.3713. 8. M. D’Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations. Nonlinear Analysis 149 (2017), 1–40. 9. M. D’Abbicco, M. R. Ebert, T. Picon, Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation, J. Pseudo-Differ. Oper. Appl. 7 (2016), 261–293. 10. M. D’Abbicco, M. Reissig, Semilinear structural damped waves, Math. Methods in Appl. Sc., 37 (2014), 1570–1592. 11. C. Fefferman and E. Stein, H p spaces of several variables, Acta Math. 129 (1972) 137–193. 12. G. A. Mangin, Physical and mathematical models of nonlinear waves in solids, A. Jeffrey, J. Engelbrecht (Eds.), NonlinearWaves in Solids, Elsevier, 1994.

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13. A. Miyachi, On some Fourier multipliers for H p , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 157–179. 14. A. Miyachi, On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 267–315. 15. N. Polat, A. Ertas, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl. 349 (2009) 10–20. 16. A. M. Samsonov, On existence of longitudinal strain solitons in a nonlinearly elastic rod, Sov. Phys.-Dokl. 4 (1988) 298–300. 17. A. M. Samsonov, On some exact travelling wave solutions for nonlinear hyperbolic equation, D. Fusco, A. Jeffrey (Eds.), Nonlinear waves and Dissipative Effects, Pitmann Research Notes in Mathematics Series, vol. 227, Longman Scientific & Technical; Longman, 1993, pp. 123–132. 18. A. M. Samsonov, E.V. Sokurinskaya, On the excitation of a longitudinal deformation soliton in a nonlinear elastic rod, Sov. Phys. Tech. Phys. 33 (1988) 989–991. 19. S. Wang, G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. 64 (2006) 159–173. 20. R. Xu, Y. Liu, T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal. 71 (2009) 4977–4983.

On Density Operators with Gaussian Weyl Symbols Maurice A. de Gosson

This paper is dedicated to Prof. Luigi Rodino for his 70th birthday

Abstract The notion of reduced density operator plays a fundamental role in quantum mechanics where it is used as a tool to study statistical properties of subsystems. In the present work we review this notion rigorously from a mathematical perspective using pseudodifferential theory, and we give a new necessary and sufficient condition for a Gaussian density operator to be separable.

1 Introduction Positive trace class operators on a Hilbert space play an important role in functional analysis and its applications; when having trace one they are the density operators (or “mixed states”) of quantum mechanics. In this contribution we give rigorous definitions motivated by the theory of pseudodifferential operators, and we focus on the notion of “separability” which is a very important topic in the theory of density operators, where many problems are yet unsolved. We give a simple necessary and sufficient condition for separability when the Weyl symbol of the density operator is a non-degenerate Gaussian on phase space.  Notation 1 The standard symplectic form on T ∗ Rn ≡ R2n is σ = nj=1 dpj ∧  0n×n In×n and · is dxJ ; in matrix notation σ (z, z ) = J z · z where J = −In×n 0n×n the Euclidean scalar product. Let M be a symmetric matrix on Rn ; we will use the notation Mx · x = Mx 2 . We denote by Sp(n) the symplectic group of (R2n , σ )

M. A. de Gosson () Faculty of Mathematics (NuHAG), University of Vienna, Wien, Austria e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_12

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and by Mp(n) the corresponding metaplectic group. S(Rn ) is the Schwartz space of rapidly decreasing test functions on Rn . Given a tempered distribution a ∈ S  (R2n ) we denote by OpW (a) the Weyl operator with symbol a.

2 Density Operators 2.1 Basic Definitions We recollect here the essential results from the theory of density operators we will need, and which are often dispersed in the literature. A good mathematical source is Simon’s book [23]. A concise and application-oriented treatment is given in [2], §22; also see [10] or [7], Chapter 12. Let H be a separable Hilbert space with inner product (ψ, φ) −→ (ψ|φ). A density operator on H is an operator ρ  ∈ B(H) having the following properties: (i) ρ  is of trace class: ρ  ∈ L1 (H) and has trace one: Tr( ρ ) = 1; (ii) ρ  is self-adjoint: ρ = ρ ∗ ; (iii) ρ  is positive semidefinite: ρ  ≥ 0, that is ( ρ ψ|ψ) ≥ 0 for every ψ ∈ H (notice that (iii) implies (ii) when H is a a complex Hilbert space). Every trace class operator on H being compact it readily follows from the spectral theorem that ρ  ∈ B(H) is a density operator if and only if it is a convex sum of orthogonal projectors: there exist a sequence (λj ) of nonnegative real numbers summing up to one and an orthonormal basis (ψj ) of H such that

 ψj , +  ψj ψ = (ψ|ψj )ψj . λj + (1) ρ = j

 ; since λj ∈ [0, 1] we have The number μ( ρ ) = j λ2j is called the “purity” of ρ μ( ρ ) ∈ (0, 1] and μ( ρ ) = 1 if and only if the sum in (1) reduces to a single term  ψ (“pure state”). + We assume from now on that H = L2 (Rn ). Each density operator is the product of two Hilbert–Schmidt operators; the latter are the bounded operators on L2 (Rn ) with square-integrable kernels. We can thus find K ∈ L2 (Rn × Rn ) such that ρ ψ(x) = K(x, ·), ψ; the partial Fourier transform  a(x, p) =

i

Rn

e− h¯ py K(x + 12 y, x − 12 y)dy

(2)

then defines the Weyl symbol of ρ , that is ρ  = OpW (a). By definition, the function ρ = (2π h¯ )−n a is the Wigner distribution of the density operator ρ , and we thus have the explicit formula  ρ ψ(x) =

R2n

i

e h¯ p(x−y) ρ( 12 (x + y), p)ψ(y)dpdy

(3)

On Density Operators with Gaussian Weyl Symbols

193

for all ψ ∈ S(Rn ). It follows from the spectral decomposition (1) that:   ψj on Proposition 2 The Wigner distribution of a density operator ρ  = j λj + 2 2n L (R ) is given by the formula ρ=

λj W ψj ,

j

λj = 1 , λj ≥ 0

(4)

j

where W ψj is the Wigner transform of ψj ∈ L2 (Rn ): W ψj (z) =



1 2π h¯

n  Rn

i

e−  py ψj (x + 12 y)ψj (x − 12 y)dy.

(5)

Conversely, if ρ is given by (4), then ρ  = (2π h) ¯ −n OpW (ρ) is a density operator. Proof Writing  ψj ψ(x) = +

 Rn

ψ(y)ψj (y)ψj (x)dy

 ψj is Kj = ψj ⊗ ψj ; by formula (2) its Weyl symbol is the kernel of the projector +  πj (x, p) =

Rn

i

e− h¯ py ψj (x + 12 y)ψj (x − 12 y)dy

hence (4) since ρ is (2π h) . ¯ −n times the Weyl symbol of ρ

 

Notice that the pure case μ( ρ ) = 1 corresponds to ρ = W ψ for some ψ ∈ L2 (Rn ). In physics texts one often finds the following formula relating the Wigner distribution to the trace:  ρ(z)dz. (6) Tr( ρ) = R2n

There are however many caveats when using this formula; it is general false if one does not add the requirement that ρ ∈ L1 (R2n ) (Du and Wong [4]; we have discussed these delicate issues in [7], §12.3). When the Weyl symbol a of an operator decreases sufficiently fast at infinity we have the following useful result, where it is not assumed from the beginning that we are dealing with a trace class operator. We will use the “Japanese bracket” notation for weights: z = (1 + |z|2 )1/2 .  = OpW (a) with a ∈ C ∞ (R2n ). Suppose that there exists Proposition 3 Let A m < −2n and δ ∈ (0, 1] such that for every α ∈ N2n we can find Cα > 0 such that |∂zα a(z)| ≤ Cα zm−δ|α|

(7)

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 is of trace class; (ii) the trace of A  is given by then: (i) A n   1  a(z)dz Tr(A) = 2π h¯ R2n

(8)

The proof of (i) can be found in Shubin [24], §27; formula (8) follows since the condition m < −2n implies that a ∈ L1 (R2n ). In particular formula (6) holds. For products we have the following result: B  ∈ L1 (L2 (Rn )) (or, more generally, are Hilbert–Schmidt Proposition 4 If A, operators), have Weyl symbols a and b then n   1   a(z)b(z)dz. (9) Tr(AB) = 2π h¯ R2n

For a proof of (iii) see [7] (Prop. 284) and the references quotes there. Suppose, conversely, that an operator ρ  ∈ B(L2 (Rn )) is defined by (3) where 2 n n ρ ∈ L (R × R ). While the properties (i) and (ii) listed above are in general relatively straightforward to verify, the positivity property (iii) is usually much more problematic. Proving the non-negativity of Weyl operators has always been a difficult business, and has led to a number of partial results in the pseudo-differential literature in the late 1970s and early 1980s; for trace-class operators on L2 (Rn ) the following criterion is known:  = OpW (a) be a trace class operator on Rn : A  ∈ L1 (L2 (Rn )). Proposition 5 Let A  ≥ 0 if and only if the symplectic Fourier transform We have A n   i  1 Fσ a(z) = 2π h¯ e− h¯ σ (z,z ) a(z )dz (10) R2n

of the Weyl symbol is continuous and of h-positive type, that is, if for every finite ¯ sequence z1 , z2 , . . . , zN of points in R2n the N × N matrix (N ) with entries i

j k = e 2h¯ σ (zj ,zk ) Fσ a(zj − zk )

(11)

is positive semi-definite. The conditions above on the matrices (N ) are usually called the “KLM conditions” in reference to the seminal work of Kastler [17], and Loupias and Miracle-Sole [20, 21]; the proof they give uses techniques from C ∗ -algebra theory; we have given an alternative and somewhat simpler proof using harmonic analysis in [3]. In the same paper the KLM conditions are refined using Gabor frame theory.

2.2 Reduced Density Operators Let us introduce some notation. Let A ∪ B be a symplectic partition of {1, 2, . . . ., 2n}. We identify R2n with direct sum R2nA ⊕R2nB and write indifferently

On Density Operators with Gaussian Weyl Symbols

195

z = zA ⊕ zB or z = (zA , zB ) and σ = σA ⊕ σB where σA is the standard symplectic form on R2nA and σB the one on R2nB . Let ρ  = (2π h) ¯ n OpW (ρ) be a density operator on L2 (Rn ) ≡ L2 (RnA ) ⊗ 2 n B L (R ). We assume that Shubin’s condition (7) in Proposition 5 holds for ρ so that the trace formula (6) is valid. The partial mappings zA −→ ρ(zA ⊕ zB ) and zB −→ ρ(zA ⊕ zB ) are absolutely integrable; setting  ρ A (zA ) =

R nB

(12)

ρ(zA , zB )dzB

ρ ) of ρ  with respect to RnA we define the reduced density operator ρ A = TrA ( (or “partial trace”) by n A ρ A = (2π h) ¯ A OpW (ρ ).

(13)

That this definition makes sense follows from: Proposition 6 The operator ρ A defined by (13) is a density operator on L2 (RnA ). Proof We first observe that since the Wigner distribution ρ is real so is ρ A and the Weyl operator ρ A is thus self-adjoint. Next, ρ A is indeed a trace class-operator: in view of the assumption (7) there exists m < −2nA − 2nB such that the derivatives of ρ satisfy |∂zα ρ(z)| ≤ Cα zm−δ|α| . Choose mA < −2nA and mB < −2nB such that m = mA + mB ; we have for all α ∈ NnA |∂zαA ρ(z)| ≤ Cα zA mA −δ|α| zB mB and hence |∂zαA ρ(z)| ≤ Cα zA mA −δ|α|

 R nB

zB mB dzB ;

the integral over RnB is convergent since mB < −2nB and hence there exists Cα > 0 such that |∂zαA ρ(z)| ≤ Cα zA mA −δ|α| and we may apply Proposition 5. Let us check the positivity condition ρ A ≥ 0. The A symplectic Fourier transform FσA ρ is given by FσA ρ A (zA ) = =

 

1 2π h¯ 1 2π h¯

nA  R2nA

nA 

R2nA

i



  e− h¯ σA (zA ,zA ) ρ A (zA )dzA

e

 ) − hi¯ σA (zA ,zA

 R nB



 ρ(zA , zB )dzB dzA

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that is, by Fubini’s theorem, FσA ρ A (zA ) =



1 2π h¯

nA 



i

R2n

e− h¯ σA (zA ,zA ) ρ(z)dz

= (2π h) ¯ Fσ ρ(zA , 0). nB

type hence FσA ρ A It suffices now to apply Proposition 5: Fσ ρ is of h-positive ¯ must also be of h-positive type as is seen by restricting the KLM conditions to ¯ lattices of points (zA , 0). There remains to check that we have Tr( ρ A ) = 1, but this follows from formulas (12) and (8), or, alternatively from evaluating FσA ρ A (zA ) at zA = 0.   The following result is often taken as the definition of reduced density operators in physically oriented texts; proofs are usually given only in the case of finitedimensional Hilbert spaces. Proposition 7 The operator ρ A is the unique trace class operator such that A ⊗ IB )) A ) = Tr( ρAX ρ (X TrA ( A ∈ B(L2 (RnA )) (TrA being the trace for operators on R2nA ). for all X Proof Recall that the space L1 (H) of trace class operators on a Hilbert space H A ∈ L1 (L2 (RnA )), and we have, is a two-sided ideal in B(H). It follows that ρ A X A A  writing X = OpW (ξ ), A ) = TrA ( ρA X

 R2nA

ρ A (zA )ξ A (zA )dzA

Suppose that there exists ρ A ∈ L1 (RnA ) such that A ⊗ IB )); A ) = Tr( ρ A X ρ (X TrA ( A ) = 0 for all in view of the additivity of the trace we then have TrA (( ρA − ρ A )X 2 n A A A  XA ∈ B(L (R )) and hence ρ  =ρ  .  

2.3 Gaussian Symbols We assume from now on that the functions ψj in (4) are in Feichtinger’s modulation space M 1 (Rn ) = S 0 (Rn ) [14, 16]. This allows us to define the covariance matrix of ρ : it is the positive-definite real symmetric 2n × 2n real matrix  =

 (z − z¯ )(z − z¯ ) ρ(z)d z , z¯ = T

2n

zρ(z)d 2n z.

(14)

On Density Operators with Gaussian Weyl Symbols

197

It is well-known (but not quite trivial to prove [5, 7, 25]) that the covariance matrix must satisfy the “quantum condition” +

i h¯ J ≥0 2

(15)

that is the Hermitian matrix + i2h¯ J must have all its eigenvalues ≥ 0; this condition is necessary for the positivity requirement ρ  ≥ 0 but not in general sufficient [11, 12]. From now we assume that the Wigner distribution of ρ  is a non-degenerate Gaussian: + 1 −1 2 ρ(z) = (2π )−n det  −1 e− 2  (z−z0 ) (16) where  is a positive definite real symmetric 2n × 2n matrix; it is straightforward to check that  indeed is the covariance matrix (14). The Weyl operator ρ  = (2π h¯ )n OpW (ρ) is of trace class in view of Proposition 3 since Shubin’s condition (7) holds for all m ∈ R; a straightforward calculation of Gaussian integrals shows that we have Tr( ρ ) = 1. Since ρ is real, the operator ρ  is self-adjoint. It turns out that for Gaussians (16) the quantum condition (15) is not only necessary, but also sufficient. It is in fact equivalent to the uncertainty principle in its strong Robertson–Schrödinger form [13]. Let us reformulate it in terms of the symplectic eigenvalues of ; i.e. the positive numbers λσ1 , . . . , λσn such that ±iλσ1 , . . . , ±λσn are the eigenvalues of J  (that is those of the antisymmetric matrix  1/2 J  1/2 ). Lemma 8 The quantum condition (15) is equivalent to the following statement: the symplectic eigenvalues λσ1 , . . . , λσn of  are all ≥ 12 h. ¯ The purity μ( ρ ) = Tr( ρ 2 ) of the density operator ρ  is given by  n Tr( ρ 2 ) = (2π h) ¯

R2n

ρ 2 (z)dz;

evaluating the integral by a standard calculation of Gaussian integrals one finds  n h¯ μ( ρ) = det( −1/2 ). 2 It will be convenient to set M = form

h¯ −1 2

(17)

in which case (16) takes the equivalent 1

.n 1/2 − Mz ρ(z) = (π h) ¯ (det M) e h¯

2

(18)

and the quantum condition (15) becomes then M −1 + iJ ≥ 0

(19)

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M. A. de Gosson

while the purity formula (17) becomes μ( ρ ) = (det M)1/2 .

(20)

Proposition 9 The density operator ρ  with Gaussian Wigner distribution (16) has purity μ( ρ ) = 1 if and only if there exists S ∈ Sp(n) such that M = S T S; S can moreover be chosen of the lower-triangular block type  1/2 0 X (21) S= X−1/2 Y X−1/2 where X and Y are real symmetric n × n matrices with X > 0and we have in this case ρ(z) = W φX,Y with 1

−n/4 (det X)1/4 e− 2h¯ (X+iY )x φX,Y (x) = (π h) ¯

2

(22)

Proof In view of formula (20) we have μ( ρ ) = 1 if and only if det M = 1. In view of Williamson’s symplectic diagonalization theorem [6, 27] there exists S ∈ Sp(n) such that M = S T DS where D = diag(λσ1 , . . . , λσn ; λσ1 , . . . , λσn ) the λσj > 0 being the symplectic eigenvalues of M. Since SJ S T = J the quantum condition M −1 + iJ ≥ 0 is equivalent to D −1 + iJ ≥ 0 which is in turn equivalent to λj ≤ 1 for all j , but since det M = 1 we must have λj = 1 for all j which is equivalent to M = S T S. Let now φX,Y (x) be given by (22); one shows by a direct calculation [1, 9, 19] that W φX,Y (z) = (π h¯ )−n e

1 − h Gz2 ¯

(23)

where G ∈ Sp(n) is, in the canonical symplectic basis, represented by the symmetric symplectic matrix G=

 X + Y X−1 Y Y X−1 ; X−1 Y X−1

that is G = S T S where S ∈ Sp(n) is given by (21), and this determines φX,Y .

(24)  

2.4 A Lemma on Gaussians From now on we write the positive definite symmetric real matrix M in block form  MAA MAB (25) M= MBA MBB

On Density Operators with Gaussian Weyl Symbols

199

where the blocks MAA , MAB , MBA , MBB are, respectively, nA × nA , nA × nB , T ,M T T nB × nA , nB × nB matrices. Thus MAA = MAA BB = MBB and MBA = MAB . In addition MAA > 0 and MBB > 0 and are hence invertible. The matrices −1 MBA M/MBB = MAA − MAB MBB

(26)

−1 M/MAA = MBB − MBA MAA MAB

(27)

are Schur complements of the blocks MBB and MAA , respectively, of M. These matrices satisfy the relations det M = det(M/MAA ) det MAA = det(M/MBB ) det MBB

(28)

(see Zhang [28]). The following elementary result will be useful: Lemma 10 Let M be the matrix (25). We have  1 2 2 n −1/2 − h1¯ (M/MBB )zA e− h¯ Mz dzB = (π h) e . ¯ B (det MBB )

(29)

R2nB



1

R2nA

1

n −1/2 − h¯ (M/MAA )zB e− h¯ Mz dzA = (π h) e . ¯ A (det MAA ) 2

2

(30)

Proof Writing z = (zA , zB ) we have 2 2 + 2MBA zA · zB + MBB zB Mz2 = MAA zA

(31)

so that  R2nB

e

− h1¯ Mz2

dzB = e

2 − h1¯ MAA zA



1

R2nB

e− h¯ (MBB zB +2MBA zA ·zB ) dzB . 2

−1 Setting zB = uB − MBB MBA zA we have −1 2 2 MBB zB + 2MBA zA · zB = MBB u2B − MAB MBB MBA zA

and hence  R2nB

1

−1

1

e− h¯ Mz dzB = e− h¯ (MAA −MAB MBB MBA )zA 2

2



1

R2nB

e− h¯ MBB uB duB . 2

Using the well-known Gaussian integral [6]  R2nB

1

n −1/2 e− h¯ MBB uB duB = (π h) ¯ B (det MBB ) 2

we get (29). Formula (30) follows, switching the sets A and B.

 

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M. A. de Gosson

2.5 Partial Traces of Gaussian Density Operators Let φX,Y ∈ S(Rn ) be the Gaussian function defined by (22). We have ||φX,Y ||L2 (Rn ) = 1, and its Wigner transform W φX,Y (z) is given by formulas (23) and (24): W φX,Y (z) = (π h¯ )−n e

1 − h Gz2 ¯

(32)

where G ∈ Sp(n) is a symmetric symplectic positive definite matrix: G = S T S where S is given by (21) in Proposition 9. We define the functions  W φX,Y (zA , zB )dzB . (33) ρ A (zA ) = R2nB

and, similarly,  ρ B (zB ) =

R2nA

(34)

W φX,Y (zA , zB )dzA .

Writing G in AB-block-form (25) that is  G=

GAA GAB GBA GBB



it immediately follows from Lemma 10 that ρ A and ρ B can be expressed in terms of the Schur complements G/GBB and G/GAA : 1

2

1

2

ρ A (zA ) = (π h¯ )−nA det(G/GBB )1/2 e− h¯ (G/GBB )zA ρ B (zB ) = (π h¯ )−nB det(G/GAA )1/2 e− h¯ (G/GAA )zB .

(35) (36)

In fact, the matrix M in (25) is the symplectic matrix G defined by (24), hence det M = 1 and formula (35) follows from (29); formula (36) is proven in a similar way. Notice that it follows from Proposition 6 that ρ A and ρ B indeed are density operators on L2 (RnA ) and L2 (RnB ), respectively and hence the quantum conditions −1 (G/GBB )−1 + i hJ ¯ A ≥ 0 and (G/GAA ) + i hJ ¯ B ≥0

(37)

must hold. We will denote WA (resp. WB ) the Wigner transform in the zA (resp. zB ) variables). Proposition 11 We have ρ A = WA φXA ,YA for some Gaussian function φXA ,YA on RnA if and only if det GBB = 1.

On Density Operators with Gaussian Weyl Symbols

201

Proof (i) Suppose that ρ A = WA φXA ,YA for some pair (XA , YA ). Then the purity (20) of ρ A is μ( ρA ) = det G/GBB = 1; in view of formula (28) we have det(G/GBB ) det GBB = 1

(38)

hence det GBB = 1. Suppose conversely that det GBB = 1. The quantum condition (37) for G/GBB is equivalent to saying that the symplectic eigenvalues λσ1A , . . . , λσnAA of G/GBB are ≤ 1 hence det G/GBB ≤ 1 with equality if and only if λσ1A = · · · = λσnAA = 1. In view of (38) the condition det GBB = 1 implies λσ1A = · · · = λσnAA = 1 and hence G/GBB = SAT DA SA = SAT SA which proves the sufficiency of the condition det GBB = 1.

 

3 Separability of Gaussian Density Operators 3.1 The Notion of Separability We will say that the density operator ρ  = (2π h) ¯ n OpW (ρ) is AB-separable if we can write

λj ρ jA ⊗ ρ jB (39) ρ = j ∈I

 jA and ρ jB are density operators on L2 (RnA ) and where λj ≥ 0, j λj = 1 and ρ L2 (RnB ), respectively. The Wigner function ρ is in this case of the type ρ=

λj WA ψjA ⊗ WA ψjB ;

(40)

j ∈I

it is defined on the phase space R2nA ⊕R2nB equipped with the symplectic form σ = σA ⊕ σB where σA (resp. σB ) is the standard symplectic form on R2nA (resp. R2nB ). In the particular case where ρ = W φX,Y (z) as studied above we have ρ = WA φXA ,YA ⊗ WB φXB ,YB

(41)

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M. A. de Gosson

if and only if GAB = 0 and det GAA = det GBB = 1

(42)

as follows from Proposition 11. (ii) If (41) holds, then ρ A = WA φXA ,YA and ρ B = WB φXB ,YB so we must we (42) in view of (i). Suppose conversely that (42) holds. In view of the identity (28) we have det(G/GAA ) det GAA = det(G/GBB ) det GBB = 1. It is easy to find a necessary condition for a density operator to be separable (it is known in physics as the “PPT criterion”, from Peres [22] and the Horodeckis [15]). We denote by IA the identity in R2nA and by I B the involution (xB , pB ) −→ (xB , −pB ) of R2nB . We set I AB = IA ⊕ I B . Proposition 12 Let ρ  = (2π h) ¯ n OpW (ρ) be an arbitrary (not necessarily Gaussian) density operator on L2 (Rn ). If ρ  is AB-separable, then ρ TB = n (2π h) ρ ) = Tr( ρ TB ). ¯ OpW (ρ ◦ I AB ) is also a density operator, and we have Tr( Proof Using the spectral decomposition theorem and bilinearity it is sufficient to assume that

ρ= λj WA ψjA ⊗ WB ψjB . j

We have ρ(I AB z) =

λj WA ψjA (zA )WB ψjB (I B zB );

j

using the trivial relation W ψ B (I B zB ) = W ψ B (zB ) we thus have ρ ◦ I AB =

λj W (ψjA ⊗ ψ B )(zB )

j

hence OpW (ρ ◦I AB ) is also a positive semidefinite trace class operator; that we have Tr( ρ ) = Tr( ρ TB ) is obvious.   The operator ρ TB is called in physics the partial transpose of ρ  with respect to the B variables; the acronym PPT means “positive partial transpose”. Here is another necessary condition for separability due to Werner and Wolf [26]; it is particularly simple to state because it is a condition on the covariance matrix of ρ . Proposition 13 If the density operator ρ  on L2 (Rn ) with covariance matrix  is AB-separable then there exist symmetric nA × nA and nB × nB matrices A and

On Density Operators with Gaussian Weyl Symbols

203

B such that  ≥ A ⊕  B , A +

i h¯ i h¯ JA ≥ 0 , B + JB ≥ 0 2 2

(43)

where JA and JB are the standard nA × nA and nB × nB symplectic matrices.  

Proof See [26], Prop. 1.

Finding sufficient conditions for separability is a much more difficult endeavor, to which much effort is still being devoted. The only well understood case is that of density operators with Gaussian symbols, in which case Werner and Wolf’s conditions (43) are sufficient, as we will see below (Proposition 14).

3.2 The Gaussian Case: Necessary and Sufficient Conditions Using of techniques in quantum harmonic analysis developed by Werner in [25], Werner and Wolf [26] have proven the following general criterion for separability for general Gaussians (16). Proposition 14 Assume that ρ is a Gaussian: ρ(z) =



1 2π

n

1

(det )−1/2 e− 2 

−1 z2

(44)

satisfying the quantum condition (15). The density operator ρ  is AB-separable if there exist positive symmetric matrices A and B of dimensions nA and nB such that condition (43) hold. We note that Lami et al. [18] (Theorem 5) have proved a refinement of this result. We are going to prove that ρ  is AB-separable if and only if ρ dominates, up to a factor, a tensor product of Gaussians. We will use again the more convenient form (18), that is 1

ρ(z) = (π h¯ )−n (det M)1/2 e− h¯ Mz

2

with M = h2¯  −1 . Proposition 15 The Gaussian density operator ρ  is separable if and only one of the two equivalent conditions is satisfied: Theorem 16 (i) There exist SA ∈ Sp(nA ) and SB ∈ Sp(nB ) such that M ≤ SAT SA ⊕ SBT SB

(45)

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M. A. de Gosson

that is, using the covariance matrices, ≥

h¯ T (S SA )−1 ⊕ (SBT SB )−1 . 2 A

(46)

A B (ii) There exist Gaussians φX and φX such that A ,YA B ,YB A B ⊗ WB φX ρ ≥ μ( ρ )WA φX A ,YA B ,YB

(47)

where μ( ρ ) = (det M)1/2 is the purity of ρ . These Gaussians are given by A B = SA φ0A and φX = SB φ0B φX A ,YA B ,YB

where  SA ∈ Mp(nA ) and  SB ∈ Mp(nB ) are metaplectic operators covering 2 SA and SB , respectively, and φ0A (xA ) = (π h¯ )−nA /4 e−|xA | /2h¯ and φ0B (xB ) = 2 (π h¯ )−nB /4 e−|xB | /2h¯ . Proof (i) Let us prove that condition (45) is equivalent to separability. That it is sufficient follows from Proposition 14 taking .A = h2¯ (SAT SA )−1 and B = h2¯ (SBT SB )−1 and observing that the quantum conditions A + i2h¯ JA ≥ 0 and B + i2h¯ JB ≥ 0 are satisfied since they are equivalent to the trivial inequalities IA + iJA ≥ 0 and IB + iJB ≥ 0 (IA and IB the identity matrices on R2nA and R2nB , respectively). Let us prove that condition (45) is also necessary. The condition A + i2h¯ JA ≥ 0 in (43) means that the symplectic eigenvalues λσ1A , . . . , λσnAA of A all are ≥ 12 h¯ (Lemma 8). In view of Williamson’s symplectic diagonalization theorem there exists SA ∈ Sp(nA ) such that A = (SAT )−1 DA SA−1 where DA = diag(λσ1A , . . . , λσnAA ; λσ1A , . . . , λσnAA ). We must thus have A ≥ h2¯ (SAT SA )−1 . Similarly there exists SB ∈ Sp(nA ) such that B ≥ h2¯ (SBT SB )−1 and hence the inequality (45) must hold. (ii) Suppose that (47) A B and φX ; then, by definition of ρ and recalling holds for some Gaussians φX A ,YA B ,YB that nA + nB = n 1

1 T

1 T

μ( ρ )e− h¯ Mz ≥ e− h¯ SA SA zA e− h¯ SB SB zB 2

2

2

We have [7, 9] 1

2

1

2

−nA − h¯ |zA | WA φ A (SA−1 zA ) = (π h) e ¯

−nB − h¯ |zB | WB φ B (SB−1 zB ) = (π h) e ¯

On Density Operators with Gaussian Weyl Symbols

205

and hence ρ(z) ≥ (det M)1/2 WA φ A (SA−1 zA )WB φ B (SB−1 zB )

(48)

which shows that the inequality (47) must hold if the state ρ  is separable. Suppose conversely that this inequality holds. Then we must have 1

1 T

1 T

e− h¯ Mz ≤ e− h¯ SA SA zA e− h¯ SB SB zB 2

2

2

which is equivalent to our condition (45), and the latter implies separability.

 

The theorem above has the following interesting consequence which shows that all Gaussian density operators are separable up to a conjugation with unitary operators: Corollary 17 Let ρ  be a Gaussian density operator as above. There exists  S ∈ Mp(n) such that  S ρ S −1 is separable. Proof We recall [7, 8] the symplectic covariance formula  S ρ S −1 = ρ ◦ S −1

(49)

for Weyl operators valid for all  S ∈ Mp(n) covering S ∈ Sp(n). Let M = S T DS be a Williamson symplectic diagonalization of the M = h2¯  −1 ; we have S ∈ Sp(n) and D(x, p) = (x, p) where  = diag(λσ1 , . . . , λσn ). In view of the quantization condition (19) we have λσj ≤ 1 for all j = 1, . . . , n. It follows that (S T )−1 MS −1 z2 ≤ |z|2 and hence 1

(ρ ◦ S −1 )(z) ≥ (π h¯ )−n (det M)1/2 e− h¯ |z|

2

= μ( ρ )φ0A (xA )φ0B (xB ); the proof now follows from Theorem 15 using (49).

 

Acknowledgements This work has been financed by the Austrian Research Foundation FWF (Grant number P27773). It is our pleasure to thank a Referee for very useful remarks and for having pointed out inaccuracies in a first version of this work.

References 1. M.J. Bastiaans, Wigner distribution function and its application to first-order optics, JOSA 69(12), 1710–1716 (1979) 2. P. Blanchard and E. Brüning, Mathematical methods in Physics: Distributions, Hilbert space operators, variational methods, and applications in quantum physics. Vol. 69. Birkhäuser, 2015

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On the Solvability of a Class of Second Order Degenerate Operators Serena Federico and Alberto Parmeggiani

Dedicated to Luigi Rodino

Abstract In this paper we will be concerned with the problem of solvability of second order degenerate operators that are not of principal type. We will describe some recent results we have obtained about local solvability in the Sobolev spaces of a class of degenerate operators which is an elaboration of the class considered by Colombini-Cordaro-Pernazza (in turn, an elaboration of the adjoint of the Kannai operator). Keywords Local solvability · A priori estimates · Degenerate second order operators 2010 Mathematics Subject Classification Primary 35A01; Secondary 35B45, 35A30

1 Introduction In this paper we will survey some results concerning the solvability (in L2 -based Sobolev spaces) of an interesting class of degenerate operators, whose symbol may be complex valued with a real part which may change sign. Such a class is interesting and natural, for it is built upon the operator P = Dx2 x1 Dx2 + iDx1 ,

(x1 , x2 ) ∈ R2 ,

D = −i∂,

(1)

S. Federico · A. Parmeggiani () Department of Mathematics, University of Ghent, Ghent, Belgium e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_13

207

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which is the (formal) adjoint of the important example of Kannai [18] K = P ∗ = Dx2 x1 Dx2 − iDx1 , of an operator which is hypoelliptic but not locally solvable at x1 = 0. The class of operators of the kind (1) was then extended by the adjoints of the class considered by Beals and Fefferman in [2] (in their study of hypoellipticity of degenerate operators) and next by Colombini, Cordaro and Pernazza in [4] (in their study of the solvability of operators of the form X(x, D)∗ f X(x, D)+iY (x, D)+a0 , where iX, iY are real vector-fields, f is real analytic and a0 is smooth). The class we consider contains operators of the kind (1) but also operators whose formal adjoint is not hypoelliptic (see [29]) so that the solvability is not a consequence of the hypoellipticity of the adjoint. The study of solvability of linear degenerate PDEs (even after Dencker’s resolution of the Nirenberg-Treves conjecture on condition (), see [16, 17]) is still largely open and unsettled, especially for operators not of principal type. Many are the examples, coming from several complex variables or linearization of nonlinear operators involved in physical and geometrical problems, of degenerate operators that are interesting to study. One may look at [20] for some history and basic problems on local solvability and at [29] for some history, survey, bibliography and considerations related to the solvability of degenerate operators along with some results (see also [28]) related to the solvability of operators with multiple transversal symplectic characteristics. It is important to keep in mind that the hypoellipticity of an operator P implies the local solvability of P ∗ (or tP ), thus the issue of local solvability is very much related to that of hypoellipticity and hence also to that of propagation of singularities (see, e.g., [7, 14, 16, 17, 21, 28, 29]). However, Kannai’s example shows that there are operators that are locally solvable but not hypoelliptic, and that the operation of taking adjoints may preserve local solvability but may also destroy hypoellipticity. Interesting solvability results for degenerate operators of the form P1 P2 + Q (where P1 , P2 , Q are first order operators) with double characteristics are given in a paper by Helffer [13] (in which he actually studies the problem of the hypoellipticity with a loss of 1 derivative) and by Treves [33] (in which he studies the solvability of an operator of the form X1 (x, D)X2 (x, D) + iY (x, D) + a0 , where iX1 , iX2 , iY are real vector fields, proving that under certain conditions one has solvability with of one derivative), and for operators of the form sums of squares N a loss ∗ X by Kohn [19], in which the vector fields involved are complex (see X j j =1 j also Treves [34] for the study of the solvability of vector fields with critical points). Furthermore, it is important to mention the recent work due to Dencker [5, 6] concerned with necessary conditions for the solvability of degenerate operators whose principal symbol may be complex but with a non-radial involutive doublecharacteristic set (that is, the characteristic points where the principal symbol and its differential vanish is a non-radial manifold which is involutive), based on the behavior of limit bicharacteristics and the so-called sub- condition that were introduced by Mendoza and Uhlmann in [23] (see also [22]), and the work of

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Müller (see [24, 25]) for operators whose principal symbol is complex with double characteristics, where in [25] the necessary conditions for solvability are described in terms of “dissipative pairs” (a condition related to the Hessian of the principal symbol only, at double characteristic points; hence the condition does not “see” the information carried by the other invariants one has at a double point) and in [24] he shows the sufficiency of such condition in the important instance of leftinvariant second order operators on the Heisenberg group (thus having a particular algebraic structure on the lower-order terms). For sums of squares of left-invariant vector fields on a Lie group, one has extensive work by Müller, Ricci and Peloso (see for instance [26, 27, 31]), for operators with double involutive characteristics one has an interesting result by Popivanov [32], and for the semi-global solvability of operators with transversal multiple symplectic characteristics one has the results by Parenti-Parmeggiani (see [28] and also [29]). We next introduce the class of operators we shall be considering here. The class is subdivided in three types, that will be described in the subsequent sections. The first kind of operators, which is a direct generalization of the class considered by Colombini, Cordaro and Pernazza, was introduced in [11] and studied also in [12]. Notice that an interesting and meaningful variation of it, with non-smooth coefficients and invariant under affine transformations, was studied by Federico in [8]. The other two kinds have been introduced in [12] and in [10]. Let  ⊂ Rn be open and let N ≥ 1 be an integer. We consider the following operators P1 =

N

Xj∗ f Xj + XN +1 + iX0 + a0 ,

(MT)

j =1

P2 =

N

Xj∗ fj Xj + XN +1 + a0 ,

(ST)

j =1

P3 =

N

Xj∗ fj Xj + XN +1 + iX0 + a0 ,

(MST)

j =1

where (MT) stands for “mixed type”, (ST) for “Schrödinger type” and (MST) for “mixed Schrödinger type”, respectively. The above operators are constructed from a given system (X0 , X1 , . . . , XN +1 ) of first order homogeneous partial differential operators Xj (x, D) (that we shall also call, somewhat improperly, “vector fields”; as a matter of fact, the iXj are indeed vector fields). The symbols of XN +1 and X0 will be always supposed to be real, whereas those of X1 , . . . , XN will be supposed to be real in the (MT) and (MST) cases, respectively, and complex in the (ST) case. We shall denote by Xj (x, ξ ) = αj (x), ξ  the symbols of the Xj , where αj ∈ C ∞ (; Rn ), 0 ≤ j ≤ N + 1, in the (MT) and (MST) cases, and αN +1 ∈ C ∞ (; Rn ) and αj ∈ C ∞ (; Cn ), 1 ≤ j ≤ N , in the (ST) case. The functions f, f1 , . . . , fN ∈ C ∞ (; R) are assumed to be smooth, and may vanish

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and/or change sign somewhere on , where in particular for f we assume that

f −1 (0) = ∅ and df f −1 (0) = 0. Finally a0 ∈ C ∞ (; C). Notice that a main difference among the operators P1 , P2 and P3 is in the symbol of the first order part (the subprincipal symbol sub(P )(x, ξ ), that at points (x, ξ ) such that Xj (x, ξ ) = 0 for all 1 ≤ j ≤ N is given by XN +1 (x, ξ ) + iX0 (x, ξ )): in the (MT) case sub(P1 ) is complex so that P1 is a sort of parabolic-Schrödinger operator, in the (ST) case sub(P2 ) is real so that P2 is a sort of degenerate Schrödinger operator, and in the (MST) case sub(P3 ) is again complex but with a degeneracy in the principal part which may depend on the various functions fj and may be thought of as a blend of the previous two types. Our interest in these classes, that are invariantly defined, comes from the interplay between the degeneracy due to the vanishing, and the (assumed or possible) change of sign, of the various f and fj involved, and the characteristic set  of the system of vector fields (X0 , X1 , . . . , XN ), defined as =

N 

j ⊂ T ∗  \ 0,

j = {(x, ξ ) ∈ T ∗  \ 0; Xj (x, ξ ) = 0}, 0 ≤ j ≤ N

j =0

(2) (recall that T ∗  \ 0 denotes the cotangent bundle of  with the zero-section removed). Note that in our setting  does not depend on the characteristics of XN +1 . The reason why this is the case will be made clear in the sequel. In the first, second and third section, respectively, we will state the hypotheses on the class of operators P1 , P2 and P3 , respectively, and state the related solvability results, explaining the main solvability estimates that give, in some cases, “better” solvability results (if compared to L2 to L2 local solvabiity, see Definition 2 below). For each class of operators we shall also give a number of examples. We remark once more that our classes of operators contain operators whose formal adjoint is not hypoelliptic (see [29]; see also [1, 35] for the study of hypoellipticity of degenerate operators whose coefficients may change sign) so that our solvability results are not a consequence of the hypoellipticity of the adjoints. We close the introduction by recalling the definition of local solvability and by  giving the definition of H s to H s local solvability we will be interested in, where H s = H s (Rn ) is the L2 -based Sobolev space of order s ∈ R, whose norm will be denoted by || · ||s . Definition 1 (Local Solvability) Let P be an mth-order partial differential operator with smooth coefficients on an open set  ⊂ Rn . We say that P is locally solvable at x0 ∈  if there exists a neighborhood V ⊂  of x0 such that for all v ∈ C ∞ () there is u ∈ D  () satisfying P u = v in V . 

Definition 2 (H s to H s Local Solvability) Let P be an mth-order partial differential operator with smooth coefficients on an open set  ⊂ Rn . Given s, s  ∈ R  and x0 ∈  we say that P is H s to H s locally solvable near x0 if there is a ˚ (the interior of K) such that for all v ∈ H s () compact K ⊂  with x0 ∈ K loc

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s  () with P u = v in K. ˚ We will call the number s − s  the there exists u ∈ Hloc  gain of smoothness (near x0 ) of the solution. We will say that P is H s to H s locally  solvable near V ⊂  if P is H s to H s locally solvable near x0 for all x0 ∈ V .  When one has H s to H s local solvability for all s ∈ R where s  = s + m − r, then one calls r the loss of derivatives.  ˚ of a Recall that to obtain the H s to H s local solvability in the interior K compact K, by the Hahn-Banach Theorem one needs to establish the a priori estimate

∃C > 0 such that

||ϕ||−s ≤ C||P ∗ ϕ||−s  ,

∀ϕ ∈ C0∞ (K).

Throughout the paper {·, ·} denotes the Poisson bracket and π : T ∗  \ 0 −→  the canonical projection. By (·, ·) we will denote the L2 -scalar product. Finally, given A, B ≥ 0 we will write A  B (or B  A) if there is C > 0 such that A ≤ CB.

2 The Mixed-Type Case In this section we introduce the following set of hypotheses on the operator P1 =

N

Xj∗ f Xj + XN +1 + iX0 + a0

j =1

of the kind (MT), where f ∈ C ∞ (; R), with f −1 (0) = ∅ and df f −1 (0) = 0. We will write dXj = −idiv(αj ) for the “divergence” of Xj . Notice that in this case the principal symbol of P1 is real and changing sign across f −1 (0). Hypotheses (HM1) to (HM5)

(HM1) iX0 f f −1 (0) > 0; (HM2) For all compact K ⊂  there exists C > 0 such that for all j = 1, . . . , N +1 {Xj , X0 }(x, ξ ) ≤ C 2

N

Xk (x, ξ )2 ,

∀(x, ξ ) ∈ K × Rn ;

(3)

k=0

(HM3) For all compact K ⊂  there exists C > 0 such that |(Im dX0 (x))XN +1 (x, ξ )| ≤ C

N 

Xk (x, ξ )2

1/2

,

∀(x, ξ ) ∈ K × Rn ;

k=0

(4)

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(HM4) For ρ ∈  (see (2)) let HXj (ρ) =

n 

∂Xj k=1

∂ξk

(ρ)

∂Xj ∂ ∂  − (ρ) ∂xk ∂xk ∂ξk

be the Hamilton vector-field of Xj at ρ, let V (ρ) := Span{HX0 (ρ), . . . , HXN (ρ)}, let J (ρ) ⊂ {0, . . . , N} be a set of indices for which {HXj (ρ)}j ∈J (ρ) is a basis of V (ρ), and let M(ρ) = [{Xj , Xj  }(ρ)]j,j  ∈J (ρ) be the r × r matrix of the Poisson brackets of the corresponding symbols Xj , Xj  , where r = card(J (ρ)). We say that hypothesis (HM4) is fulfilled at x0 ∈ f −1 (0) if π −1 (x0 ) ∩  = ∅ and rank M(ρ) ≥ 2,

∀ρ ∈ π −1 (x0 ) ∩ ;

(5)

(HM5) Let Lk (x) = SpanR {iX0 , . . . , iXN and their commutators up to length k at x} (recall that iXj has length 1 and [iXj , iXj  ] has length 2, and so on). We say that hypothesis (HM5) is fulfilled at x0 ∈ f −1 (0) if π −1 (x0 ) ∩  = ∅ and there exists k ≥ 1 such that dim Lk (x0 ) = n.

(6)

Remark 1 Note that if condition (5) holds at x0 then there is a neighborhood Vx0 of x0 such that the condition holds for all ρ ∈ π −1 (Vx0 ) ∩ . Since the subprincipal  ∗ symbol of N j =0 Xj Xj is zero (here the symbols Xj are real) one has (see [11]) that condition (5) amounts to Melin’s strong Tr+ condition (see [15]) N   sub Xj∗ Xj (ρ) + Tr+ FN

∗ j =0 Xj Xj

j =0

(ρ) > 0,

∀ρ ∈ π −1 (Vx0 ) ∩ ,

so that for all compact K ⊂ Vx0 one has the sharp Melin inequality [15]: There are constants cK , CK > 0 such that N N

( Xj∗ Xj u, u) = ||Xj u||20 ≥ cK ||u||21/2 − CK ||u||20 , j =0

∀u ∈ C0∞ (K).

(7)

j =0

Remark 2 Condition (6) yields the Rothschild-Stein sharp subelliptic estimate in a sufficiently small neighborhood Vx0 of x0 , that is, for any given compact K ⊂ Vx0 there exist cK , CK > 0 such that (

N

j =0

Xj∗ Xj u, u) =

N

j =0

||Xj u||20 ≥ cK ||u||21/k − CK ||u||20 ,

∀u ∈ C0∞ (K).

(8)

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One may prove that when hypothesis (HM4) is fulfilled then also hypothesis (HM5) is fulfilled with k = 2 (see Federico [9]). However, it is still interesting to distinguish the two cases, because of the fact that the subprincipal symbol and the positive trace are symplectic invariants of an operator with double characteristics. Notice also that having dim L1 (x0 ) = n is equivalent to saying that the system (X0 , X1 , . . . , XN ) is elliptic near x0 , that is, π −1 (Vx0 ) ∩  = ∅ for some neighborhood Vx0 of x0 , so that inequality (8) becomes the well-known Gårding inequality. For the class of operators (MT) we have the following solvability result (see [12]) near S := f −1 (0) (which is the region of interest for us). Theorem 1 Supposing hypotheses (HM1), (HM2) and (HM3), one has: (i) For all x0 ∈ S the operator P1 is L2 to L2 locally solvable at x0 ; (ii) If x0 ∈ S is such that π −1 (x0 ) ∩  = ∅ and (HM4) holds at x0 , then P1 is H −1/2 to L2 locally solvable at x0 ; (iii) If x0 ∈ S is such that π −1 (x0 ) ∩  = ∅ and (HM5) holds at x0 for some k ≥ 2, then P1 is H −1/k to L2 locally solvable at x0 ; (iv) If x0 ∈ S is such that π −1 (x0 ) ∩  = ∅, then P1 is H −1 to L2 locally solvable at x0 . Remark 3 Notice that the operator given in (1) falls in case (iv) of the theorem. Of course, we won’t be giving the proof of the theorem (which can be found in [12]). Instead, to explain the role of the assumptions we next recall the main estimate needed to prove the theorem, that is: For all sufficiently small compact K ⊂  with ˚ there are constants cK , CK > 0 such that x0 ∈ K 2 Re(P1∗ u, −iX0 u)

≥ cK

N

j =0

3 ||Xj u||20 + ||X0 u||20 − CK ||u||20 , 2

∀u ∈ C0∞ (K). (9)

A fundamental step to obtain the main estimate (9) is the use the Fefferman-Phong inequality for the operator ε,γ := P

N   1

ε Xj∗ Xj − [Xj , X0 ]∗ [Xj , X0 ] + Y, γ γ j =0

where   Y = −Re (Im dX0 )XN +1 , ε = ||f ||L∞ (K) → 0 when K 2 {x0 } and γ is an auxiliary parameter to be picked. Hence, hypotheses (HM1), (HM2) and (HM3) and the fact that x0 ∈ S allow one to choose K sufficiently small containing x0 (in its interior) so as to have, by then

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picking γ , the Fefferman-Phong estimate, that is, the existence of CK > 0 such that ε,γ u, u) ≥ −CK ||u||20 , (P

∀u ∈ C0∞ (K).

Such control allows one to bound from below Re(P1∗ u, −iX0 u), u ∈ C0∞ (K), by the right-hand side of (9), provided K is sufficiently small about x0 . Once the main estimate is obtained (which, remark, holds under the assumptions (HM1) to (HM3)), one gets, according to hypotheses (HM4), or (HM5), or π −1 (x0 ) ∩  = ∅, by virtue of the Melin, or the Rothschild-Stein, or the Gårding estimates, respectively, the control from below ||P1∗ u||20 ≥ c0 ||X0 u||20 + c1 ||u||2s − C2 ||u||20 ,

∀u ∈ C0∞ (K),

where c0 , c1 , C2 > 0 (depending on K) and where • s = 0 when only (HM1), (HM2) and (HM3) hold; • s = 1/2 when, in addition to the first three hypotheses, (HM4) holds; • s = 1/k when, in addition to the first three hypotheses, (HM5) holds for some k ≥ 2; • s = 1 when, in addition to the first three hypotheses, π −1 (x0 ) ∩  = ∅. One finally gets rid of the L2 -error terms by using the Poincaré inequality for the term ||X0 u||20 and by possibly shrinking further the compact K (keeping x0 in its interior). At last, the solvability estimate ||P1∗ u||20 ≥ CK (||u||2s + ||u||20 ),

∀u ∈ C0∞ (K),

is obtained and an application of the Hahn-Banach Theorem gives the result. We next give a few examples of operators P1 in the class (MT) for which we can conclude local solvability near quite degenerate points.

2.1 Example Let x = (x1 , x2 ) be coordinates in R2 , let g(x2 ) = 1+x22 , f (x) = x1 −(x2 +x23 /3). Let ) * 1 g(x2 ) A(x2 ) = . 1 1/g(x2 ) We have dim KerA(x2 ) = 1 for all x2 . Consider P1 =

2

j1 ,j2 =1

  Dj1 f (x)aj1 j2 (x2 )Dj2 + X3 + iX0 + a0 ,

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where X3 (x, ξ ) = μ1 (x)X(x, ξ ) + μ2 (x)X0 (x, ξ ), with X(x, ξ ) = g(x2 )ξ1 + ξ2 ,

X0 (x, ξ ) = αξ1 +

1 ξ2 , g(x2 )

where α > 1 is a constant and μ1 , μ2 are smooth real-valued functions. Then, putting X1 (x, ξ ) =

+

X(x, ξ ) g(x2 ) + , 1 + g(x2 )2

X2 (x, ξ ) = √

1 X(x, ξ ) + , g(x2 ) 1 + g(x2 )2

gives that P1 may be written in the form P1 =

2

Xj∗ f Xj + X3 + iX0 + a0 ,

j =1

where conditions (HM1), (HM2) and (HM3) hold, but (HM4) and (HM5) (including the case k = 1) do not, since X1 (x, ξ ), X2 (x, ξ ) and {X, X0 }(x, ξ ) are always proportional to X(x, ξ ). Therefore P1 is L2 to L2 locally solvable near f −1 (0).

2.2 Example The next example deals with a situation in which one has better (than L2 to L2 ) local solvability. Let x = (x1 , x2 , x3 ) ∈ R3 and let k ≥ 0 be an integer. Take f (x) = x2 , and the following system of vector fields X1 = Dx1 ,

X2 = x1k Dx3 ,

X3 = β(x)Dx1 ,

X0 = Dx2 ,

where β ∈ C ∞ (R3 ; R). Let P1 =

2

Xj∗ f Xj + X3 + iX0 + a0 .

j =1

Then it is readily seen that dX0 ≡ 0, that {Xj , X0 } = 0 for j = 1, 2 and that |{X0 , X3 }(x, ξ )|2  X1 (x, ξ )2 for all (x, ξ ) (locally for x in compact sets), so that

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hypotheses (HM1), (HM2) and (HM3) are fulfilled. We therefore have that P is H −1/(k+1) to L2 locally solvable near x2 = 0 with k + 1 given by • k + 1 = 1, that is in the case  = ∅ (whence π −1 (x0 ) ∩  = ∅ for all x0 ∈ f −1 (0)); • k + 1 = 2, that is in the case in which (HM4) is fulfilled; • k + 1 ≥ 2, that is in the case in which (HM5) is fulfilled.

2.3 Example In this example, we show that condition (HM4) might not always be satisfied at f −1 (0) so that the gain of derivatives may vary depending on the position of π −1 (x0 ), x0 ∈ f −1 (0), with respect to . Let x = (x1 , x2 , x3 ) ∈ R3 and let  ⊂ R3 be an open set such that  ∩ {x; x1 = −1} = ∅. Let ± := {x ∈ ; x1 ≷ −1} and f (x) = x2 + x23 /3 − x1 x3 . Introduce the following system of vector fields X1 (x, ξ ) = ξ1 − x3 ξ3 , X3 (x, ξ ) =

X2 (x, ξ ) = (1 + x1 )ξ3 ,

X0 (x, ξ ) = ξ2 − x1 ξ3 ,

2  

βj (x)Xj (x, ξ ) + γ (x){X0 , Xj }(x, ξ ) , j =0

where βj , γ ∈ C ∞ (; R). Let P1 =

2

Xj∗ f Xj + X3 + iX0 + a0 .

j =1

Since dX0 = 0 and {X1 , X0 } = −X2 ,

{X1 , X2 } = (2 + x1 )ξ3 ,

{X2 , X0 } = 0,

one has that hypotheses (HM1), (HM2) and (HM3) are satisfied. Therefore P1 is always L2 to L2 locally solvable near f −1 (0). However, since  = {(x, ξ ); ξ1 = x3 ξ3 , (1 + x1 )ξ3 = 0, ξ2 = x1 ξ3 , ξ = 0}, so that ξ3 = 0 and therefore also x1 = −1 when (x, ξ ) ∈ , it follows that  = {(x, ξ ); x1 = −1, ξ2 + ξ3 = 0, ξ1 = x3 ξ3 , ξ3 = 0}.

Solvability of Degenerate Operators

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At any given ρ = (x, ξ ) ∈  we have ⎤ 0 ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢1⎥ HX0 (ρ) = ⎢ ⎥ , ⎢ ξ3 ⎥ ⎢ ⎥ ⎣0⎦ 0 ⎡

⎤ 1 ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ −x3 ⎥ HX1 (ρ) = ⎢ ⎥, ⎢ 0 ⎥ ⎥ ⎢ ⎣ 0 ⎦ ξ3 ⎡

⎤ 0 ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0 ⎥ HX2 (ρ) = ⎢ ⎥, ⎢ −ξ3 ⎥ ⎥ ⎢ ⎣ 0 ⎦ 0 ⎡

whence dim V (ρ) = 3 (here J (ρ) = {0, 1, 2}), ⎤ ⎡ ⎤ ⎡ 0 0 0 0 0 X2 (x, ξ )   ⎥ ⎢ ⎥ ⎢ M(ρ) := {Xj , Xj  }(ρ) 0≤j,j  ≤2 =⎣−X2 (x, ξ ) 0 (2 + x1 )ξ3 ⎦=⎣0 0 ξ3 ⎦ , 0 −(2 + x1 )ξ3 0 0 −ξ3 0

has rank 2 for all ρ ∈  and condition (HM4) is fulfilled at x0 = π(ρ) (when π(ρ) ∈ f −1 (0)). Equivalently, condition (HM5) with k = 2 is fulfilled, for one has at x0 = (−1, x20 , x30 ) = π(ρ), ρ ∈ , L2 (x0 ) = SpanR {iX0 (x0 , D), iX1 (x0 , D), [iX1 , iX2 ](x0 , D)}  ∂ ∂ ∂ ∂ ∂  = R3 . + , − x30 , = SpanR ∂x2 ∂x3 ∂x1 ∂x3 ∂x3 In general π −1 (± ) ∩  = ∅, and when x0 = (−1, x20 , x30 ) then π −1 (x0 ) ∩  = ∅, so that we have the following different cases: • when x0 ∈ f −1 (0) ∩ ± then P1 is H −1 to L2 locally solvable near x0 (in fact, in this case we have π −1 (x0 ) ∩  = ∅); • when x0 = (−1, x20 , x30 ) ∈ f −1 (0) ∩  then π −1 (x0 ) ∩  = ∅ and (HM4) holds (equivalently, (HM5) with k = 2 holds), whence P1 is H −1/2 to L2 locally solvable near x0 .

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2.4 Example: A Mildly Complex Case We conclude the section by briefly discussing an example involving a complex operator of mixed type. In [11] we considered P =

N

Zj∗ f Zj + iZ0 + a0 ,

j =1

where Z1 , . . . , ZN have a complex symbol and Z0 has a real symbol. For it, under suitable assumptions, similar in nature to (HM1) through (HM5), we could give a −1 H −1/k to L2 local solvability result near We consider Nf (0).  it a mildly∗ complex case, because the subprincipal parts of j =1 Zj∗ Zj and of N j =1 [Zj , Z0 ] [Zj , Z0 ] N are in addition assumed (see [11]) to be controlled by k=0 |Zk |2 (and hence to  −1 vanish on N j =0 Zj (0)). For instance, an example of operator that can be considered is given by P = Z1∗ f Z1 + Z2∗ f Z2 + iZ0 + a0 , where, for x = (x1 , x2 , x3 , x4 ) ∈ R4 , Z1 (x, ξ ) = ξ1 + ix2k ξ3 ,

Z2 (x, ξ ) = eig(x1 ,x2 ) ξ2 ,

Z0 (x, ξ ) = ξ4 ,

with g ∈ C ∞ (R2x1 ,x2 ; R) is such that ∂g/∂x2 = 0 and f (x) = x4 + f˜(x1 , x2 , x3 ). Then P is H −1/(k+1) to L2 locally solvable near f −1 (0).

3 The Schrödinger-Type Case We next turn our attention to the Schrödinger-type case (ST). Recall that in this case P2 =

N

Xj∗ fj Xj + XN +1 + a0 ,

j =1

where f1 , . . . , fN ∈ C ∞ (; R) (notice that X0 ≡ 0). Notice that also in this case the principal symbol of P2 is real and may change sign. Here we make the following assumptions. Hypotheses (HS1) and (HS2) (HS1) The operators X1 , . . . , XN have complex coefficients; (HS2) For all x0 ∈  there exists a connected neighborhood Vx0 ⊂  of x0 and g ∈ C ∞ (Vx0 ; R) such that

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(i) Xj g = 0 on Vx0 , for all 1 ≤ j ≤ N ; (ii) XN +1 g = 0 on Vx0 . Remark 4 Note that, since we are interested in a degenerate setting where the vanishing of the functions fj has an interplay with the degeneracies of the operators X1 , . . . , XN appearing in the second order part of P2 , we do not assume any nondegeneracy conditions on the Xj for 1 ≤ j ≤ N (in the sense that iXj may have a critical point at some x, i.e. αj (x) = 0). At the same time, since we prove solvability by a priori estimates, we need to put a nondegeneracy condition somewhere on P2 , which is indeed placed on the first order part XN +1 (condition (HS2-(ii)). We also remark that, since the sets fj−1 (0) may be disjoint, the local solvability of P2 is studied at each point of  and not near any particular zero-set fj−1 (0). In this case we have the following result (see [12]). Theorem 2 In the above hypotheses the operator P2 is L2 to L2 locally solvable at any given x0 ∈ . Note that now the way the functions fj may degenerate in  is no longer important, neither does the characteristic set of the system (X0 = 0, X1 , . . . , XN ) play any role. The point now is to estimate, given x0 ∈ Vx0 ⊂  and a compact K ⊂ Vx0 containing x0 , the quantity Im(eλg P ∗ u, eλg u) for λ ≥ 1 large and u ∈ C0∞ (K). By the hypotheses, this is indeed possible. It is worth mentioning that the estimate follows from a general framework. In fact, let B : C0∞ (Vx0 ) −→ C0∞ (Vx0 ) be a 0th-order properly supported ψdo, such that B ∗ = B + R, where R is a smoothing operator. Then Im(P ∗ ϕ, Bϕ) =

N

j =1

1 Im(Xj ϕ, [fj , B]Xj ϕ) 2 N

Im(Xj ϕ, fj [Xj , B]ϕ)+

j =1

(10) +Im(ϕ, [XN +1 , B]ϕ) + O(||ϕ||20 ),

∀ϕ ∈

C0∞ (Vx0 ),

where in O(||ϕ||20 ) we gathered the contributions of [R, Xj ]ϕ, [R, XN +1 ]ϕ and of [B, dXN+1 ]ϕ. The first two terms on the right-hand side of (10) are delicate, in that we cannot control terms like ||Xj ϕ||0 and are able to control only the third term, and this is where we make use of assumption (HS2)–(ii), a reasonable choice of B being eλg . Then one obtains the estimate (c0 > 0)   ||dXN+1 ||L∞ (K) Im(eλg P ∗ u, eλg u) ≥ λc0 − −||a0 ||L∞ (K) ||eλg u||20 , 2

∀u ∈ C0∞ (K),

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whence by picking λ > 0 sufficiently large and by using the Cauchy-Schwarz inequality, one obtains the L2 solvability estimate ||P ∗ u||0  ||u||0 , for all u ∈ C0∞ (K). We next wish to give a few examples of operators in the class (ST) of the P2 .

3.1 Example In Rt × Rnx × Rm y we may consider the operators P2 = −x ± y + Dt , or P2 = f1 (t)x + f2 (t)y + Dt , where f1 , f2 are smooth, real-valued (and not identically zero). In all cases, P2 is L2 to L2 locally solvable.

3.2 Example This example is related to the so-called Mizohata structures. Let 0 ⊂ Rnx × Ry be open. Then  := Rt × 0 ⊂ Rt × Rnx × Ry is open. Take Q = Q(x) to be a real-valued quadratic form and let Xj = Dxj − i

∂Q (x)Dy , ∂xj

1 ≤ j ≤ n.

Let Y = Y (x, y, Dx , Dy ) be a first order homogeneous operator with real symbol and let XN +1 = Dt +Y. One immediately has that g = g(t) = t satisfies hypotheses (HS1) and (HS2) whence P2 =

n

Xj∗ fj Xj + XN +1 + a0

j =1

is L2 to L2 locally solvable near any given point of , regardless the choice of the (non indentically zero) functions fj ∈ C ∞ (; R) (and, of course, of a0 ∈ C ∞ (; C)).

3.3 Example Another example is the following. Let x = (x1 , x2 , x3 , x4 ) ∈ R4 and let  ⊂ R4 be open. Let X1 = Dx1 − i

x2 Dx3 , 2

X2 = Dx2 + i

x1 Dx3 , 2

X3 = Dx4 + α(x)Dx3 ,

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where α ∈ C ∞ (; R). We then choose g = g(x) = x4 and have that, whatever the (non indentically zero) functions f1 , f2 ∈ C ∞ (; R) (and of course a0 ∈ C ∞ (; C)) the operator P2 = X1∗ f1 X1 + X2∗ f2 X2 + X3 + a0 is L2 to L2 locally solvable near any given point of . Remark 5 It may be interesting to think of the operators P2 appearing in Examples 3.2 and 3.3 as evolution operators in the direction iXN +1 associated with the involutive/hypoanalytic structure (see [3]) on the leaves g −1 (c), spanned by the system (iX1 , . . . , iXN ) tangential to g −1 (c) (c near some regular value of g).

4 The Mixed-Schrödinger-Type Case We finally consider the case (MST), that is, recall, an operator of the kind P3 =

N

Xj fj Xj + XN +1 + iX0 + a0 ,

j =1

where the novelty with respect to the class (MT), where we had only one f and had the presence of X0 , and with respect to the class (ST), where we had many smooth real-valued fj but no X0 , lies in the fact that we may now allow the presence of many fj and a non-zero X0 . Notice that also in this case the principal symbol of P3 is real and may change sign. The hypotheses to deal with such a case are the following. Hypotheses (HMS1), (HMS2) and (HMS3) (HMS1) X0 = 0 throughout  (i.e. iX0 has no critical points), and iX0 fj ≥ 0 on , 1 ≤ j ≤ N ; (HMS2) {X0 , Xj }(x, ξ ) = 0 for all (x, ξ ) ∈ T ∗  and all j = 1, . . . , N; (HMS3) For all x0 ∈  there exists a compact K ⊂  with non-empty interior containing x0 , and a positive constant CK such that for all (x, ξ ) ∈ T ∗ K |{X0 , XN +1 }(x, ξ )|2 ≤ CK

N   (iX0 fj (x))Xj (x, ξ )2 + X0 (x, ξ )2 . j =1

Remark 6 In this case, as well as in the (ST) case above, the presence of several possibly vanishing functions fj (which may have nonintersecting zeros) in  motivates the study of the local solvability of P3 at any point of . This explains why our conditions are given on  and, especially, why the nondegeneracy condition (HMS1) is required on  (and not on any particular fj−1 (0)). In this regard, note that in the present (MST) case we assume iX0 fj ≥ 0 on  for all 1 ≤ j ≤ N and not iX0 f |f −1 {0} > 0 as in the (MT) case.

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In this case we have the following result (see [10]). Theorem 3 Suppose hypotheses (HMS1) to (HMS3) hold. Then P3 is L2 to L2 locally solvable near any given x0 ∈ . The main point, as before, is to estimate from below the quantity   Re P ∗ u, −iX0 u ,

u ∈ C0∞ (K).

One obtains the main estimate, with cK , CK > 0, 2Re(P ∗ u, −iX0 u) ≥ (P0 u, u) + cK ||X0 u||20 − CK ||u||20 ,

u ∈ C0∞ (K),

where P0 =

N

[iX0 , Xj∗ fj Xj ] + X02 − ![X0 , XN +1 ]∗ [X0 , XN +1 ]

(11)

j =1

(! > 0 is small depending on a compact K0 containing K), which then satisfies the Fefferman-Phong inequality (by virtue of hypotheses (HMS1) to (HMS3)), so that one gets the inequality (c, C > 0) ||P ∗ u||20 ≥ c||X0 u||20 − C||u||20 ,

u ∈ C0∞ (K).

Then using, as in the (MT) case, the Poincaré inequality for the term ||X0 u||20 makes it possible to get rid of the term −C||u||20 (it is here that one exploits hypothesis (HMS1) and has possibly to shrink the compact set K keeping x0 in its interior, process that does not change the above constants c, C > 0), whence the L2 to L2 local solvability estimate ||P ∗ u||20  ||u||20 ,

∀u ∈ C0∞ (K).

Notice that requiring stronger assumptions on the vector fields yields stronger properties of P0 , whence the possibility of exploiting the Gårding inequality, or the Melin inequality, or the Rothschild-Stein inequality to obtain improved solvability results (i.e. with a better gain of derivatives).

4.1 Example Consider in Rn , n ≥ 3, the operator P3 = x1 (D12 −D22 )+i(D1 +D2 )+X3 (x, D) = D1 x1 D1 −D2 x1 D2 +iD2 +X3 (x, D),

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223

with X3 (x, D) = g1 (x)D1 + g2 (x)D2 +

n

gj (x)Dj ,

j =3

where the gj , 1 ≤ j ≤ n, are smooth real-valued, and where g1 and gj , 3 ≤ j ≤ n, are independent of x2 . In this case X0 = D2 ,

X1 = D1 ,

X2 = D2

f1 (x) = f2 (x) = x1 .

Since we then have {X0 , X3 }(x, ξ ) = −i

∂g2 (x)X0 (x, ξ ), ∂x2

{X0 , X1 } = {X0 , X2 } = 0,

for such an operator conditions (HMS1) to (HMS3) are satisfied and P is L2 to L2 locally solvable in Rn .

4.2 Example Consider next the operator in Rnx × Rt P3 =

M

Dj xjm1 Dj ±

j =1

n

Dk xkm2 Dk + ig(t)Dt +

k=M+1

n

gh (x)Dh ,

h=1

where 1 ≤ M ≤ n − 1, m1 , m2 ≥ 1, and g and the gh , 1 ≤ h ≤ n, are smooth real-valued functions, with g nowhere vanishing. Once more, since in this case (here N = n) X0 = g(t)Dt ,

XN +1 =

n

gh (x)Dh ,

h=1

conditions (HMS1) to (HMS3) are satisfied and P is L2 to L2 locally solvable in Rn × R.

4.3 Example In this final example one sees that there are cases in which an operator of the kind (MST) can be locally solvable with a better gain of derivatives. Consider in fact in R2 the operator P3 = D1 x1 D1 − D2 x2 D2 + i(D1 − D2 ) + x2 D1 = x1 D12 − x2 D22 + x2 D1 ,

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in which case X1 = D1 , X2 = D2 , X0 = D1 − D2 , X3 = x2 D1 , f1 (x) = x1 , f2 (x) = − x2 . In this case the associated operator P0 (see (11)) is P0 = D12 + D22 + (D1 − D2 )2 − !D12 , whence for 0 < ! < 1 the operator P0 is elliptic and one can use the Gårding inequality in place of the Fefferman-Phong inequality, thus obtaining an H −1 to L2 local solvability result near any given point of R2 .

5 Concluding Remarks There is a number of problems raised by the study of this class of degenerate operators. Among them, we wish to mention the following two. Problems 1 For the operators considered here one should study whether given any s ∈ R one has H s to H s+2−r local solvability, where r is the loss of derivatives. Of course, this problem is very difficult because the operator is very degenerate and even microlocalization gives lower order terms that may be too big to control. It might very well be the case that the Sobolev regularity s cannot range freely in R but that there might be thresholds due to the kernel of P ∗ . 2 One should study semi-global solvability (see [16]) for these operators, and for that one needs to understand the propagation of singularities. One should expect that things abruptly change, depending on the different classes (mixedtype, Schrödinger-type, mixed-Schrödinger-type, respectively), depending on the  ∗ X when hitting the sets f −1 (0), behavior of the bicharacteristics of N X j =0 j j f1−1 (0), . . . , fN−1 (0). In addition, as we saw in Example 2.3 of Sect. 2, the gain of regularity may wildly change depending on the position of π −1 (S) with respect to  (recall that S = f −1 (0)), whence an approach based on the  propagation of Sobolev ∗ microlocal regularity along the bicharacteristics (of P or N j =0 Xj Xj ) might be the appropriate one. A first important step in this direction was taken by Parenti and Rodino in [30], where they studied the hypoellipticity and microlocal hypoellipticity of a class of anisotropic operator in terms of the Lascar anisotropic wave-front set.

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References 1. T. Akamatsu. Hypoellipticity of a second order operator with a principal symbol changing sign across a smooth hypersurface. J. Math. Soc. Japan 58 (2006), 1037–1077. 2. R. Beals, C.. Fefferman. On hypoellipticity of second order operators. Comm. Partial Differential Equations 1 (1976), 73–85. 3. S. Berhanu, P. Cordaro, J. Hounie. An introduction to involutive structures. New Mathematical Monographs, 6. Cambridge University Press, Cambridge, 2008. xii+392 pp. 4. F. Colombini, P. Cordaro, L. Pernazza. Local solvability for a class of evolution equations. J. Funct. Anal. 258 (2010), 3469–3491. 5. N. Dencker. Solvability and limit bicharacteristics. J. Pseudo-Differ. Oper. Appl. 7 (2016), no. 3, 295–320, doi: https://doi.org/10.1007/s11868-016-0164-x 6. N. Dencker. Operators of subprincipal type. Anal. PDE 10 (2017), no. 2, 323–350, doi: https:// doi.org/10.2140/apde.2017.10.323 7. J. J. Duistermaat, L. Hörmander. Fourier integral operators, II. Acta Math. 128 (1972), 184–269. 8. S. Federico. A model of solvable second order PDE with non smooth coefficients. Journal of Mathematical Analysis and Applications 440 (2016), no. 2, 661–676, doi: https://doi.org/10. 1016/j.jmaa.2016.03.056 9. S. Federico. Local Solvability of a Class of Degenerate Second Order Operators. Ph.D. Thesis, University of Bologna (2017) 10. S. Federico. Sufficient conditions for local solvability of some degenerate pdo with complex subprincipal symbol To appear in Journal of Pseudo-Differential Operators and Applications (2018), doi: https://doi.org/10.1007/s11868-018-0264-x 11. S. Federico, A. Parmeggiani. Local solvability of a class of degenerate second order operators. Comm. Partial Differential Equations 41 (2016), no. 3, 484–514, doi: https://doi.org/10.1080/ 03605302.2015.1123273 12. S. Federico, A. Parmeggiani. Local solvability of a class of degenerate second order operators. Comm. Partial Differential Equations 43 (2018), no. 10, 1485–1501, doi: https://doi.org/10. 1080/03605302.2018.1517789 13. B. Helffer. Construction de paramétrixes pour des opérateurs pseudodifférentiels caractéristiques sur la réunion de deux cônes lisses. Bull. Soc. Math. France Suppl. Mém. No. 51–52 (1977), 63–123. 14. L. Hörmander. Propagation of singularities and semiglobal existence theorems for (pseudo)differential operators of principal type. Ann. of Math. 108 (1978), 569–609. 15. L. Hörmander. The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators. Grundlehren der Mathematischen Wissenschaften 274. Springer-Verlag, Berlin, 1985. viii+525 pp. 16. L. Hörmander. The analysis of linear partial differential operators. IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften, 275. Springer-Verlag, Berlin, 1985. vii+352 pp. 17. L. Hörmander. Unpublished manuscripts-from 1951 to 2007. Springer, Cham, 2018. x+357 pp. 18. Y. Kannai, An unsolvable hypoelliptic differential operator. Israel J. Math. 9 (1971), 306–315. 19. J. J. Kohn. Loss of derivatives. From Fourier analysis and number theory to radon transforms and geometry, 353–369, Dev. Math., 28, Springer, New York, 2013. 20. N. Lerner. When is a pseudo-differential equation solvable? Ann. Inst. Fourier 50 (2000), 443–460. 21. N. Lerner. Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, 3. Birkhäuser Verlag, Basel, 2010. xii+397 pp. 22. G. A. Mendoza. A necessary condition for solvability for a class of operators with involutive double characteristics. In Microlocal analysis (Boulder, Colo., 1983), 193–197, Contemp. Math., 27, Amer. Math. Soc., Providence, RI, 1984.

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Small Data Solutions for Semilinear Waves with Time-Dependent Damping and Mass Terms Giovanni Girardi

Abstract We consider the following Cauchy problem for a wave equation with time-dependent damping term b(t)ut and mass term m(t)2 u, and a time-dependent non-linearity h = h(t, u): " utt − u + b(t)ut + m2 (t)u = h(t, u), u(0, x) = f (x),

t ≥ 0, x ∈ Rn ,

ut (0, x) = g(x).

Here, we consider an effective time-dependent damping term and a time-dependent mass term, in the case in which the mass is dominated by the damping term, i.e. m(t) = o(b(t)) as t → ∞. Under suitable assumptions on the non-linearity h = h(t, u) (Hypothesis 1.3), we prove the global existence of small data solutions in a supercritical range p > p, ¯ assuming small data in the energy space (f, g) ∈ H 1 × L2 . Keywords Critical exponent · Effective damping · Damped Klein-Gordon equation

1 Introduction In this paper, we consider the Cauchy problem for the dissipative wave equation with mass term " utt − u + b(t)ut + m2 (t)u = h(t, u), t ≥ 0, x ∈ Rn , (1) u(0, x) = f (x), ut (0, x) = g(x),

G. Girardi () University of Bari, Bari, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_14

227

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G. Girardi

where b(t)ut and m(t)2 u, with b(t), m(t) > 0, respectively represent a damping and a mass term. We will establish sufficient conditions on the non linearity h = h(t, u) to guarantee the existence of C([0, ∞), H 1 ) ∩ C 1 ([0, ∞), L2 ) solution to (1) for small initial data in the space H 1 × L2 . Previous papers concern with the classical semilinear problem, i.e. with the case h(t, u) = h(u) and h(u) verifies h(0) = 0,

|h(u) − h(v)|  |u − v| (|u| + |v|)p−1

(2)

for a given p > 1. In particular, in [1, 3, 6] the model without mass has been considered "

utt − u + b(t)ut = h(t, u),

u(0, x) = f (x),

t ≥ 0, x ∈ Rn ,

ut (0, x) = g(x),

(3)

in the case h(t, u) = h(u) = |u|p . Here, under the assumption of effectiveness of the damping term (see later Hypothesis 1.1), the authors prove that the critical exponent for global (in time) small data solutions to (3) remains the same as in the case b ≡ 1 (see [10–12, 15, 18, 21]). In particular, it is proved that the critical exponent is p¯ = 1 + 2/n if the initial data are assumed to be small in exponentially weighted energy spaces, under suitable sign assumption for the data [3]. The same result holds in space dimension n = 1, 2 if smallness of the data is assumed only in H 1 × L2 and in L1 . If also the additional L1 smallness is dropped, then the critical exponent becomes 1 + 4/n. In this paper, effectiveness of the damping term means that for a suitable large class of damping coefficients b(t), the estimates obtained for (3) are the same obtained for the solution to the corresponding Cauchy problem for the heat equation " b(t)vt − v = 0, v(0, x) = ϕ(x),

t ≥ 0, x ∈ Rn ,

(4)

for suitable initial data ϕ = ϕ(f, g, b) (see [20]). In the case of polynomial shape b(t) = μ(1 + t)k , the damping is effective if k ∈ (−1, 1] (see [14, 16, 19] for the corresponding global existence result). If b(t) = μ(1 + t)−1 we say that the damping term is of scale-invariant type. It is proved that in this case the critical exponent remains p¯ = 1 + 2/n if μ is large (see [2, 19]); instead the critical exponent seems to increase to max{pS (n + μ), 1 + 2/n} if μ becomes small, as conjectured in [4, 7], (see also [9, 13]) where pS is the Strauss exponent for the semilinear undamped wave equation.

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In [17] the Cauchy problem (1) is studied in the scale-invariant case b(t) = μ1 (1 + t)−1 and m(t) = μ2 (1 + t)−1 , with power non-linearity |u|p . In [5] the same Cauchy problem (1) is considered, in the case in which the damping term is effective and it dominates the mass term. In particular, the following assumptions are supposed for the coefficient b = b(t): Hypothesis 1.1 We assume that b ∈ C2 , with b(t) > 0, is monotone and it has controlled oscillations:



(k)

k = 1, 2. (5)

b (t) ≤ C b(t)(1 + t)−k , and that tb(t) → ∞,

as t → ∞,

1 ∈ L1 (R+ ). b(t)(1 + t)2

(6) (7)

Moreover, we assume that b verifies tb (t) ≤ ab(t),

for some a ∈ [0, 1),

(8)

being this latter trivially satisfied if b < 0. By virtue of Hypothesis 1.1, the function  B(t, s) = s

t

1 dτ b(τ )

is positive, monotone increasing with respect to t and decreasing with respect to s and, for any fixed s ≥ 0, it holds B(t, s) → ∞, as t → ∞. For the sake of brevity, we fix B(t) = B(t, 0). In [5] we also assume that the influence of the damping term dominates the influence of the mass term in the equation, so that the presence of the mass has a minor influence on the profile of the solution, which is mainly determined by the effective action of the damping term on the wave equation. Hypothesis 1.2 We assume that m ∈ C1 , with m(t) > 0, has controlled oscillations:



m (t) ≤ C m(t) , 1+t

(9)

and that m(t) = o(b(t)),

as t → ∞.

(10)

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Here, we show that under a simple condition on the interaction between b(t) and m(t), and assuming only small initial data in the energy space, one may find a scale of critical exponents, which continuously move from 1 + 4/n to 1, as the mass becomes more influent, with respect to the damping term. In particular, in that case for any small initial data (f, g) ∈ H 1 × L2 we obtain the following estimate for the solution to the non linear Cauchy problem (1): u(t, ·) L2 ≤ C γ (t) (f, g) H 1 ×L2 , where we define   γ (t) = exp −

t 0

m2 (τ ) dτ . b(τ )

(11)

Thus, the decreasing function γ = γ (t) in (11) describes the influence on the estimates of the mass term with respect to the damping term. In order to study the influence of the term γ (t) on the critical exponent for global in time small data energy solutions to (1), in [5] we define the parameter β = lim inf B(t) m(t)2 .

(12)

t→∞

The critical exponent for (1) only depends on this parameter β ∈ [0, ∞]. On the other hand, in [8] we consider the case in which the influence of the mass term dominates the influence of the damping term, i.e. lim inf (m(t)/b(t)) > 1/4. t→∞

Under this hypothesis, assuming the effectiveness of the damping term b(t)ut , we obtain an exponential decay rate for the solution to the linear Cauchy problem associated to (1) and so we are able to prove global existence (in time) of small data solutions to the Cauchy problem (1) for any p > 1. In this note we suppose that the non linear term h = h(t, u) satisfies the following assumption. Hypothesis 1.3 We assume that there exist p > 1 and ω ∈ [−1, ∞) such that h(t, 0) = 0;

|h(t, u) − h(t, v)| ≤ (1 + B(t, 0))ω |u − v|(|u| + |v|)p−1 .

By using the same approach used in [5] we want to prove the following result: Theorem 1 Let n ≥ 1, and assume that b = b(t) satisfies Hypothesis 1.1 and m = m(t) satisfies Hypothesis 1.2. Let β be as in (12). Moreover, assume β > −1 + n/4 + ω(n/2 − 1), if n ≥ 3. Under these assumptions, for any p > pβ (n), where " pβ (n) =

1+ 1

4(1+ω) n+4β

if β ∈ [0, ∞), if β = ∞,

(13)

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and p ≤ 1 + 2/(n − 2) if n ≥ 3, there exists !0 > 0 such that for any initial data (f, g) ∈ H 1 × L2 with (f, g) H 1 ×L2 ≤ !0 , there exists a unique energy solution u ∈ C([0, ∞), H 1 )∩C1 ([0, ∞), L2 ) to Cauchy problem (1). Moreover, it satisfies the following estimates: u(t, ·) L2 ≤ C(1 + B(t))−α (f, g) H 1 ×L2 , −α− 21

∇u(t, ·) L2 ≤ C(1 + B(t)) ut (t, ·) L2 ≤ C

(14)

(f, g) H 1 ×L2 ,

(15)

m(t)2 + (1 + B(t))−1 (1 + B(t))−α (f, g) H 1 ×L2 , b(t)

(16)

for any α < β if β ∈ (0, ∞], and with α = 0 if β = 0. Remark 1 Notice that in the special case in which γ (t) ≈ B(t)−β for some β ∈ (0, ∞), estimates (14), (15), (16) also hold for α = β. Remark 2 The presence of the time dependent coefficients in the non-linear term h = h(t, u), has only influence on the critical exponent. Whereas, the decay estimates for the solutions to (1) remain the same as in the case h(u) = |u|p obtained in [5]. It is well-known (see [1, 5, 6]) that, due to the diffusion phenomenon, the use of an additional L1 smallness of the initial data should lead to obtain an extra decay rate for the solution to (1); this allows to prove the global existence of global (in time) small data solutions for each p > pβ,1 (n), where " pβ,1 (n) =

1+ 1

2(1+ω) n+2β

if β ∈ [0, ∞), if β = ∞

.

This critical exponent is consistent with the corresponding result for m = 0 (see [1]). However, even in the case b = 1 and m = 0, the use of the smallness of initial data in the energy space and in L1 leads to a technical limit; in fact, if no further assumptions are taken for the data, it is necessary to assume p ≥ 2, but this is consistent with the critical exponent (1), only if β ≤ (1 + ω) − n/2. Having this in mind, for η ∈ [1, 2] we introduce the function space     Aη = Lη (Rn ) ∩ H 1 (Rn ) × Lη (Rn ) ∩ L2 (Rn ) , with norms (f, g) Aη = f Lη (Rn ) + f H 1 (Rn ) + g Lη (Rn ) + g L2 (Rn ) .

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By using the same approach used in these note one can prove the following result: Theorem 2 Let n ≥ 1, η ∈ [1, 2), and assume that b = b(t) satisfies Hypothesis 1.1 and m = m(t) satisfies Hypothesis 1.2. Let β be as in (12). Under these assumptions, for any p ≥ 2/η such that p > pβ,η (n), " pβ,η (n) =

1+

2η(1+ω) n+2ηβ

if β ∈ [0, ∞),

(17)

if β = ∞.

1

and p ≤ 1 + 2/(n − 2) if n ≥ 3, there exists !0 > 0 such that if (f, g) Aη ≤ !0 , then there exists a unique energy solution u ∈ C([0, ∞), H 1 ) ∩ C1 ([0, ∞), L2 ) to Cauchy problem (1). Moreover, it satisfies the following estimates: − n2



u(t, ·) L2 ≤ C(1 + B(t))

− n2

∇u(t, ·) L2 ≤ C(1 + B(t)) ut (t, ·) L2 ≤ C



1 1 η−2

 −α

(f, g) Aη ,

 1 1 1 η − 2 −α− 2

(18)

(f, g) Aη ,

m(t)2 + (1 + B(t))−1 −n (1 + B(t)) 2 b(t)



(19)  1 1 η − 2 −α

(f, g) Aη , (20)

for any α < β if β ∈ (0, ∞], and with α = 0 if β = 0. Remark 3 Notice that in the case β = 0 it holds pβ,η = 1 + 2η(1 + ω)/n, that is the same critical exponent obtained in [1] for the same Cauchy problem (1) with m = 0. Remark 4 The non existence of a solution to the Cauchy problem (1) can be investigated by considering the corresponding non linear Cauchy problem for the heat equation " b(t)vt − v + m2 (t)v = h(t, v), t ≥ 0, x ∈ Rn , v(0, x) = ϕ(x).

(21)

If ϕ(x) ∈ Lη , one can prove that for any b(t) as in Hypothesis 1.1 and m(t) as in Hypothesis 1.2, the exponent pβ,η (n) in Theorem 2 is critical for (21). In fact, one can prove the global existence of small data solutions for p > pβ,η (n), by using the same approach used to prove Theorem 2. For p < pβ,η (n), by using a modified test function method (see [3, 5]), one can prove that there exists arbitrarily small datum ϕ(x) ∈ Lη , such that there exists no global solution to (21).

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Due to the diffusion phenomenon, we expect that under a suitable sign assumption, global (in time) Sobolev solutions with Lη data may also be excluded for (1), when p < pβ,η (n), verifying the optimality of the exponent found for the global (in time) existence of small data solutions in this paper.

2 The Linear Estimates Let us denote by u lin = u lin (t, x) the solution to the linear Cauchy problem ⎧ lin lin lin 2 lin ⎪ ⎪ ⎨utt − u + b(t)ut + m (t)u = 0, lin u (0, x) = f (x), ⎪ ⎪ ⎩u lin (0, x) = g(x),

t ≥ 0, x ∈ Rn , (22)

t

and let v = v(t, s, x) the solutions to the family of parameter-dependent Cauchy problems ⎧ 2 ⎪ ⎪ ⎨vtt − v + b(t)vt + m (t)v = 0, v(s, s, x) = 0, ⎪ ⎪ ⎩v (s, s, x) = g(s, x),

t ≥ s, (23)

t

where s ≥ 0. Thanks to the Duhamel’s principle, the solution to (1) may be written has  t v(t, s, x)ds, u(t, x) = u lin (t, x) + 0

if we take g(s, x) = f (s, u(s, x)) as initial data for (23). One can derive the following useful results concerning the linear Cauchy problems (22) and (23), under the assumptions Hypothesis 1.1 for b = b(t) and Hypothesis 1.2 for m = m(t). Lemma 1 (Theorem 1 in [5]) Let n ≥ 1 and (f, g) ∈ H 1 × L2 . Then, u lin = u lin (t, x) satisfies the following decay estimates: 1 1 1 lin 1 1u (t, ·)1 2 ≤ C γ (t) (f, g) H 1 ×L2 , L 1 1 1 1 1 lin 1∇u (t, ·)1 2 ≤ C γ (t) (1 + B(t))− 2 (f, g) H 1 ×L2 , L

1 1 1 1 lin 1ut (t, ·)1

L2

≤ C γ (t)

m(t)2 + (1 + B(t))−1 (f, g) H 1 ×L2 , b(t)

where the constant C > 0 does not depend on the data.

(24) (25) (26)

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Lemma 2 (Lemma 3.1 in [5]) Let n ≥ 1 and g ∈ L2 . Then, the solution v = v(t, s, x) to (23), satisfies the following estimates: v(t, s, ·) L2 ≤ C

1 γ (t) g(s, ·) L2 , b(s) γ (s)

(27)

1 γ (t) 1 g(s, ·) L2 , (1 + B(t, s))− 2 (28) b(s) γ (s)  m(t)2 1 2 −1 γ (t) g(s, ·) L2 + b(t)(1 + b(t) B(t, s)) ≤C b(s) b(t) γ (s) (29)

∇v(t, s, ·) L2 ≤ C vt (t, s, ·) L2

where the constant C > 0 is independent of s. We notice that  γ (t) = exp − γ (s)



t

s

m2 (τ )  dτ . b(τ )

In order to prove Theorem 2, we will also make use of the following easy statement (which motivates the choice of the use of the parameter β in (12)). Lemma 3 Let β > 0 be as in (12), and fix α ∈ (0, β). Then there exists a constant C = C(α) > 0 such that, for any 0 ≤ s ≤ t, it holds   exp −

t s

m2 (τ ) dτ b(τ )



≤ C B(t)−α B(s)α .

We notice that B(t) ≥ B(s), so that the right-hand term is well-defined. Proof For any α ∈ (0, β) there exists t0 = t0 (α) such that m(τ )2 α ≥ , b(τ ) b(τ )B(τ )

∀τ ≥ t0 .

If s ≤ t ≤ t0 , the statement trivially follows by the compactness of [0, t0 ], for C ≥ C1 (t0 ). If t ≥ s ≥ t0 , the statement follows for C ≥ 1 by integrating the previous inequality, so that   exp −

t s

m2 (τ ) dτ b(τ )



  ≤ exp −α

t s

1 dτ b(τ )B(τ )



= B(t)−α B(s)α .

If s < t0 < t the statement follows by combining the previous arguments.

 

Remark 5 It is worth noticing that by Theorem 1 and Lemma 1, applying Lemma 3, it follows that the solution to the nonlinear Cauchy problem (1) satisfies the same estimates of the solution to the linear Cauchy problem (22).

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2.1 Examples The easiest class of coefficients b(t) and m(t) which can be considered are of polynomial type. Example 1 Let b(t) = μ(1 + t)k ,

m(t) = ν(1 + t) ,

for some μ, ν > 0 and k,  ∈ R. Then Hypothesis 1.1 holds if, and only if, k ∈ (−1, 1). In this case, B(t) =

(1 + t)1−k − 1 . μ(1 − k)

In particular, 1 + B(t) ≈ (1 + t)1−k . On the other hand, Hypothesis 1.2 holds if, and only if,  < k. Moreover, • if  < (k − 1)/2, then γ does not vanish as t → ∞, so that β = 0 in Theorem 1; • if  = (k − 1)/2, then   t ν2 ν2 γ (t) = exp − dτ = (1 + t)− μ , 0 μ(1 + τ ) that is, β = ν 2 /(μ(1 − k)) and estimates (18)–(19) also hold for α = β; • if  ∈ ((k − 1)/2, k), then Theorem 2 holds with β = ∞. Example 2 Let b(t) = μ(1 + t)k (log(e + t))a ,

m(t) = ν(1 + t) (log(e + t))b ,

for some μ,  > 0, and k, , a, b ∈ R. Then Hypothesis 1.1 holds if, and only if, either k ∈ (−1, 1), or k = −1 and a > 0, and we get B(t) =

(1 + t)1−k (log(e + t))−a (1 + o(1)) . μ(1 − k)

On the other hand, Hypothesis 1.2 holds if, and only if, either  < k, or  = k and b < a. Although the explicit computation of γ is not hard, it is quicker to apply the definition in (12): • if either  < (k − 1)/2 or  = (k − 1)/2 and 2b < a, then β = 0 in (12); • if  = (k − 1)/2 and a = 2b, then (12) holds with β = ν 2 /(μ(1 − k)); • if either  ∈ ((k − 1)/2, k) or  = (k − 1)/2 and a < 2b, then (12) holds with β = ∞. In particular, if  = (k − 1)/2 and 2b ∈ [a − 1, a), then γ vanishes as t → ∞, but the dissipation is too weak to modify the critical exponent for (1).

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3 Proof of Theorem 1 A function u = u(t, x) solves the Cauchy problem (1) in a suitable Sobolev solution space X if and only if u(t, x) = u lin (t, x) + (N u)(t, x), where u lin is the solution to the Cauchy problem (22), and  t Nu(t, x) = (t, s, ·) ∗(x) h(s, u(s, ·))(x)ds, 0

in X, where by (t, s, ·) ∗(x) h(s, u(s, ·))(x) we denote the solution to the Cauchy problem (23) with f = 0 and g(s, ·) = h(s, u(s, ·)). To prove Theorems 1, we will rely on a standard contraction argument in the solution space C1 ([0, ∞), H 1 ) × C([0, ∞), L2 ), equipped with a suitable norm, defined accordingly to the decay estimates for the solutions to the corresponding linear Cauchy problems with vanishing right-hand side obtained in Lemma 2. For any T > 0, we define the Banach spaces X0 (T ) = C([0, T ], H 1 ),

X(T ) = X0 (T ) ∩ C 1 ([0, T ], L2 ).

We will fix a suitable norm on X(T ) such that u lin X(T ) ≤ C (f, g) H 1 ×L2 .

(30)

Then we will prove that p

N u X(T ) ≤ C u X0 (T ) , N u − N v X(T )

 p−1 p−1  ≤ C u − v X(T ) u X0 (T ) + v X0 (T ) ,

(31) (32)

with a constant C > 0, independent of T . From condition (31) it follows that N maps X0 (T ) into X(T ). Due to the inequalities (31) and (32), we can apply the Banach’s fixed point theorem to prove that there exists a uniquely determined solution to Cauchy problem (1), in X(T ), provided that (f, g) H 1 ×L2 in (30) is sufficiently small. Since the constants in (30), (31), (32) do not depend on T , the solution is global (in time). We prove Theorem 1. Proof If β > 0 in (12), due to p > pβ (n), there exists α ∈ (0, β) such that p > pα (n). If β = 0, we fix α = 0.

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To prove Theorem 1, we equip X0 (T ) and X(T ) with the norms u X0 (T ) = sup

0≤t≤T



1

(1 + B(t))α u(t, ·) L2 + (1 + B(t))α+ 2 ∇u(t, ·) L2

u X(T ) = u X0 (T ) + sup

0≤t≤T





 (1 + B(t))α b(t)(m(t)2 + (1 + B(t))−1 )−1 ut (t, ·) L2 .

By Lemma 1, we derive (30). As a consequence of Gagliardo-Nirenberg inequality, any function u ∈ X0 (T ) verifies the inequality − n2

u(τ, ·) Lq ≤ C (1 + B(τ ))



1 1 2−q



−α

u X0 (T ) ,

(33)

for any τ ∈ [0, T ], for any q ∈ [2, ∞) if n = 1, 2 and for any q ∈ [2, 2n/(n − 2)] if n ≥ 3. Let j = 0, 1. Then  t1 1 1 1 1 j 1 1 1 j 1∇ (t, s, ·) ∗(x) h(s, u(s, ·))1 2 ds. 1∇ N u(t, ·)1 2 ≤ L

L

0

By Lemma 2, and using Lemma 3, we get  t 1 1 j 1 1 j 1 ∇ (1 + B(t, s))− 2 B(t)−α B(s)α h(s, u(s, ·)) L2 ds. N u(t, ·) ≤ C 1 1 2 L b(s) 0 Using |h(t, u)|  (1 + B(t))ω |u|p and (33), noticing that 2p ≤ 2n/(n − 2) if n ≥ 3, we may estimate n

h(s, u(s, ·)) L2  (1 + B(s))ω u(s, ·) L2p  (1 + B(s))− 4 (p−1)−pα+ω u X0 (T ) . p

p

(34) For short time, say for t ≤ 2, it is clear that 1 1 1 j 1 p 1∇ Nu(t, ·)1 2  u X0 (T ) . L

Let t ≥ 2. To estimate the integral terms we use the following properties for B(t, s) (see [6]): • B(t, s) ≈ B(t) in [0, 2t ], • B(s, 0) ≈ B(t) in [ 2t , t], as well as B(t) ≈ 1 + B(t). So, we get 1 1 1 j 1 1∇ N u(t, ·)1

 p − j2 −α u  (1 + B(t)) X0 (T ) 2

L

t/2

n 1 (1 + B(s))−( 4 +α )(p−1)+ω ds b(s) 0  t j n 1 p (1 + B(t, s))− 2 ds. + u X0 (T ) B(t)−( 4 +α )(p−1)−α+ω t/2 b(s)

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By using the change of variable ρ = B(s) in the first integral and ρ = B(t, s) in the second one, we obtain: 

t/2

0



t t/2

n 1 (1 + B(s))−( 4 +α )(p−1)+ω ds ≤ C, b(s) j j 1 (1 + B(t, s))− 2 ds  B(t/2)1− 2 . b(s)

where we used p > pα (n) in the first integral, and j/2 < 1 in the second one. Using B(t/2) ∼ 1 + B(t) for t ≥ 2, and p > pα (n) once again, we derive 1 1 1 1 j 1∇ Nu(t, ·)1

j

 u X0 (T ) (1 + B(t))− 2 −α . p

L2

We now estimate the term with the time derivative:  t ∂t (t, s, ·) ∗ h(s, u(s, ·)) L2 ds. ∂t Nu(t, ·) L2 ≤ 0

By Lemma 2, and using Lemma 3, we get  ∂t N u(t, ·) L2 ≤ C

t 0

  1  m(t)2 +b(t)(1+b(t)2 B(t, s))−1 B(t)−αB(s)α h(s, u(s, ·)) L2 ds. b(s) b(t)

Clearly, we may estimate m(t)2 b(t)



t

0

1 m(t)2 p B(t)−α B(s)α h(s, u(s, ·)) L2 ds  (1 + B(t))−α u X0 (T ) , b(s) b(t)

as we did in the previous step for j = 0. Therefore we restrict ourselves to estimate 

t

I (t) = 0

1 b(t)(1 + b(t)2 B(t, s))−1 B(t)−α B(s)α h(s, u(s, ·)) L2 ds. b(s) p

For short time, say for t ≤ 2, it is clear that I (t)  u X0 (T ) . Let t ≥ 2. Now we get  p ∂t Nu(t, ·) L2  u X0 (T ) b(t) (1 + b(t)2 B(t))−1 (1 + B(t))−α  n p + u X0 (T ) b(t) B(t)−( 4 +α )(p−1)−α+ω

0

t

t/2

t/2

n 1 (1 + B(s))−( 4 +α )(p−1)+ω ds b(s)

1 (1 + b(t)2 B(t, s))−1 ds b(s)

 u X0 (T ) b(t) (1 + b(t)2 B(t))−1 (1 + B(t))−α p

n

+ u X0 (T ) b(t) B(t)−( 4 +α )(p−1)−α+ω p

1 log(1 + b(t)2 B(t)). b(t)2

Semilinear Klein-Gordon Equation with Effective Damping

239

In the second integral, we used the change of variable ρ = b(t)2 B(t, s). By using 1 + b(t)2 B(t) ≈ b(t)2 B(t) (due to b(t)2 B(t) ≈ tb(t) → ∞), and controlling the logarithmic term with an arbitrarily small power (we also use (8)), this leads to p

I (t)  u X0 (T )

1 (1 + B(t))−1−α . b(t)

Summarizing, we proved that p

Nu X(T ) ≤ C u X0 (T ) , with C independent of T . This concludes the proof of (31). We proceed similarly to prove (32). In particular, we replace (34) by 1 1 1 1 h(s, u(s, ·)) − h(s, v(s, ·)) L2  (1 + B(s))ω 1|u(s, ·) − v(s, ·)| (|u(s, ·)|p−1 + |v(s, ·)|p−1 )1  (1 + B(s))ω u(s, ·) − v(s, ·) L2p

L2

1 1 1 1 1|u(s, ·)|p−1 + |v(s, ·)|p−1 1

L2p



n  p−1 p−1   (1 + B(s))− 4 (p−1)−pα+ω u − v X0 (T ) u X (T ) + v X (T ) . 0 0

This concludes the proof.

 

References 1. M. D’Abbicco, Small data solutions for semilinear wave equations with effective damping, Discrete and Continuous Dynamical Systems, Supplement 2013, 183–191. 2. M. D’Abbicco, The Threshold of Effective Damping for Semilinear Wave Equations, Mathematical Methods in Appl. Sci., 38 (2015), no. 6, 1032–1045, http://dx.doi.org/10.1002/mma. 3126 3. M. D’Abbicco, S. Lucente, A modified test function method for damped wave equations, Adv. Nonlinear Studies, 13 (2013), 867–892. 4. M. D’Abbicco, S. Lucente, NLWE with a special scale-invariant damping in odd space dimension, Discr. Cont. Dynamical Systems, AIMS Proceedings, 2015, 312–319, http://dx. doi.org/10.3934/proc.2015.0312. 5. M. D’Abbicco, G. Girardi, M. Reissig: A scale of critical exponents for semilinear waves with time-dependent damping and mass terms, Nonlinear Analysis, 179, 15–40 (2019), https://doi. org/10.1016/j.na.2018.08.006. 6. M. D’Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping, Chinese Ann. Math., 34B (2013), no. 3, 345–380, http://dx.doi.org/10.1007/s11401-013-07730; 7. M. D’Abbicco, S. Lucente, M. Reissig, A shift in the critical exponent for semilinear wave equations with a not effective damping, J. Differential Equations, 259 2015, 5040–5073, http:// dx.doi.org/10.1016/j.jde.2015.06.018. 8. G. Girardi, Semilinear damped Klein-Gordon models with time-dependent coefficients, New Tools for Nonlinear PDEs and Application, 2019, 203–216, https://doi.org/10.1007/978-3030-10937-0_7

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9. M. Ikeda, M. Sobajima, Life-span of solutions to semilinear wave equation with timedependent critical damping for specially localized initial data, Mathematische Annalen, 372 (2018), no. 3, 1017–1040. 10. R. Ikehata, Y. Mayaoka, T. Nakatake, Decay estimates of solutions for dissipative wave equations in RN with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), no. 2, 365–373. 11. R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in RN , J. Math. Anal. Appl., 269 (2002), 87–97. 12. T.T. Li, Y. Zhou, Breakdown of solutions to u + ut = |u|1+α , Discrete and Continuous Dynamical Systems 1 (1995), 503–520. 13. N.A. Lai, H. Takamura, K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent, J. Differential Equations, 263 (2017), 5377– 5394. 14. J. Lin, K. Nishihara, J. Zhai, Critical exponent for the semilinear wave equation with timedependent damping, Discrete and Continuous Dynamical Systems, 32 (2012), no. 12, 4307– 4320, http://dx.doi.org/10.3934/dcds.2012.32.4307. 15. A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS. 12 (1976), 169–189. 16. K. Nishihara, Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Timedependent Damping, Tokyo J. Math., 34 (2011), 327–343. 17. W. N. do Nascimento, A. Palmieri, M. Reissig, Semi-linear wave models with power nonlinearity and scale invariant time-dependent mass and dissipation, Math. Nachr. 290 (2017), 1779–1805. 18. G. Todorova, B. Yordanov, Critical Exponent for a Nonlinear Wave Equation with Damping, J. Differential Equations, 174 (2001), 464–489. 19. Y. Wakasugi, Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients, J. Math. Anal. Appl., 447 (2017), 452–487. 20. J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations, 232 (2007), 74–103. 21. Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109–114.

Integrating Gauge Fields in the ζ -Formulation of Feynman’s Path Integral Tobias Hartung and Karl Jansen

Abstract In recent work by the authors, a connection between Feynman’s path integral and Fourier integral operator ζ -functions has been established as a means of regularizing the vacuum expectation values in quantum field theories. However, most explicit examples using this regularization technique to date, do not consider gauge fields in detail. Here, we address this gap by looking at some well-known physical examples of quantum fields from the Fourier integral operator ζ -function point of view. Keywords ζ -Regularization · Feynman path integral · Gauge fields

1 Introduction Feynman’s path integral [12, 13] is a fundamental building block of modern quantum field theory. For instance, the time evolution semigroup (U (t, s))t,s∈R≥0 of a quantum field theory is a semigroup of integral operators whose kernels are given by the path integral. In terms of the Hamiltonian H of a given quantum field theory, U is the semigroup generated by − hi¯ H , i.e., U formally satisfies U (t, s) =   t Texp − hi¯ s H (τ )dτ where Texp is the time-ordered exponential for unbounded operators as to be understood in terms of the time-dependent Hille-Yosida Theorem (e.g., Theorem 5.3.1 in [31]). Furthermore, the path integral is intimately connected to vacuum expectation values which play two very crucial roles. On one hand, vacuum expectation values are physical and allow for experimental

T. Hartung () Department of Mathematics, King’s College London, London, UK e-mail: [email protected] K. Jansen NIC, DESY Zeuthen, Zeuthen, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_15

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verification and thus to test theories. On the other hand, vacuum expectation values of n field operators (so called n-point functions) uniquely determine the quantum field theory by Wightman’s Reconstruction Theorem (Theorem 3–7 in [38]). Let us consider a quantum field theory with Hilbert space H and time evolution semigroup U . Then, the vacuum expectation value A of an observable A can be expressed as A =

lim

T →∞+i0+

tr (U (T , 0)A) tr U (T , 0)

(∗)

where the denominator tr U (T , 0) is also known as the partition function. Upon closer inspection however, (∗) reveals one of the major mathematical obstacles. The traces on the right hand side of (∗) should be the canonical trace on trace-class operators S1 (H) but for a continuum theory U (T , 0) is a bounded, non-compact operator and U (T , 0)A is in general an unbounded operator on H. Vacuum expectation values are thus only generally understood in terms of discretized quantum field theories. This is the starting point of lattice quantum field theory for instance and great computational effort is necessary to extrapolate the continuum limit from these discretized vacuum expectation values. If we wish to understand (∗) in the continuum however, the traces need to be constructed in such a way that they coincide with the canonical trace on S1 (H) provided U (T , 0), U (T , 0)A ∈ S1 (H). One such trace construction technique are operator ζ -functions. They were introduced by Ray and Singer [34, 35] for pseudo-differential operators and first proposed as a regularization method for path integrals in perturbation theory by Hawking [20]. The Fourier integral operator ζ -function approach generalizes the pseudo-differential framework to non-perturbative settings with general metrics (Euclidean and Lorentzian) and includes special cases like Lattice discretizations in a Lorentzian background. Given an operator A and a trace τ for which we want to define τ (A), we construct a holomorphic family A such that A(0) = A and there exists a maximal open and connected subset  of C for which A maps  into the domain of τ . In general, we construct A such that  contains a half-space C6(·)0 e

r z+N r dr



i R>0 e

2π T X

r z+N −1 r dr

N −1

N  N   i 2πXT r z+N −1 N vol∂B e r dr N N ∈N R R>0

 −N z−N 2 −1 N  N −1 i 2π T N vol∂B (z + N + 1)(z + N ) N N ∈N R X N    2 −N z−N  2π N vol∂BRN N i 2π T (z + N ) N ∈N X X

 = lim lim

z→0 T →∞

shows that we have not completely ζ -regularized since the series might not be convergent for sufficiently small 6(z). Instead we need to introduce a regularization for the summation over N as well. For instance, let  N z 2π T N z z  z αN := (z + N + 1)−1 (z + N)−N i N vol∂BRN . X 0 = 1 and we obtain Then ∀N ∈ N : αN

Hn ζ  = lim lim

z→0 T →∞

Ś

 N N ∈N R



Ś

z i 2πXT N∈N αN e



N N ∈N R

N n=1

1 1 1 1 1ξN,n 1 CN 1ξN,m 1z dξ m=1 N 1 CN 1 1ξN,m 1z dξ n=1 ξN,n

ξN,n 2π N

z i 2πXT N∈N αN e

X

n=1

m=1

 −N 2 −1 + N )−1 i 2πXT = lim lim  −N 2   z→0 T →∞  1+z vol∂B N N(1+z) (z + N + 1)−1 i 2π T N∈N N R X I JK L    N(1+z)  2π N 1+z vol∂BRN (z N∈N X

∈O

=0

which coincides with the Hn .

1 T

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249

Regarding Hζ , let G be a gauged Fourier integral operator such that G(0) = 1. iT π Then, e− 6X eiT Hn G(0) = eiT Hζ and   iT π ζ e− 6X eiT Hn GHζ   (z) Hζ ζ = lim lim π z→0 T →∞ ζ e− iT 6X e iT Hn G    iT π   π e− 6X ζ eiT Hn GHn − 6X ζ eiT Hn G (z) = lim lim   iT π z→0 T →∞ e− 6X ζ eiT Hn G π =Hn ζ − 6X π . implies Hζ ζ = Hζ  = − 6X

2.2 The N → ∞ Particle Limit Alternatively, we can consider the Hamiltonian Hn≤N =

N



∂j

j =1

N in the “up to D N particle

! Hilbert D space” L2 ((R/XZ) ) where |P0 , . . . , Pk−1  is

embedded as j ∈k Pj ⊗ j ∈N −k |0. Physically, taking the limit N → ∞ says that we are only considering states that have finitely many particles. The ζ -regularized vacuum energy is then computed as

lim Hn≤N ζ

N →∞

= lim lim lim

 RN

ei

N →∞ z→0 T →∞

= lim lim lim

2π X

N →∞ z→0 T →∞



N

N →∞ z→0 T →∞

= lim lim lim

2π T X

2π N X

N

CN z n=1 ξn n=1 ξn dξ N CN z n=1 ξn n=1 ξn dξ

n=1 ξn 2π

N

X

ei CN

RN

2π T X



i 2πXT ξ ξ z+δm,n n=1 m=1 RN e N C N  i 2πXT ξ ξ z dξ n=1 m=1 RN e

 RN

ei

2π T X

dξ

 N −1 2π T ξ z+1 dξ RN ei X ξ ξ z dξ  N 2π T N RN ei X ξ ξ z dξ ξ

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= lim lim lim

2π X

 R>0

ei

2π T X



N →∞ z→0 T →∞

= lim lim lim

r z+N r dr R>0

2π X (z

N →∞ z→0 T →∞

ei



2π T X

R>0

ei

2π T X

r z+N −1 r dr

r z+N −1 r dr

N −1

N

 −N z−N 2 −1 + N + 1)(z + N)N −1 i 2πXT  −N z−N 2 (z + N)N i 2πXT

=0 in this setting.

3 Free Complex Scalar Quantum Fields Complex scalar fields are generalizations of real scalar fields which allow for the creation of antiparticles. More precisely, in a real scalar field the particle is its own antiparticle. The distinction between particles and antiparticles for the complex scalar field becomes obvious once they are quantized. Writing a complex scalar field +iϕ2 ψ = ϕ1√ as the sum of two real scalar fields ϕ1 and ϕ2 with creation operators 2

b† and c† , and expanding the field operator as a sum of planar waves yields (x) =

+

  1 bp eipx + cp† e−ipx 2XEp

+

  1 bp† e−ipx + cp eipx 2XEp

p∈M

 † (x) =

p∈M

on R/XZ where M = 2π X Z \ {0} is the set of momenta. This furthermore implies the conjugate momentum +(x) =

p∈M

+ (x) = †

p∈M

B i

 Ep  † −ipx ipx b e − c e p p 2XN B

(−i)

 Ep  ipx † −ipx b . e − c e p p 2XN

 If we consider the charge operator Q = i +(x)(x)− ∗ (x)+∗ (x)dx we directly obtain

Q= cp cp† − bp† bp p∈M

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251

which is not normally ordered. The normally ordered charge Qn is thus given by Qn =

cp† cp − bp† bp .

p∈M

This again can be explained using a ζ -argument and the commutator relation  [cp , cp† ] = 1. More precisely, we need to ζ -regularize the series p∈M 1 which is nothing other than tr id on R/XZ. Since id has no critical degree of homogeneity, tr id := ζ (z → |∇|z )(0) exists and is a well-defined constant (in fact, it is 2ζR (0) = −1 where ζR is the Riemann ζ -function), i.e., Qζ = Qn + tr id = Qn − 1. As for the Hamiltonian, we repeat the same calculation we did in the real case but with Lagrangian ∂ μ ϕ ∗ ∂μ ϕ instead of ∂ μ ϕ∂μ ϕ and obtain the normally ordered Hamiltonian

Ep (bp† bp − cp† cp ) Hn = p∈ 2π X Z

which differs from the ζ -regularized Hamiltonian by a constant again. This also shows the interesting effect that antiparticles appear with negative energy in the theory which allows us to reproduce the Feynman-Stückelberg interpretation of antiparticles. Considering the wave propagator under time-reversal exp(itHn )  exp(−itHn ) we obtain an algebraically equivalent theory with reversed roles for bp and cp . In other words, antiparticles are particles that move backwards in time and creation and annihilation of particle-antiparticle pairs can be seen as a particle reversing the direction it travels through time. In any case, the negative energies yield the up to N particle and N anti-particle Hamiltonian Hn≤N =

N



 

∂1,j − ∂2,j

j =1

on L2 ((R/XZ)2N ) which directly implies that limN →∞ Hn≤N ζ = Hn ζ = 0 since the degrees of homogeneity are identical to the ones in the real scalar field case.

4 The Dirac Field The free Dirac field is closely related to the complex scalar field but we are now considering spinor valued fields, assume that the creation and annihilation operators satisfy the canonical anticommutator relations, and possibly introduce a

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mass term m. Hence, our fields on the spatial torus (R/XZ)N are (x) =



:

p∈M s∈S

 † (x) =

p∈M s∈S

:

1 2XN Ep 1 2XN Ep

  bps usp eipx + cps† vps e−ipx   −ipx s† ipx bps† us† e + c v e p p p

where S is the set of spins, M the set of momenta, and u and v are spinors, i.e., they satisfy (i) (γ μ pμ − m)usp = 0 (ii) (γ μ pμ + m)vps = 0 r† s s rs (iii) ur† p up = vp vp = 2Ep δ r† s s (iv) ur† p v−p = vp u−p = 0

+ where (pμ )μ = (Ep , p)T , Ep = p, p + m2 , and the γ -matrices are given   1 0 0 σk in the Dirac basis γ 0 = with the Pauli matrices and γ k = −σ k 0 0 −1    01 0 −i 1 0 σ1 = , σ2 = , and σ 3 = . Plugging everything into the 10 i 0 0 −1 Dirac Hamiltonian density  † γ 0 (−iγ j ∂j + m) and integrating then yields H =



Ep (bps† bps + cps cps† )

p∈M s∈S

and cps cps† = 1 − cps† cps yields the normally ordered Hamiltonian Hn =



Ep (bps† bps − cps† cps ).

p∈M s∈S

For m = 0 this is precisely the same situation we had for the complex scalar field just with an additional summation over spins. For m > 0 we still have the question whether we can normally order the Hamiltonian using a ζ -argument again. + In other words, we need to ζ -regularize the trace of an operator with kernel ξ 2 + m2 but for ξ > m we observe the asymptotic expansion :

ξ 2

+ m2

=

1 2

j ∈N0

j

ξ 1−2j m2j

Integrating Gauge Field in the ζ -Path Integral

253

which has a degree of homogeneity −N if and only if N is odd. In particular,  12  N +1 the residue trace is given by N+1 m vol∂BRN . Hence, ζ -regularization fails to 2

normally order this Hamiltonian. However, this is no problem in the light of vacuum expectation values as we are taking quotients of ζ -functions. Hence, the presence of poles simply means that the GA) value of ζζ(U (U G) (0) is given by the quotient of residues rather than the quotient of constant Laurent coefficients.

5 Coupling a Fermion of Mass m to Light in 1 + 1 Dimensions Coupling light to matter in 1 + 1 dimensions is one of the text-book examples of ζ -regularization in the physical literature because it is a toy model for QED. In particular, the Schwinger model which has m = 0 has been studied extensively (cf. e.g. [21]). Here, we will show how the well-known applications of ζ -regularization tie into the framework of ζ -regularized vacuum expectation values as discussed in [16–18, 22]. In order to consider coupling a fermion to a gauge field (Aμ )μ , we will restrict our considerations to a fermion in 1 + 1 dimensions with a constant background field. This ignores the self-interaction of the gauge field which gives an additional term to the Hamiltonian that has already been discussed in [18, 22]. In the present case, and using the temporal gauge A0 = 0, A := A1 , the (fermionic coupling) Hamiltonian on R/XZ is given by  HF =

X

(x)† ((i∂ − eA)σ3 + m) (x)dx

0

 1 0 where e is the coupling constant, σ3 = , and  is the spinor field which we 0 −1 endow with anti-periodic boundary conditions (x + X) = −(x) (this is allowed because  is an auxiliary field; all physical quantities are composed of sesquilinear forms in  which are periodic). To study this system, we will first expand  into eigenmodes of (i∂ −eA)σ3 +m, i.e., we are looking to solve   ψ+ ψ+ + (i∂ − eA)σ3 =ε − m 0 0

 and

0 (i∂ − eA)σ3 ψ−



= −ε− − m

These imply   x 1 ± A(y)dy − i(ε ∓ m)x ψ (x) = √ exp −ie X 0 ±

0 . ψ−

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where e−iπ −2iπ n (x) = −(x) = (x + X) implies that ε± has to satisfy  − iπ − 2iπ n − ie

x

A(y)dy − i(ε± ∓ m)x

0



x

= − ie

A(y)dy − i(ε± ∓ m)x − ie

0



X

A(y)dy − i(ε± ∓ m)X.

0

In other words, the eigenvalues are given by ∀n ∈ Z :

εn±

M M  e A e A 2π 2π 1 π + n− ±m= n + ± mX − := X X X X 2 2π

M M X e A where A := 0 A(y)dy. For brevity, we will write C ± := 2π ∓ mX. First quantization of , then introduces annihilation operators an and bn for the upper and lower components of  with {am , an† } = {bm , bn† } = δm,n and  is given by

(x) =

ψ + (x)an n∈Z

n ψn− (x)bn

    1 exp −ie 0x A(y)dy − i(εn+ − m)x an x   =√ . − X n∈Z exp −ie 0 A(y)dy − i(εn + m)x bn

In particular, this implies  X

HF = (x)† ((i∂ − eA)σ3 + m) (x)dx = (εn+ an† an − εn− bn† bn ). 0

n∈Z

At this point, we will split our considerations into the positive (an ) and negative (bn ) chirality sectors. The positive sector has the Hamiltonian H+ := n∈Z εn+ an† an and  † chiral charge Q+ := n∈Z an an . Since there is no minimum energy, we define + the N -vacuum of the positive chirality sector by filling all states with energies εn+ where n < N + .  + To compute the N + -chiral charge Q+ N + = n∈Z 0 only have different estimates in the closed subsectors S θ  —but the mere analyticity in Sθ is unaffected by the translation by ωI . One therefore has the following improved version of [5, Prop. 1]: Proposition 2 If a C0 -semigroup etA of type (M, ω) on a complex Banach space B has an analytic extension ezA to Sθ for θ > 0, then ezA is injective for every z ∈ Sθ .

Well-Posed Final Value Problems

263

Proof Let ez0 A u0 = 0 hold for some u0 ∈ B and z0 ∈ S. Analyticity of ezA in Sθ carries over by the differential calculus in Banach spaces to the map f (z) = ezA u0 . So for z in a suitable open ball B(z0 , r) ⊂ Sθ , a Taylor expansion and the identity f (n) (z0 ) = An ez0 A u0 for analytic semigroups (cf. [24, Lem. 2.4.2]) give f (z) =

∞ ∞

1 1 (z − z0 )n f (n) (z0 ) = (z − z0 )n An ez0 A u0 ≡ 0. n! n! n=0

(12)

n=0

Hence f ≡ 0 on Sθ by unique analytic extension. Now, as etA is strongly continuous, u0 = limt→0+ etA u0 = limt→0+ f (t) = 0. Thus the null space of ez0 A is trivial.   Remark 1 The injectivity in Proposition 2 was claimed by Showalter [27] for z > 0, θ ≤ π/4 and B a Hilbert space (with a flawed proof, as noted in [5, Rem. 1], cf. details on the counter-example in Lemma 3.1 and Remark 3 in [17]). A local version for the Laplacian on Rn was given by Rauch [25, Cor. 4.3.9]. As a consequence of the above injectivity, for an analytic semigroup etA we may consider its inverse that, consistently with the case in which etA forms a group in B(B), may be denoted for t > 0 by e−tA = (etA )−1 . Clearly e−tA maps its domain D(e−tA ), which is the range R(etA ), bijectively onto H , and it is in general an unbounded, but closed operator in B. ∗ Specialising to a Hilbert space B = H , then also (etA )∗ = etA is analytic, so its ∗ tA −tA null space Z(e ) = {0} by Proposition 2, whence D(e ) is dense in H . A partial group phenomenon and commutation properties are restated here: Proposition 3 ([5, Prop. 2]) The above inverses e−tA form a semigroup of unbounded operators in H , e−sA e−tA = e−(s+t)A

for t, s ≥ 0.

(13)

This extends to (s, t) ∈ R× ]−∞, 0], where e−(t+s)A may be unbounded for t +s > 0. Moreover, as unbounded operators the e−tA commute with esA ∈ B(H ), i.e., esA e−tA ⊂ e−tA esA

for t, s ≥ 0,

(14)



and have a descending chain of domains, D(e−t A ) ⊂ D(e−tA ) ⊂ H for 0 < t < t  . The above domains serve as basic structures for the final value problem (1).

1.3 The Abstract Final Value Problem The basic analysis is made for a Lax–Milgram operator A defined in H from a V elliptic sesquilinear form a(·, ·) in a Gelfand triple, i.e., in a set-up of three Hilbert

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spaces V → H → V ∗ having the norms · , | · | and · ∗ , respectively. Hereby V is the form domain of a. Specifically there are constants Cj such that, for all u, v ∈ V , one has v ∗ ≤ C1 |v| ≤ C2 v and |a(u, v)| ≤ C3 u v and 6a(v, v) ≥ C4 v 2 . In fact, D(A) consists of those u ∈ V for which a(u, v) = ( f | v ) for some f ∈ H holds for all v ∈ V ; then Au = f . It is also recalled that there is a bounded bijective extension A : V → V ∗ given by Au, v = a(u, v) for u, v ∈ V . (The reader may consult [11, Ch. 12], [13] or [5] for more details on the set-up and basic properties of the unbounded, but closed operator A in H .) In this framework, the general final value problem is as follows: for given data f ∈ L2 (0, T ; V ∗ ) and uT ∈ H , determine the u ∈ D (0, T ; V ) such that "

∂t u + Au = f u(T ) = uT

in D (0, T ; V ∗ ), in H.

(15)

By definition of the vector distribution space D (0, T ; V ∗ ), cf. [26], the above equation means that u, −ϕ   + Au, ϕ = f, ϕ holds as an identity in V ∗ for every scalar test function ϕ ∈ C0∞ (]0, T [). A wealth of parabolic Cauchy problems with homogeneous boundary conditions have been treated via triples (H, V , a) and the D (0, T ; V ∗ ) set-up in (15); cf. the work of Lions and Magenes [21], Tanabe [28], Temam [29], Amann [2]. To compare (15) with the Cauchy problem for u + Au = f obtained from the initial condition u(0) = u0 ∈ H , for some f ∈ L2 (0, T ; V ∗ ), it is recalled that there is a unique solution u in the Banach space   X =L2 (0, T ; V ) C([0, T ]; H ) H 1 (0, T ; V ∗ ),   T  T 1/2 2 2 u(t) dt + sup |u(t)| + ( u(t) 2∗ + u (t) 2∗ ) dt . u X = 0

0≤t≤T

0

(16) For (15) it would thus be natural to envisage solutions u in the same space X. This turns out to be true, but only under substantial further conditions on the data (f, uT ). To formulate these, it is exploited that A = −A generates an analytic semigroup e−zA in B(H ), where z ∈ Sθ for θ = arccot(C3 /C4 ). This is classical, but crucial for the entire analysis ([5, Lem. 4] has a verification of (i), hence of (ii), in Proposition 1). By Proposition 2, it therefore has the inverse (e−tA )−1 = etA for t > 0. Its domain is the Hilbert space D(etA ) = R(e−tA ) with u = (|u|2 + tA |e u|2 )1/2 . In [5, Prop. 11] it was shown that a non-empty spectrum, σ (A) = ∅, yields strict inclusions, as one could envisage, 

D(et A )  D(etA )  H

for 0 < t < t  .

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Well-Posed Final Value Problems

265

This follows from the injectivity of e−tA , using some well-known result for semigroups that may be found in [24]; cf. [5, Thm. 11] for details. For t = T these domains enter decisively in the well-posedness result below, where condition (20) is a fundamental clarification for the class of final value problems (15). But it also has important implications for parabolic differential equations. Another ingredient is the full yield yf of the source term f , namely  yf =

T

e−(T −s)A f (s) ds.

(18)

0

Hereby it is used that e−tA extends to an analytic semigroup in V ∗ , as the extension A ∈ B(V , V ∗ ) is an unbounded operator in the Hilbertable space V ∗ satisfying the necessary estimates; cf. [5, Lem. 4]. Hence yf is a priori a vector in V ∗ , but it belongs to H in view of (16), as it is the final state of a solution of the Cauchy problem with u0 = 0. Moreover, the Closed Range Theorem implies, cf. [5, Prop. 5], that the operator f → yf is a continuous surjection yf : L2 (0, T ; V ∗ ) → H . These remarks on yf make it clear that the difference in (20) is meaningful in H : Theorem 1 Let A be a V -elliptic Lax–Milgram operator defined from a triple (H, V , a) as above. Then the abstract final value problem (15) has a solution u(t) belonging the space X in (16), if and only if the data (f, uT ) belong to the subspace Y ⊂ L2 (0, T ; V ∗ ) ⊕ H

(19)

defined by the condition 

T

uT −

e−(T −t)A f (t) dt ∈ D(eT A ).

(20)

0

In the affirmative case, the solution u is uniquely determined in X and 



T

u X ≤ c |uT | + 2

0

f (t) 2∗ dt



T A uT − + e

T 0



2  12 e−(T −t)A f (t) dt

= c (f, uT ) Y . (21) whence the solution operator (f, uT ) → u is continuous Y → X. Moreover, u(t) = e−tA eT A (u(T ) − yf ) +



t

e−(t−s)A f (s) ds,

0

where all terms belong to X as functions of t ∈ [0, T ].

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The norm on the data space Y in (21) is seen at once to be the graph norm of the composite map eT A



L2 (0, T ; V ∗ ) ⊕ H −−−−→ H −−−−−→ H

(23)

given by (f, uT ) → uT − yf → eT A (uT − yf ). In fact, (20) means that the operator eT A  must be defined at (f, uT ), so the data space Y is its domain. Being an inverse, eT A is a closed operator, and so is eT A ; hence Y = D(eT A ) is complete. Consequently Y is a Hilbertable space (like V ∗ ). Thus the unbounded operator eT A  is a key ingredient in the rigorous treatment of (15). In control theoretic terms its role is to provide a unique initial state u(0) = eT A (f, uT )

(24)

that is steered by f to the final state u(T ) = uT at time T ; cf. (25) below. Because of e−(T −t)A and the integral over [0, T ], condition (20) clearly involves non-local operators in both space and time as an inconvenient aspect—which is exacerbated by the abstract domain D(eT A ) that for longer lengths T of the time interval gives increasingly stricter conditions; cf. (17). Anyhow, (20) is a compatibility condition on the data (f, uT ), and thus the notion of compatibility is generalised. For comparison it is recalled that Grubb and Solonnikov [12] systematically investigated a large class of initial-boundary problems of parabolic (pseudo-)differential equations and worked out compatibility conditions, which are necessary and sufficient for well-posedness in full scales of anisotropic L2 -Sobolev spaces. Their conditions are explicit and local at the curved corner ∂ × {0}, except for half-integer values of the smoothness s that were shown to require so-called coincidence, which is expressed in integrals over the product of the two boundaries {0} ×  and ]0, T [ × ∂; hence it also is a non-local condition. However, whilst their conditions are decisive for the solution’s regularity, the above condition (20) is crucial for the existence question; cf. the theorem. Previously, uniqueness in (15) was shown by Amann [2, Sect. V.2.5.2] in a tdependent set-up, but injectivity of u(0) → u(T ) was proved much earlier for problems with t-dependent sesquilinear forms by Lions and Malgrange [20]. Showalter [27] strived to characterise the possible uT via Yosida approximations for f = 0 and A having half-angle π4 . Invertibility of e−tA was claimed for this purpose in [27] for such A (but, as mentioned, not quite obtained). To make a few more remarks, it is noted that the proof given below exploits that the solution u also in this set-up necessarily is given by Duhamel’s principle, or the variation of constants formula, for the analytic semigroup e−tA in V ∗ , u(t) = e

−tA



t

u(0) + 0

e−(t−s)A f (s) ds.

(25)

Well-Posed Final Value Problems

267

For t = T this yields a bijection u(0) ↔ u(T ) between the initial and terminal states; in particular backwards uniqueness of the solutions holds in the large class X. Of course, this relies crucially on the invertibility of e−tA in Proposition 2. Now, (25) also shows that u(T ) consists of two parts, that differ radically even when A has nice properties: First, the integral amounts to yf for t = T , and by the mentioned surjectivity this terms can be anywhere in H . Secondly, e−tA u(0) solves u + Au = 0, and for u(0) = 0 there is the precise property in non-selfadjoint dynamics that the “height” function h(t) = |e−tA u(0)| is strictly positive (h > 0), strictly decreasing (h < 0) and strictly convex.

(26)

Whilst this holds if A is self-adjoint or normal, it was emphasized in [5] that it suffices that A is just hyponormal. Recently this was followed up by the author in [17], where the stronger logarithmic convexity of h(t) was proved equivalent to the weaker property that 2(6( Ax | x ))2 ≤ 6( A2 x | x )|x|2 + |Ax|2 |x|2 for x ∈ D(A2 ). The stiffness inherent in strict convexity reflects that u(T ) = e−T A u(0) is confined to a dense, but very small space, as by the analyticity u(T ) ∈



n n∈N D(A ).

(27)

For u +Au = f , the possible final data uT will hence be a sum of an arbitrary vector yf in H and a term e−T A u(0) of great stiffness, cf. (27). Thus uT can be prescribed in the affine space yf + D(eT A ). As any yf = 0 will shift D(eT A ) ⊂ H in some arbitrary direction, u(T ) can be expected anywhere in H (unless yf ∈ D(eT A ) is known). So neither u(T ) ∈ D(eT A ) nor (27) can be expected to hold for yf = 0— not even if |yf | is much smaller than |e−T A u(0)|. In view of this conclusion, it seems best for final value problems to consider inhomogeneous problems from the very beginning.

2 Proof of Theorem 1 The point of departure is the following well-known result, which is formulated as a theorem here only to indicate that it is a cornerstone in the proof. It is also emphasized that the Lax–Milgram operator A need not be selfadjoint. Theorem 2 Let V be a separable Hilbert space with V ⊆ H algebraically, topologically and densely, and let A denote the Lax–Milgram operator induced by a V -elliptic sesquilinear form, as well as its extension A ∈ B(V , V ∗ ), cf. Sect. 1.3. When u0 ∈ H and f ∈ L2 (0, T ; V ∗ ) are given, then the Cauchy problem "

∂t u + Au = f u(0) = u0

in D (0, T ; V ∗ ), in H,

(28)

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J. Johnsen

has a uniquely determined solution u(t) belonging to the space X in (16). This is a special case of a classical result of Lions and Magenes [21, Sect. 3.4.4] on t-dependent forms a(t; u, v). The conjunction of the properties u ∈ L2 (0, T ; V ) and u ∈ L2 (0, T ; V ∗ ), which appears in [21], is clearly equivalent to the property in (16) that u belongs to the intersection of L2 (0, T , V ) and H 1 (0, T ; V ∗ ). To clarify a redundancy, it is first noted that in Theorem 2 the solution space X is a Banach space, which can have its norm in (16) written in the form 1/2  . u X = u 2L2 (0,T ;V ) + sup |u(t)|2 + u 2H 1 (0,T ;V ∗ )

(29)

0≤t≤T

Here there is a well-known inclusion L2 (0, T ; V ) ∩ H 1 (0, T ; V ∗ ) ⊂ C([0, T ]; H ) and an associated Sobolev inequality for vector functions ([5] has an elementary proof) sup |u(t)| ≤ (1 + 2

0≤t≤T

C22 C12 T



T

) 0



T

u dt + 2

0

u 2∗ dt.

(30)

Hence one can safely omit the space C([0, T ]; H ) in (16). Likewise sup |u| can be T removed from · X , as one obtains an equivalent norm (similarly 0 u(t) 2∗ dt is redundant in (16)). Thus X is more precisely a Hilbertable space; but (16) is kept as stated in order to emphasize the properties of the solutions. However, two refinements of the above theory is needed. For one thing, the next result yields well-posedness of (28), which is a well-known corollary to the proofs in [21]. But a short explicit argument is also possible: Proposition 4 The unique solution u of (28), given by Theorem 2, depends as an element of X continuously on the data (f, u0 ) ∈ L2 (0, T ; V ∗ ) ⊕ H , i.e. u 2X ≤ c(|u0 |2 + f 2L2 (0,T ;V ∗ ) ).

(31)

Here and in the sequel, c denotes as usual a constant in ]0, ∞[ of unimportant value, which moreover may change from place to place. Proof As u ∈ L2 (0, T ; V ) while u , f and Au belong to the dual space L2 (0, T ; V ∗ ), one has the identity 6∂t u, u + 6Au, u = 6f, u in L1 . Now, a classical regularisation yields ∂t |u|2 = 26∂t u, u, so by Young’s inequality and the V -ellipticity, ∂t |u|2 + 2C4 u 2 ≤ 2|f, u| ≤ C4−1 f 2∗ + C4 u 2 .

(32)

Using again that |u(t)|2 and ∂t |u(t)|2 are in L1 (0, T ), integration yields 

t

|u(t)| + C4 2

0

u(s) 2 ds ≤ |u0 |2 + C4−1 f 2L2 (0,T ;V ∗ ) .

(33)

Well-Posed Final Value Problems

269

This gives u 2L2 (0,T ;V ) ≤ C4−1 |u0 |2 + C4−2 f 2L2 (0,T ;V ∗ ) for the first term in u X . As u solves (28), it is clear that ∂t u(t) 2∗ ≤ ( f (t) ∗ + Au ∗ )2 , hence 

T 0

where

T 0

 ∂t u(t) 2∗ dt

T

≤2 0

 f (t) 2∗ dt

+ 2 A 2B(V ,V ∗ )

T

u 2 dt,

(34)

0

u 2 dt is estimated above. Finally sup |u| can be covered via (30).

 

Secondly, as e−tA extends to an analytic semigroup in V ∗ , cf. [5, Lem. 5], so that is defined, Theorem 2 can be supplemented by the explicit expression:

e−tA f (t)

u(t) = e−tA u0 +



t

e−(t−s)A f (s) ds

for 0 ≤ t ≤ T .

(35)

0

This is of course the Duhamel formula, but even for analytic semigroups the proof that it does give a solution requires f (t) to be Hölder continuous ([24, Cor. 4.3.3] is slightly more general, though), whereas above f ∈ L2 (0, T ; V ∗ ). As the present space X even contains non-classical solutions, (35) requires a new proof here—but it suffices to reinforce the usual argument by the injectivity of e−tA in Proposition 2: Theorem 3 The unique solution u in X provided by Theorem 2 is given by (35), where each of the three terms belongs to X. Proof Once (35) has been shown, Theorem 2 yields for f = 0 that u(t) ∈ X, hence e−tA u0 ∈ X. For general (f, u0 ) the last term containing f is then also in X. To obtain (35) in the present context, note that all terms in ∂t u + Au = f are in L2 (0, T ; V ∗ ). Therefore e−(T −t)A applies to both sides as an integration factor, so ∂t (e−(T −t)A u(t)) = e−(T −t)A ∂t u(t) + e−(T −t)A Au(t) = e−(T −t)A f (t).

(36)

Indeed, e−(T −t)A u(t) is in L1 (0, T ; V ∗ ) and its derivative in D (0, T ; V ∗ ) follows a Leibniz rule, as one can prove by regularisation since u(t) ∈ V = D(A) for t a.e. The right-hand side above is in L2 (0, T ; V ∗ ), hence in L1 (0, T ; V ∗ ), so when the Fundamental Theorem for vector functions (cf. [29, Lem. III.1.1]) is applied and followed by commutation of e−(T −t)A with the integral (via Bochner’s identity), e

−(T −t)A

u(t) = e

−T A

 u0 +

t

e−(T −s)A f (s) ds

0

=e

−(T −t)A −tA

e

u0 + e

−(T −t)A



(37)

t

e

−(t−s)A

f (s) ds.

0

Since e−(T −t)A is injective, cf. Proposition 2, (35) now results at once.

 

270

J. Johnsen

As all terms in (35) are in C([0, T ]; H ), we may safely evaluate at t = T , which in view of (18) gives that u(T ) = e−T A u(0)+yf ; this is the flow map u(0) → u(T ). Owing to the invertibilty of e−T A once again, this flow is inverted by u(0) = eT A (u(T ) − yf ).

(38)

In other words, the solutions in X to u + Au = f are for fixed f parametrised by the initial states u(0) in H . Departing from this observation, one may give an intuitive Proof (Proof of Theorem 1) When (15) has a solution u ∈ X, then u(T ) = uT is reached from the initial state u(0) determined from the bijection in (38). This gives that uT − yf = e−T A u(0) ∈ D(eT A ), so (20) is necessary. In case uT , f do fulfill (20), then u0 = eT A (uT − yf ) is a well-defined vector in H , so Theorem 2 yields a function u ∈ X solving u + Au = f and u(0) = u0 . By (38) this u has final state u(T ) = e−T A eT A (uT − yf ) + yf = uT , hence solves (15). In the affirmative case, one obtains (22) by insertion of formula (38) for u(0) into (35). That each term in (22) is a function belonging to X was seen in Theorem 3. Uniqueness of u in X is obvious from the right-hand side of (22). The solution can hence be estimated in X by insertion of (38) into the inequality in Proposition 4, which gives u 2X ≤ c(|eT A (uT − yf )|2 + f 2L2 (0,T ;V ∗ ) ). Here one may add |uT |2 on the right-hand side to obtain the expression for (f, uT ) Y in Theorem 1.   Remark 2 The above arguments seem to extend to Lax–Milgram operators A that are only V -coercive, i.e. fulfil 6a(u, u) ≥ C4 u 2 − k|u|2 for u ∈ V . In fact, it was observed already in [21] that Theorem 2 holds verbatim for such A, for since A + k is V -elliptic, the unique solvability in X of v  + (A + k)v = e−kt f , v(0) = u0 yields a unique solution u = ekt v of u + Au = f , u(0) = u0 . Since the same translation trick gave the improved version of Proposition 2, also V -coercive A generate analytic semigroups of injections; so the proofs of the Duhamel formula and of Theorem 1 seem applicable in their present form. However, the estimates in Proposition 4 need to be modified using Grönwall’s lemma. The details of this are left for future work. (Added in proof: An elaboration of the results indicated in this remark will appear in the forthcoming paper [18].) Remark 3 Recently Almog, Grebenkov, Helffer, Henry [1, 8, 9] studied variants of the complex Airy operator via triples (H, V , a), and Theorem 1 is expected to extend to final value problems for those of their realisations that have non-empty spectrum.

Well-Posed Final Value Problems

271

3 The Heat Problem with Final Time Data To follow up on Theorem 1, it is now applied to the heat equation and its final value problem. In the sequel  stands for a C ∞ smooth, open bounded set in Rn , n ≥ 2 as described in [11, App. C]. In particular  is locally on one side of its boundary  = ∂. For such sets we consider the problem of finding the u satisfying ⎧ ∂t u(t, x) − u(t, x) = f (t, x) ⎪ ⎪ ⎨ γ0 u(t, x) = g(t, x) ⎪ ⎪ ⎩ rT u(x) = uT (x)

in Q =]0, T [×, on S =]0, T [×,

(39)

at {T } × .

Hereby the trace of functions on  is written in the operator notation γ0 u = u| . Similarly γ0 is also used for traces on S, while rT denotes the trace at t = T . Moreover, H01 () is the subspace obtained by closing C0∞ () in the Sobolev space H 1 (). Dual to this one has H −1 (), which identifies with the set of restrictions to  from H −1 (Rn ), endowed with the infimum norm; Chapter 6 and Remark 9.4 in [11] could be references for this and basic facts on boundary value problems.

3.1 The Boundary Homogeneous Case In case g ≡ 0 in (39), the main result in Theorem 1 applies for V = H01 (),

H = L2 (),

V ∗ = H −1 ().

(40)

Indeed, the boundary condition γ0 u = 0 is then imposed via the condition that u(t) ∈ V for all t, or rather through use of the Dirichlet realization of the Laplacian −γ0 (denoted by −D in the introduction), which is the Lax–Milgram operator A induced by the triple (L2 (), H01 (), s) for s(u, v) =

n

( ∂j u | ∂j v )L2 () .

(41)

j =1

In fact, Poincaré’s inequality yields that s(u, v) is H01 ()-elliptic and symmetric, so A = − γ0 is a selfadjoint unbounded operator in L2 (), with D(− γ0 ) ⊂ H01 (). Hence −A = γ0 generates an analytic semigroup et γ0 in B(L2 ()), and the bounded bijective extension  : H01 () → H −1 () induces the analytic semigroup et  on V ∗ = H −1 (). As done previously, we set (et γ0 )−1 = e−t γ0 .

272

J. Johnsen

Moreover, when g = 0 in (39), then the solution and data spaces amount to   X0 = L2 (0, T ; H01 ()) C([0, T ]; L2 ()) H 1 (0, T ; H −1 ()), (42)

 

Y0 = (f, uT ) ∈ L2 (0, T ; H −1 ()) ⊕ L2 () uT − yf ∈ D(e−T γ0 ) . (43) Here, with yf given in (18), the data norm from Theorem 1 specialises to  (f, uT ) 2Y0

T

= 0

 f (t) 2H −1 () dt

+

(|uT |2 + |e−T γ0 (uT − yf )|2 ) dx.



(44) Now Theorem 1 straightforwardly gives the following result, which first appeared in [5] even though the problem is entirely classical: Theorem 4 Let A = −γ0 be the Dirichlet realization of the Laplacian in  and − its extension, as introduced above. When g = 0 in the final value problem (39) and f ∈ L2 (0, T ; H −1 ()), uT ∈ L2 (), then there exists a solution u in X0 of (39) if and only if the data (f, uT ) are given in Y0 , i.e. if and only if  uT −

T

e(T −s)  f (s) ds

belongs to

D(e−T γ0 ).

(45)

0

In the affirmative case, u is uniquely determined in X0 and u X0 ≤ c (f, uT ) Y0 . Furthermore the difference in (45) equals eT γ0 u(0) in L2 ().

3.2 The Inhomogeneous Case When g = 0 on the surface S, cf. (39), then the solution u(t, ·) belongs to the full Sobolev space H 1 () for each t > 0, so here the solution space is denoted by X1 , X1 = L2 (0, T ; H 1 ())



C([0, T ]; L2 ())



H 1 (0, T ; H −1 ()).

(46)

Clearly X1 is a Banach space when normed analogously to (29), u X1 = ( u 2L

2 (0,T ;H

1 ())

+ sup u(t) 2L2 () + u 2H 1 (0,T ;H −1 ()) )1/2 . 0≤t≤T

(47) Here H 1 , H −1 are not dual on , so the previously mentioned redundancy does not extend to the term sup[0,T ] u L2 above.

Well-Posed Final Value Problems

273

For g = 0 the standard strategy is, of course, to (try to) reduce to a semihomogeneous problem by choosing w so that γ0 w = g on S, using the classical Lemma 1 γ0 : H 1 (Q) → H 1/2 (S) is a continuous surjection having a bounded right inverse K˜ 0 , that is, γ0 K˜ 0 g = g for every g ∈ H 1/2 (S) and w H 1 (Q) = K˜ 0 g H 1 (Q) ≤ c g H 1/2 (S) .

(48)

In lack of a reference with details, the reader is referred to [5] for a sketch of how this lemma follows using standard techniques. Another preparation is based on the fine theory of the elliptic problem − u = f,

γ0 u = g.

(49)

1 Indeed, it exploited below that Q = I −−1 γ0  is a well-known projection in H (), along H01 () and onto the closed subspace of harmonic H 1 -functions,

Z(−) = { u ∈ H 1 () | −u = 0 }.

(50)

  , with To recall this, (49) may conveniently be treated via the matrix operator − γ 0 f   −1  − K that applies to the data g , for then the basic an inverse in row form 0 γ0 composition formulae appear in two identities on H 1 () and H −1 () ⊕ H 1/2 (),  −   −1 = −1 I = −γ0 K0 γ0  +K0 γ0 , γ0     −1  K   I 0 −  0 γ 0 = . −−1 γ0 K0 = 0I −γ0 −1 γ0 γ0 γ0 K0

(51)

(52)

1 In particular the first formula yields that Q = I − −1 γ0  = K0 γ0 on H (). However, it should be emphasized that the simplicity of the formulas (51) and (52) relies on a specific

choice of K0 , which is recalled here: 2 As γ0 =  H 1 holds in the distribution sense, P = −1 γ0  clearly fulfils P = 0

P, it is bounded H 1 → H01 and equals I on H01 , so P is the projection onto H01 () along its null space, which clearly is the space in (50). Therefore H 1 is a direct sum, H 1 () = H01 () Z(−),

(53)

1 so that Q = I − P = I −−1 γ0  is the projection onto Z(−) along H0 (). 1 1 1/2 Since γ0 : H () → H () is surjective with null-space H0 , it has an inverse K0 on the complement Z(−), which by the open mapping principle is bounded K0 : H 1/2 () → Z(−) → H 1 (). Since in this construction

274

J. Johnsen

K0 = (γ0 |Z(−) )−1 , it is also a right-inverse, i.e. γ0 K0 = IH 1/2 () . The rest of (52) now follows at once. Moreover, since γ0 P = γ0 −1 γ0  = 0, the definition of Q gives (51) thus: K0 γ0 = K0 γ0 (P + Q) = K0 γ0 Q = IZ(−) Q = Q = I −−1 γ0  .

(54)

Remark 4 K0 is an example of a Poisson operator; these are amply discussed within the pseudo-differential boundary operator calculus, for example in [10]. Remark 5 The H s (S)-norm can be chosen so that this is a Hilbert space; cf. the use of local coordinates in [11, (8.10)]. The vast subject of Sobolev spaces on C ∞ surfaces generally requires distribution densities as explained in [14, Sect. 6.3] (cf. [11, Sect. 8.2] for a concise review). But the surface measure on S induces a well-known identification of densities with distributions on S, and within this framework, a systematic exposition can be found in [19, Sect. 4], albeit in an Lp s,a setting with anisotropic mixed-norm Triebel–Lizorkin spaces Fp,q (S), which are the correct boundary data spaces for parabolic problems having different integrability properties in x and t. Cf. also the application to the heat problem (39) in [19, Sect. 6.5], and the more detailed discussion in [23, Ch. 7]. Now, when splitting the solution of (39) as u = v+w for w = K˜ 0 g, cf. Lemma 1, then v should satisfy v  − v = f˜, γ0 v = 0 and rT v = u˜ T for data f˜ = f − (∂t w − w),

u˜ T = uT − rT w.

(55)

For this problem to be solved by v, (45) stipulates that D(e−T γ0 ) should contain u˜ T − yf3 = uT − yf − (rT w − y∂t w− w ).

(56)

But the presence of the term rT w − y∂t w− w makes it impossible just to transfer condition (45) of being a member of D(e−T γ0 ) from u˜ T − yf3 to uT − yf . Thus the compatibility condition (45) destroys the trick of reducing to homogeneous boundary conditions, despite the linearity of the problem. To find the correct compatibility conditions on (f, g, uT ), the strategy in [5] was (with hindsight) to use Lemma 1 to get a solution formula for the corresponding linear initial value problem instead. This is motivated by the fact that, for the present space X1 of low regularity, no compatibility condition is needed for this: ∂t u − u = f

in Q,

γ0 u = g

on S,

r0 u = u0

at {0} × .

(57)

(For general background material on (57) the reader could consult Section III.6 in [3]; and [12] for the fine theory including compatibility conditions.) Similarly to Theorem 2 and Proposition 4, one may depart from well-posedness of (57). While this is well known per se, an explanation is given to account below for the crucial existence of an improper integral showing up when g = 0 in (39).

Well-Posed Final Value Problems

275

Proposition 5 The heat initial value problem (57) has a unique solution u ∈ X1 for given f ∈ L2 (0, T ; H −1 ()), g ∈ H 1/2 (S), u0 ∈ L2 (), and there is an estimate u 2X1 ≤ c( u0 2L2 () + f 2L

2 (0,T ;H

−1 ())

+ g 2H 1/2 (S) ).

(58)

Proof Let I = ]0, T [ . Setting w = K˜ 0 g by means of Lemma 1, we tentatively write u = v + w for some v ∈ X0 solving (57) for data f˜ = f − (∂t − )w,

g˜ = 0,

u˜ 0 = u0 − w(0).

(59)

Here w(0) is well defined, as w ∈ H 1 (Q) ⊂ H 1 ([0, T ]; L2 ()) ⊂ C([0, T ]; L2 ()) by a Sobolev embedding, which (e.g. as in [5, Rem. 4]) also gives the estimate sup w(t) L2 () ≤ c( w L2 (I ;L2 ()) + ∂t w L2 (I ;L2 ()) ) ≤ c w 2H 1 (Q) .

0≤t≤T

(60) To show that w ∈ X1 and w 2X1 ≤ c w 2H 1 (Q) , it is noted that estimates of the two remaining terms in w 2X1 , cf. (47), can be read off from the obvious inequalities w  2L

2 (I ;H

−1 )

+ w 2L

2 (I ;H

−1 )

≤ c w 2H 1 (I ;L ) + c w 2L

2 (I ;H

2

1)

≤ c w 2H 1 (Q) . (61)

This also entails f˜ ∈ L2 (0, T ; H −1 ()), and u˜ 0 ∈ L2 (), so by Theorem 2, the boundary homogeneous problem for v has a solution in X0 ; cf. (42). Hence (57) has the solution u = v + w in X1 ; its uniqueness is easily carried over from Theorem 2. Finally the estimate in Theorem 4 (that is a consequence of Proposition 4) gives u 2X1 ≤ 2( v 2X0 + w 2X1 ) ≤ c( u˜ 0 2L2 () + f˜ 2L

2 (I ;H

≤ c( u0 2L2 () + f 2L

2 (I ;H

−1 ())

+ w  − w 2L

−1 ())

2 (I ;H

+ w 2X1 )

−1 )

+ w 2H 1 (Q) ), (62)

which via (61) and (48) entails the stated estimate (58).

 

Obviously the formula in Theorem 3 now applies directly to the function v = u − w in the above proof, which as a crucial addendum to Proposition 5 yields 

t

u(t) − w(t) = etγ0 (u0 − w(0)) + 0

e(t−s)  (f − (∂s − )w) ds.

(63)

276

J. Johnsen

The next step is to rewrite the contributions from w so that g = γ0 w can be reintroduced. First a regularisation of w in H 1 (0, t; L2 ()) leads to the Leibniz rule ∂s (e(t−s)γ0 w(s)) = e(t−s)γ0 ∂s w(s) − γ0 e(t−s)γ0 w(s).

(64)

Here the last term is only integrable on [0, t − ε] for ε > 0, as it has a singularity at s = t; cf. Proposition 1. As a remedy, one can use the improper Bochner integral  t  − γ0 e(t−s)γ0 w(s) ds = lim

ε→0 0

0

t−ε

γ0 e(t−s)γ0 w(s) ds.

(65)

Lemma 2 For every w ∈ H 1 (Q) the limit (65) exists in L2 () and 

t

w(t) = etγ0 w(0) + 0

 t e(t−s)γ0 ∂s w(s) ds − − γ0 e(t−s)γ0 w(s)) ds.

(66)

0

Proof When applied to (64), the Fundamental Theorem for vector functions gives  s=t−ε [e(t−s)γ0 w(s)]s=0 =

0

t−ε

 (−γ0 )e(t−s)γ0 w ds +

t−ε

e(t−s)γ0 ∂s w ds.

0

(67) The left-hand side converges, for etγ0 is of type (M, 0) and the proof above gave w ∈ C([0, T ], L2 ()), so by bilinearity e(t−s)) γ0 w(s) → w(t) in L2 () for s → t −. Moreover, by dominated convergence the last integral converges in L2 () for  t−ε ε → 0+ , whence 0 γ0 e(t−s)γ0 w(s) ds does so. Then (66) results.   By substituting (66) for w(t) in (63), one obtains, as terms with ∂s w cancel, 

t

u(t) = et γ0 u0 +



t

e(t−s)  f ds +

0

0

 t e(t−s)   w ds − − γ0 e(t−s) γ0 w ds. 0

(68) While the last two integrals look highly similar, a further reduction is possible since 1  = γ0 −1 γ0  holds on H (). Hence they combine into a single improper integral,  t − − γ0 e(t−s) γ0 (I − −1 γ0 )w(s) ds.

(69)

0

Because of (54) we have (I − −1 γ0 )w = Qw = K0 γ0 w = K0 g as γ0 w = g, and when this is applied via (69) in (68), we finally arrive at the desired solution formula:

Well-Posed Final Value Problems

277

Proposition 6 If u ∈ X1 denotes the unique solution to the initial boundary value problem (57) provided by Proposition 5, then u fulfils the identity  u(t) = etγ0 u0 + 0

t

 t e(t−s)  f (s) ds − − γ0 e(t−s) γ0 K0 g(s) ds,

(70)

0

where the improper Bochner integral converges in L2 () for every t ∈ [0, T ]. Remark 6 Despite the classical context, (70) seemingly first appeared in [5]. T For t = T the second term in (70) gives back yf = 0 e(T −s)  f (s) ds, cf. (18). But the full influence of the boundary data g on u(T ) is contained in the third term,  T zg = −  e(T −s) γ0 K0 g(s) ds.

(71)

0

Even the basic fact that g → zg is a well-defined map is a non-trivial result; it results at once for t = T in Proposition 6. The map is clearly linear by the calculus of limits. In case f = 0, u0 = 0 it is seen from (70) that zg = −u(T ), so the estimate in Proposition 5 yields that zg L2 () ≤ supt u(t) L2 () ≤ c g H 1/2 (S) . This proves Lemma 3 The linear operator g → zg is bounded H 1/2 (S) → L2 (). Moreover, Proposition 6 gives for an arbitrary solution in X1 of the heat equation u − u = f with γ0 u = g on S that there is a bijection u(0) ↔ u(T ) given by u(T ) = eT γ0 u(0) + yf − zg .

(72)

Indeed, the above breaks down as application of the bijection eT γ0 , cf. Proposition 2 followed by translation in L2 () by the fixed vector yf − zg . The above considerations suffice for a proof of unique solvability in X1 of the inhomogeneous final value problem (39) for suitable data (f, g, uT ). Both the result and the proof are highly similar to the abstract Theorem 1, but one should note, of course, the new clarification that the boundary data g do appear in the compatibility condition, and only via zg : Theorem 5 For given data (f, g, uT ) in L2 (0, T ; H −1 ()) ⊕ H 1/2 (S) ⊕ L2 (), the final value problem (39) is solved by a function u ∈ X1 , whereby X1 = L2 (0, T ; H 1 ())



C([0, T ]; L2 ())



H 1 (0, T ; H −1 ()),

(73)

if and only if the data in terms of (18) and (71) satisfy the compatibility condition uT − yf + zg ∈ D(e−T γ0 ).

(74)

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J. Johnsen

In the affirmative case, u is uniquely determined in X1 and has the representation u(t) = etγ0 e−T γ0 (uT − yf + zg ) +



t 0

 t e(t−s)  f ds − − γ0 e(t−s) γ0 K0 ds, 0

(75) where the three terms all belong to X1 as functions of t ∈ [0, T ]. Proof Given a solution u ∈ X1 , the bijection (72) yields uT = eT γ0 u(0)+yf −zg , so that (74) necessarily holds. Inserting its inversion u(0) = e−T γ0 (uT − yf + zg ) into the solution formula from Proposition 6 yields (75); thence uniqueness of u. If (74) does hold, u0 = e−T γ0 (uT − yf + zg ) is a vector in L2 (), so the initial value problem with data (f, g, u0 ) can be solved using Proposition 5. This yields a u ∈ X1 that also solves (39), since u(T ) = uT holds by (72) and the choice of u0 . The final regularity statement follows from the fact that X1 also is the solution space for the Cauchy problem in Proposition 5: the improper integral in (75) is a solution in X1 to (57) for data (f, g, u0 ) = (0, g, 0), according to Proposition 6; the integral containing f solves (57) for data (f, 0, 0), hence is in X1 ; the first term in (75) solves (57) for data (0, 0, e−T γ0 v) for the vector v = uT − yf + zg .   Exploiting the above theorem, we let Y1 stand for the set of admissible data corresponding to X1 . Within the broader space L2 (0, T ; H −1 ()) ⊕ H 1/2 () ⊕ L2 (), the data space Y1 is the subspace given via the map 1 (f, g, uT ) = uT − yf + zg as

 

Y1 = (f, g, uT ) uT − yf + zg ∈ D(e−T γ0 ) = D(e−T γ0 1 ).

(76)

Naturally Y1 is endowed with the graph norm of e−T γ0 1 , that is, of the composite map (f, g, uT ) → e−T γ0 (uT − yf + zg ). Taking t = 0 in (75), it is clear that e−T γ0 1 (f, g, uT ) is the initial state u(0) steered by f , g to the final one u(T ) = uT . In some details, the above-mentioned graph norm is given by (f, g, uT ) 2Y1 = uT 2L2 () + g 2 wH 1/2 (Q) + f 2L (0,T ;H −1 ()) 2 

 T  T   2

−T γ0 uT − + e(T −s)  f ds + − γ0 e(T −s) γ0 K0 g ds dx.

e 

0

0

(77) Hereby e−T γ0 (uT −yf +zg ) 2L2 () is written with explicit integrals to emphasize the complexity of the fully inhomogeneous boundary and final value problem (39). Completeness of Y1 follows from continuity of 1 , cf. Lemma 3 concerning zg . Indeed, its composition to the left with the closed operator e−T γ0 in L2 () (cf. Proposition 3) is also closed. Therefore its domain D(e−T γ0 1 ) = Y1 is

Well-Posed Final Value Problems

279

complete with respect to the graph norm in (77). This is induced by an inner product −1 1/2 , and when when H −1 () is given the equivalent norm |||f |||∗ = s(−1 γ0 f, γ0 f ) 1/2 H (Q) is normed as in Remark 5. Hence Y1 is a Hilbertable space. Theorem 6 The unique solution u of problem (39) lying in the Banach space X1 depends continuously on the data (f, g, uT ) in the Hilbert space Y1 , when these are given the norms in (47) and (77), respectively. Proof Boundedness of the solution operator (f, g, uT ) → u is seen by inserting the expression u0 = e−T γ0 (uT − yf + zg ) from (72) into the estimate in Proposition 5, u 2X1 ≤ c( e−T γ0 (uT − yf + zg ) 2L2 () + f 2L

2 (0,T ;H

−1 ())

+ g 2H 1/2 (S) ). (78)

Adding uT 2L2 () on the right-hand side, one arrives at (f, g, uT ) 2Y1 .

 

Taken together, Theorems 5 and 6 yield the final result: Theorem 7 When  is a C ∞ smooth, open bounded set in Rn for n ≥ 2, and T > 0, then the fully inhomogeneous final value heat conduction problem ⎧ ∂t u − u = f in ]0, T [ ×, ⎪ ⎪ ⎨ γ0 u = g on ]0, T [ ×, ⎪ ⎪ ⎩ u(T ) = uT in ,

(79)

is well posed with solutions u and data (f, g, uT ) belonging to the spaces X1 and Y1 defined in (46), (76), and normed as in (47), (77), respectively.

References 1. Y. Almog and B. Helffer, On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent, Comm. PDE 40 (2015), no. 8, 1441–1466. 2. H. Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995, Abstract linear theory. 3. W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, second ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. 4. A.-E. Christensen and J. Johnsen, On parabolic final value problems and well-posedness, C. R. Acad. Sci. Paris, Ser. I 356 (2018), 301–305. 5. A.-E. Christensen and J. Johnsen, Final value problems for parabolic differential equations and their well-posedness, Axioms 7 (2018), article no. 31; 1–36. 6. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. 7. E. B. Davies, One-parameter semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, Inc., London-New York, 1980.

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8. D. S. Grebenkov and B. Helffer, On the spectral properties of the Bloch–Torrey operator in two dimensions, SIAM J. Math. Anal. 50 (2018), no. 1, 622–676. 9. D.S. Grebenkov, B. Helffer, and R. Henry, The complex Airy operator on the line with a semipermeable barrier, SIAM J. Math. Anal. 49 (2017), no. 3, 1844–1894. 10. G. Grubb, Functional calculus of pseudo-differential boundary problems, second ed., Progress in Mathematics, vol. 65, Birkhäuser, Boston, 1996. 11. G. Grubb, Distributions and operators, Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009. 12. G. Grubb and V. A. Solonnikov, Solution of parabolic pseudo-differential initial-boundary value problems, J. Differential Equations 87 (1990), 256–304. 13. B. Helffer, Spectral theory and its applications, Cambridge Studies in Advanced Mathematics, vol. 139, Cambridge University Press, Cambridge, 2013. 14. L. Hörmander, The analysis of linear partial differential operators, Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin, 1983, 1985. 15. V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences, vol. 127, Springer-Verlag, New York, 1998. 16. F. John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. (4) 40 (1955), 129–142. 17. J. Johnsen, Characterization of log-convex decay in non-selfadjoint dynamics, Elec. Res. Ann. Math., 25 (2018), 72–86. 18. J. Johnsen, Well-posed final value problems and Duhamel’s formula for coercive Lax–Milgram operators, Elec. Res. Arch. 27 (2019), 20–36. 19. J. Johnsen, S. Munch Hansen, and W. Sickel, Anisotropic Lizorkin–Triebel spaces with mixed norms—traces on smooth boundaries, Math. Nachr. 288 (2015), 1327–1359. 20. J.-L. Lions and B. Malgrange, Sur l’unicité rétrograde dans les problèmes mixtes parabolic, Math. Scand. 8 (1960), 227–286. 21. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. 22. W. L. Miranker, A well posed problem for the backward heat equation, Proc. Amer. Math. Soc. 12 (1961), 243–247. 23. S. Munch Hansen, On parabolic boundary problems treated in mixed-norm Lizorkin–Triebel spaces, Ph.D. thesis, Aalborg University; Aalborg, Denmark, 2013. 24. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. 25. J. Rauch, Partial differential equations, Springer, 1991. 26. L. Schwartz, Théorie des distributions, revised and enlarged ed., Hermann, Paris, 1966. 27. R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl. 47 (1974), 563–572. 28. H. Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman, Boston, Mass., 1979, Translated from the Japanese by N. Mugibayashi and H. Haneda. 29. R. Temam, Navier–Stokes equations, theory and numerical analysis, Elsevier Science Publishers B.V., Amsterdam, 1984, (Third edition). 30. K. Yosida, Functional analysis, sixth ed., Springer-Verlag, Berlin-New York, 1980.

Localization of a Class of Muckenhoupt Weights by Using Mellin Pseudo-Differential Operators Yu. I. Karlovich

Abstract Let  be a finite or infinite interval of R, p ∈ (1, ∞), and let w ∈ Ap () be a Muckenhoupt weight. Relations of the weighted singular integral operator wSR+ w −1 I on the space Lp (R+ ) and Mellin pseudo-differential operators with non-regular symbols are studied. A localization of a class of Muckenhoupt weights to power weights at finite endpoints of , which is related to the Allan-Douglas local principle, is obtained by using quasicontinuous functions and Mellin pseudodifferential operators with non-regular symbols. Keywords Weighted singular integral operators · Muckenhoupt weight · Quasicontinuous function · Mellin pseudo-differential operator · Local study

1 Introduction Let B(X) denote the Banach algebra of all bounded linear operators acting on a Banach space X, let K(X) be the closed two-sided ideal of all compact operators in B(X), and let Bπ (X) = B(X)/K(X) be the Calkin algebra of the cosets Aπ := A + K(X), where A ∈ B(X). An operator A ∈ B(X) is said to be Fredholm, if its image is closed and the spaces ker A and ker A∗ are finite-dimensional (see, e.g., [9] and [16]). Equivalently, A ∈ B(X) is Fredholm if and only if the coset Aπ is invertible in the quotient algebra Bπ (X). Let  be an interval of R or the unit circle T ⊂ C. A measurable function w :  → [0, ∞] is called a weight if the preimage w−1 ({0, ∞}) of the set {0, ∞} has

Yu. I. Karlovich () Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos, Cuernavaca, Mexico e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_17

281

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Yu. I. Karlovich

measure zero. For p ∈ (1, ∞), a weight w belongs to the Muckenhoupt class Ap () if  cp,w := sup I

1 |I |

1/p 

 w p (τ )|dτ | I

1 |I |



w −q (τ )|dτ |

1/q < ∞,

I

where 1/p + 1/q = 1, and supremum is taken over all intervals I ⊂  of finite length |I |. In what follows we assume that p ∈ (1, ∞), w ∈ Ap (), and consider the weighted Lebesgue space Lp (, w) equipped with the norm  f

Lp (,w)

1/p

:=

|f (τ )| w (τ )|dτ | p

p

.



As is known (see, e.g., [5, 15]), the Cauchy singular integral operator S given by 1 ε→0 π i



(S f )(t) = lim

\(t,ε)

f (τ ) dτ (t ∈ ), τ −t

(1)

where (t, ε) :=  ∩ {z ∈ C : |z − t| < ε}, is bounded on every space Lp (, w) if and only if p ∈ (1, ∞) and w ∈ Ap (). For p ∈ (1, ∞) and w ∈ Ap (T), let H p (w) denote the weighted Hardy space of all functions in Lp (T, w) whose negative Fourier coefficients vanish. Let QC and P QC be, respectively, the C ∗ -algebras of quasicontinuous and piecewise quasicontinuous functions on T (see Sect. 2), and let M(QC) be the maximal ideal space of QC. A Fredholm criterion and an index formula for Toeplitz operators with P QC symbols on the Hardy space H 2 were obtained by D. Sarason [32]. A. Böttcher and I.M. Spitkovsky [10] established a Fredholm criterion for Toeplitz operators with P QC symbols on the weighted Hardy spaces H p (-), where p ∈ (1, ∞), -(t) =

m G

|t − tj |μj

(t ∈ T),

j =1

t1 , . . . , tm are pairwise distinct points on T and μj ∈ (−1/p, 1/q). Fredholm symbol calculi for several Banach subalgebras of B(Lp (T, -)) generated by P QC multiplications, the Cauchy singular integral operator ST and a Carleman shift were elaborated in [8] by applying the Calkin images, the Allan-Douglas local principle (see, e.g., [14, Theorem 7.47] and [9, Theorem 1.35]) and the two idempotents theorem (see, e.g., [4, 5, 30] and the references therein). In particular, in [8] (see also [10]), for the Banach algebra A of singular integral operators with P QC coefficients on the space Lp (T, -) and for the Toeplitz operators T (a) = P aP + Q ∈ A with symbols a ∈ P QC, where P = (I + ST )/2 and Q = (I − ST )/2, the local spectra sp[T (a)]πξ for the cosets [T (a)]πξ := [T (a)]π + Jξπ in the quotient Banach algebra Aπξ := Aπ /Jξπ were described for all ξ ∈ M(QC), where Jξπ are the closed

Localization of Muckenhoupt Weights

283

, two-sided ideals of Aπ generated by the maximal ideals Iπξ := [aI ]π : a ∈ , QC, a(ξ ) = 0 of the commutative Banach algebra Zπ := [aI ]π : a ∈ QC . On the other hand, the description of such local spectra in the setting of weighted Lebesgue spaces with general Muckenhoupt weights is an open problem (see [11, p. 106]). Some results in this direction were obtained earlier in [22, 23]. A first step for identifying the local spectra consists in a possible simplification of corresponding cosets. The present paper is devoted to localizing sufficiently general Muckenhoupt weights by using Mellin pseudo-differential operators and quasicontinuous functions, and simplifying the key cosets related to those that arise in applications of the Allan-Douglas local principle and the two idempotents theorem to the Fredholm study of the Banach algebras A of singular integral operators with P QC coefficients on weighted Lebesgue spaces with Muckenhoupt weights. 

Consider the C ∗ -algebra QC of quasicontinuous functions on R := R ∪ {∞}, 



QC := (H ∞ + C(R)) ∩ (H ∞ + C(R)),

(2)

where H ∞ is the closed subalgebra of L∞ (R) that consists of all functions being non-tangential limits on R of bounded analytic functions defined on the upper halfplane. Let P QC be the C ∗ -subalgebra of L∞ (R) generated by the C ∗ -algebras QC 

and P C, where P C consists of all piecewise continuous functions on R, that is, the 

functions having finite one-sided limits at each point t ∈ R. Given  = [0, δ] ⊂ R or  = R+ ∪ {0}, let QC() and P QC() be the restrictions to  of QC and P QC, respectively. For p ∈ (1, ∞) and w ∈ Ap (), we consider the Banach algebra , Ap,w () := alg aI, S : a ∈ P QC() ⊂ B(Lp (, w))

(3)

generated by all multiplication operators aI (a ∈ P QC()) and by the Cauchy singular integral operator S given by (1). The algebra Ap,w () contains the ideal K(Lp (, w)) (see the proof of [30, Theorem 4.1.5]). The Fredholm study of the algebra Ap,w () defined by (3) is equivalent to that for the Banach algebra , 3 Ap,w () := alg aI, wS w −1 I : a ∈ P QC() ⊂ B(Lp ()).

(4)

For p ∈ (1, ∞), let p () be the Banach algebra of all operators A ∈ B(Lp ()) for which the commutators [aI, A] for all a ∈ QC() are compact operators on the space Lp (), and let πp () := p ()/K(Lp ()). Then the Banach algebra 3 Ap,w () is contained in the Banach algebra p (), which follows from Corollary 1 below. Let M(QC()) be the maximal ideal space of the commutative C ∗ -algebra QC(). For every ξ ∈ M(QC()), we consider the closed two-sided ideal Jπp,ξ,

284

Yu. I. Karlovich

, of the Banach algebra πp () generated by the maximal ideal Iπp,ξ, := [aI ]π : , a ∈ QC(), a(ξ ) = 0 of the central Banach subalgebra Zπp () := [aI ]π : a ∈ Ap,w () QC() of πp (). By the Allan-Douglas local principle, an operator A ∈ 3 is Fredholm on the space Lp () if and only if for every ξ ∈ M(QC()) the coset Aπp,ξ, := Aπ + Jπp,ξ, is invertible in the quotient Banach algebra πp,ξ () := πp ()/Jπp,ξ, . Our aim is to study the cosets [wS w −1 I ]πp,ξ, for points ξ ∈ M(QC()) associated with endpoint 0 of . Applying the ideals Jπp,ξ, ⊂ πp () instead of π the closed two-sided ideals Jp,ξ, ⊂3 Ap,w () generated by the ideals Iπp,ξ, allows one to simplify essentially the cosets [wS w −1 I ]πp,ξ, . The paper is organized as follows. In Sect. 2 the C ∗ -algebra SO  of slowly 

oscillating functions on R and the C ∗ -algebra QC of quasicontinuous functions 

on R are considered and their maximal ideal spaces are described. Similarly to [32], each fiber Mt (QC) of the maximal ideal space M(QC) of QC for t ∈ R is the union 3t− (QC), Mt0 (QC) and M 3t+ (QC). In Sect. 3, for of the three pairwise disjoint sets M p ∈ (1, ∞) and w ∈ Ap (T), the maximal ideal space of the Banach algebra , Zπp,w := [aI ]π : a ∈ QC(T) ⊂ Bπ (Lp (T, w))

(5)

is identified and the compactness of the commutators [aI, ST ] ∈ B(Lp (T, w)) for all a ∈ QC(T) is shown. This implies that [aI, wS w −1 I ] ∈ K(Lp ()) for a ∈ QC(). In Sect. 4, modifying [22, Section 6], we develop a procedure of smoothness improvement for the Muckenhoupt weights satisfying assumption (A) (see Sect. 4.4). Section 5 deals with Mellin pseudo-differential operators with nonregular symbols, their special symbols and their applications to studying operators wSR+ w −1 I ∈ B(Lp (R+ )) for p ∈ (1, ∞) and some Muckenhoupt weights w ∈ Ap (R+ ). Section 6 is devoted to a useful localization of a sufficiently general class of Muckenhoupt weights w ∈ Ap () at points ξ ∈ M00 (QC()), where  = [0, δ] ⊂ R+ , by using Mellin pseudo-differential operators with non-regular symbols and quasicontinuous functions in QC(), with a subsequent simplification of the cosets [wS w −1 I ]πp,ξ, for ξ ∈ M00 (QC()). As a result, this localization reduces considered Muckenhoupt weights w ∈ Ap () to power weights wξ ∈ Ap () parameterized by points ξ ∈ M00 (QC()).

Localization of Muckenhoupt Weights

285

2 The C ∗ -Algebras SO  and QC 2.1 The C ∗ -Algebra SO  of Slowly Oscillating Functions 



For a bounded measurable function f : R → C and a set I ⊂ R, let , osc (f, I ) = ess sup |f (t) − f (s)| : t, s ∈ I . Following [1, Section 4], we say that a function f ∈ L∞ (R) is slowly oscillating at 

a point λ ∈ R if for every r ∈ (0, 1) or, equivalently, for some r ∈ (0, 1),   lim osc f, λ + ([−x, −rx] ∪ [rx, x]) = 0 if λ ∈ R,

x→+0

  lim osc f, [−x, −rx] ∪ [rx, x] = 0 if λ = ∞.

x→+∞ 

For every λ ∈ R, let SOλ denote the C ∗ -subalgebra of L∞ (R) defined by    SOλ := f ∈ Cb (R \ {λ}) : f slowly oscillates at λ , 



where Cb (R \ {λ}) := C(R \ {λ}) ∩ L∞ (R). Let SO  be the minimal C ∗ -subalgebra 

of L∞ (R) that contains all the C ∗ -algebras SOλ with λ ∈ R. In particular, SO  

contains the C ∗ -algebra C := C(R). Given a commutative unital C ∗ -algebra A, we denote by M(A) the maximal 

ideal space of A. Identifying the points λ ∈ R with the evaluation functionals δλ , 

δλ (f ) = f (λ) for f ∈ C, we see that M(C) maximal ideal space  = R and the  M(SO  ) of SO  is of the form M(SO  ) =  Mλ (SO ), where λ∈R

, Mλ (SO  ) := ξ ∈ M(SO  ) : ξ |C = δλ 

are fibers of M(SO  ) over points λ ∈ R. Applying [22, Proposition 2.1] and [2, 

Proposition 5], we infer that for every λ ∈ R, ∗ R) \ R, Mλ (SO  ) = Mλ (SOλ ) = M∞ (SO∞ ) = (closSO∞ ∗ R is the weak-star closure of R in SO ∗ , the dual space of SO∞ . where closSO∞ ∞ For a segment  ⊂ R, let SO  () be the restriction of SO  to .

286

Yu. I. Karlovich

2.2 The C ∗ -Algebra QC of Quasicontinuous Functions 



Consider the C ∗ -algebra QC = QC(R) of quasicontinuous functions on R, which is defined by (2). Since C ⊂ QC, we conclude that M(QC) =



Mt (QC),

, Mt (QC) := ξ ∈ M(QC) : ξ |C = t ,

(6)



t∈R 

where Mt (QC) are fibers of the maximal ideal space M(QC) over points t ∈ R. For each (λ, t) ∈ (0, 1) × R, the map δλ,t

1 : QC → C, f →  δλ,t (f ) := 2λ



t+λ

f (x)dx, t−λ

defines a linear functional in QC ∗ , which is identified with the point (λ, t). For t ∈ R, let Mt0 (QC) := Mt (QC) ∩ closQC ∗ ((0, 1) × {t}) denote the set of functionals in Mt (QC) that lie in the weak-star closure of the set (0, 1) × {t}. For every t ∈ R, we also consider the sets , Mt+ (QC) := ξ ∈ Mt (QC) : ξ(f ) = 0 if f ∈ QC and lim sup |f (z)| = 0 , z→t +

,

Mt− (QC) := ξ ∈ Mt (QC) : ξ(f ) = 0 if f ∈ QC and lim sup |f (z)| = 0 . z→t −

For each t ∈ R, it follows from [32, Lemma 8] that Mt+ (QC) ∩ Mt− (QC) = Mt0 (QC),

Mt+ (QC) ∪ Mt− (QC) = Mt (QC).

Hence, the fiber Mt (QC) given by (6) splits into the three disjoint sets: Mt0 (QC), 3t+ (QC) := Mt+ (QC) \ Mt0 (QC), M

3t− (QC) := Mt− (QC) \ Mt0 (QC). M

It is clear that the C ∗ -algebra QC = QC(T) of quasicontinuous functions on T can be defined as 

QC(T) = {f ◦ β : f ∈ QC(R)}, 

(7)

where β : T → R, β(t) = i(t + 1)/(1 − t). Similarly, one can define the C ∗ algebra P QC = P QC(T) of piecewise quasicontinuous functions on T on the 

basis of P QC(R).

Localization of Muckenhoupt Weights

287 

It follows from [32, p. 823] and (7) that for QC = QC(R) and every t ∈ R, the set Mt0 (QC) can be identified with the fiber Mt (SOt ) = Mt (SO  ). Hence, for every ξ ∈ Mt0 (QC) and every a ∈ QC, there exists a slowly oscillating function b ∈ SOt and a point  ξ ∈ Mt (SOt ) such that a(ξ ) = b( ξ ). Let  be a finite segment of R. For each interval I ⊂  and each f ∈ L1 (), the  −1 average of f over I is given by I (f ) := |I | I f (x)dx, where |I | is the length of I . A function f ∈ L1 () is said to have vanishing mean oscillation on  if  lim

δ→0

sup

I ⊂T, |I |≤δ

1 |I |

|f (τ ) − I (f )| |dτ | = 0.

 I

The set of functions of vanishing mean oscillation on  is denoted by V MO(). According to [31] and [32], the C ∗ -algebra QC() of quasicontinuous functions on  can be defined as QC() := V MO() ∩ L∞ ().

(8)

3 The Banach Algebras Zp,w and Zπp,w Letting Bp,w := B(Lp (T, w)) and Kp,w := K(Lp (T, w)) for p ∈ (1, ∞) and w ∈ Ap (T), we consider the Banach algebra , Zp,w := aI : a ∈ QC ⊂ Bp,w , where QC := QC(T) is the C ∗ -algebra of quasicontinuous functions on T. Along with Zp,w , we consider the quotient Banach algebra Zπp,w given by (5) and consisting of the cosets [aI ]π := aI + Kp,w for all a ∈ QC. Lemma 1 If p ∈ (1, ∞) and w ∈ Ap (T), then the maximal ideal spaces M(Zπp,w ) and M(QC) can be identified: M(Zπp,w ) = M(QC).

(9)

Proof If a ∈ QC is invertible in L∞ (T), then the function 1/a belongs to the C ∗ -algebra QC. Hence the coset [(1/a)I ]π is the inverse of the coset [aI ]π in the Banach algebra Zπp,w , which implies that sp [aI ]π ⊂ sp a for every a ∈ QC,

(10)

where sp x denotes the spectrum of an element x in a unital Banach algebra. If the coset [aI ]π is invertible in the Banach algebra Zπp,w , then there exists a function b ∈ QC and a compact operator K ∈ Kp,w such that (ab − 1)I = K.

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Hence ab = 1 because Zp,w ∩ Kp,w = {0}, which implies the inclusion sp a ⊂ sp [aI ]π for every a ∈ QC.

(11)

sp a = sp [aI ]π for every a ∈ QC.

(12)

By (10) and (11),

Moreover, the map a → [aI ]π is a bijection of QC onto Zπp,w . Since [aI ]π :=

inf

K∈Kp,w

aI + K Bp,w ≤ aI Bp,w = a L∞ (T)

and since, in view of (12), a L∞ (T) = r(a) = r([aI ]π ) ≤ [aI ]π , where r(x) denotes the spectral radius of an element x, we conclude that [aI ]π = a L∞ (T) , and therefore the map a → [aI ]π is an isometric isomorphism of QC onto Zπp,w . This allows one to identify the maximal ideal spaces of QC and Zπp,w by the formula 3 μ([aI ]π ) = μ(a), where a ∈ QC, μ ∈ M(QC) and 3 μ ∈ M(Zπp,w ), which gives (9).   Similarly to the proof of [25, Theorem 3.2], the Stein-Weiss interpolation theorem (see, e.g., [3, Corollary 5.5.4]) implies the following weighted analogue of the Krasnoselskii theorem [26, Theorem 3.10] on interpolation of compactness. Theorem 1 Let 1 < pi < ∞, wi ∈ Api (T) and T ∈ B(Lpi (T, wi )) for i = 1, 2. If the operator T is compact on the space Lp1 (T, w1 ), then T is compact on every space Lp (T, w) where 1 1−θ θ = + , p p1 p2

w = w11−θ w2θ ,

0 < θ < 1.

Note that Theorem 1 also follows from the Stein-Weiss interpolation theorem (see [3, Corollary 5.5.2]) and the one-sided compactness result obtained by the real interpolation method in [13, Theorem 1.1] (see also [12]). Theorem 2 Let p ∈ (1, ∞), w ∈ Ap (T) and a ∈ L∞ (T). The commutator [aI, ST ] = aST − ST aI is compact on the space Lp (T, w) if and only if a ∈ QC. Proof Combining the compactness criteria from [17] and Sarason’s representation of quasicontinuous functions     QC = H ∞ (T) + C(T) ∩ H ∞ (T) + C(T) immediately implies the criterion: for a ∈ L∞ (T), the commutator Ta := [aI, ST ] is compact on the space L2 (T) if and only if a ∈ QC (see also [28, Section 2]).

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289

Since each commutator Ta for a ∈ L∞ (T) is bounded on all the spaces Lp (T, w) for p ∈ (1, ∞) and w ∈ Ap (T), we infer from Theorem 1 (or from the complex extrapolation compactness results in [13, Theorem 2.1] or [12, Theorem 5.3]) that the compactness of Ta on any of the spaces Lp (T, w) is equivalent to the compactness of Ta on L2 (T), which in its turn is equivalent to the fact that a ∈ QC.   By analogy with Theorem 2, we obtain the following. Corollary 1 If p ∈ (1, ∞),  is a finite segment of R, and w ∈ Ap (), then [aI, wS w −1 I ] ∈ K(Lp ()) for all a ∈ QC().

4 Muckenhoupt Weights 4.1 Submultiplicative Functions and Their Indices By [5, p. 13], a function - : (0, ∞) → (0, ∞) is called regular if it is bounded from above in some open neighborhood of the point 1. A function - : (0, ∞) → (0, ∞) is called submultiplicative if -(x1 x2 ) ≤ -(x1 )-(x2 ) for all x1 , x2 ∈ (0, ∞). For every regular submultiplicative function - : (0, ∞) → (0, ∞), there exist its lower index α(-) and its upper index β(-) defined by log -(x) log -(x) = lim , x→0 log x x∈(0,1) log x

α(-) := sup

log -(x) log -(x) β(-) := inf = lim , x→∞ x∈(1,∞) log x log x

(13)

and −∞ < α(-) ≤ β(-) < +∞ (see, e.g., [5, Theorem 1.13]). Let  := [0, δ], where 0 < δ < ∞. Following [5, p. 15], with every continuous function ψ :  \ {0} → (0, ∞), we associate two functions W0 ψ, W00 ψ : (0, ∞) → (0, ∞] given by "   sup0 0 and x ∈ (x0 , ∞). Then it follows from (25) that min

y∈[x −1 R,R]

   3 (y) ≤ yV

1 log x



R

3 (y) yV

x −1 R

3(R) − V 3(x −1 R) V dy = , y log x

and therefore, by the first relations in (29), lim sup R→0

3 3 −1    3 (y) ≤ lim sup V (R) − V (x R) ≤ β(W00 ψ)+ε. yV log x y∈[x −1 R,R] R→0 min

(30)

3 (y) is in SO0 (), it follows from (18) that Since the function y → y V lim sup R→0

min

y∈[x −1 R,R]

   3 (y) = lim sup yV R→0

max

y∈[x −1 R,R]

      3 (y) = lim sup y V 3 (y) . yV y→0

   3 (y) ≤ β(W 0 ψ)+ε, which implies in view of (28) Hence, by (30), lim supy→0 y V 0 and the arbitrariness of ε > 0 the equality    3 (y) = β(W 0 ψ). lim sup y V 0

(31)

y→0

Analogously, fix ε > 0 and x ∈ (0, x0 ). Then it follows from (25) that    3 (y) ≥ max y V

y∈[xR,R]

1 log x



xR

3 (y) yV

R

3(xR) − V 3(R) V dy = , y log x

and therefore, by the second relations in (29), 3 3    3 (y) ≥ lim inf V (xR) − V (R) yV R→0 y∈[xR,R] R→0 log x   3(xR) − V 3(R) lim supR→0 V = ≥ α(W00 ψ) − ε. log x

lim inf max

3 (y) is in SO0 (), we infer from (18) that As the function y → y V lim inf max

R→0 y∈[xR,R]

         3 (y) = lim inf min y V 3 (y) = lim inf y V 3 (y) . yV R→0 y∈[xR,R]

y→0

(32)

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   3 (y) ≥ α(W 0 ψ) − ε, which gives in view of (28) Hence, by (32), lim infy→0 y V 0 the equality    3 (y) = α(W00 ψ). (33) lim inf y V y→0

It remains to prove the regularity of the function W0 ψ. It again suffices to consider ψ : (0, δ) → (0, ∞). By (25) and (26), we obtain      log ψ(xR)/ψ(R)    3 3 (y) if x ∈ (0, 1), ≤ sup y V inf y V (y) ≤ y∈(0,δ) log x y∈(0,δ)      log ψ(R)/ψ(x −1 R)    3 3 (y) if x ∈ (1, ∞). inf y V (y) ≤ ≤ sup y V y∈(0,δ) log x y∈(0,δ) 3 (y) that This implies by (14) and the boundedness of the function y → y V −∞ < inf



y∈(0,δ)

      log (W0 ψ)(x)  3 3 (y) < +∞. ≤ sup y V y V (y) ≤ log x y∈(0,δ)

Thus, the function W0 ψ is regular, which together with Theorem 3 immediately imply the assertion of the theorem, where (23) follows from (16), (31) and (33).  

4.4 Weights Locally Equivalent to Slowly Oscillating Muckenhoupt Weights Let 1 < p < ∞,  = [0, δ] ⊂ R and w = ev ∈ Ap (). Then v = ln w ∈ BMO() (see, e.g., [15, p. 258] or [5, Theorem 2.5]) and the function V given by (19) belongs to C( \ {0}). In what follows we assume that (A) there is a closed semi-neighborhood γ ⊂  of zero such that the function x → xV  (x) belongs to the C ∗ -algebra QC(γ ) := V MO(γ ) ∩ L∞ (γ ). Since QC(γ ) ⊂ BMO(γ ) and since xV  (x) = v(x) − V (x) for almost all x ∈ ,

(34)

we infer from (A) that the function V is in BMO(γ ) along with v ∈ BMO(γ ). By (A) and the definition of V MO(γ ) in [15, Chapter VI], the function x → xV  (x) is in V MO0 (γ ) ∩ L∞ (γ ). Hence, by Lemma 2 and (22), the function x →

1 x



x

τ V  (τ )dτ (x ∈ γ \ {0})

0

N0 (γ ) ∩ L∞ (γ ). belongs to the set SO0 (γ ) = SO

Localization of Muckenhoupt Weights

295

Following [22], we say that a weight w is locally equivalent to a weight W on  if w/W, W/w ∈ L∞ (γ ) on a small neighborhood γ of zero. Setting W = eV on  \ {0}, we deduce from (A) and (34) that the weight w = ev is locally equivalent to the weight W = eV on . For x ∈  \ {0}, we also define the functions 3(x) := 1 V x



x

V (τ )dτ, 0

(x) := 1 V x



x

3(τ )dτ. V

(35)

0

In view of (19) and (34)–(35), we infer that 3(x) = 3 (x) = V (x) − V xV

1 x



x

τ V  (τ )dτ for all x ∈ γ \ {0},

(36)

0

3 where  the function x → x V (x) belongs to SO0 (γ ) along with the function x → 1 x x 0 τ V (τ )dτ . Then the weights w and W are locally equivalent on  to the weight 3 belong to Ap (γ ). 3 = eV3 . Since w ∈ Ap (), it follows that the weights W and W W 3 It is easily seen that W ∈ Ap (γ ) if and only if W ∈ Ap (γ ). 3 is continuously differentiable on γ \ {0} and since the Since the function V 3 (x) is in SO0 (γ ), we deduce from Theorem 4 (cf. [5, function σ : x → x V 3 = eV3 is in Ap (γ ) if and only if (23) holds. Theorem 2.36]) that the weight W  given by (35), we get By analogy with (36), for the function V  (x) = V 3(x) − V (x) = xV

1 x



x

3 (τ )dτ for all x ∈ γ \ {0}. τV

(37)

0

 (x) belongs to SO0 (γ ) along with the function x → Hence the function x → x V  3 (x) in view of Lemmas 3 and 2. Moreover, by (37) and [18, Proposition 3.4], xV     (x) − x V 3 (x) = 0. Then (23) holds for V  in place of V 3. limx→0 x V x Repeating the procedure V → x1 0 V (τ )dτ several times, we can attain an arbitrary smoothness of locally equivalent weights under condition (A) (see [22, Section 6.2]), where the obtained weights are in Ap (γ ) along with initial one.

5 Mellin Pseudo-Differential Operators and Their Applications 5.1 Boundedness and Compactness of Mellin Pseudo-Differential Operators If a is an absolutely continuous function of finite total variation on R, then a  ∈ L1 (R) and V (a) = R |a  (x)|dx (see, e.g., [27, Chapter VIII, § 3; Chapter IX, § 4]). The set V (R) of all absolutely continuous functions a of finite total variation on R

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forms a Banach algebra when equipped with the norm a V := a L∞ (R) + V (a). Let R+ = (0, ∞). Following [19, 20], let Cb (R+ , V (R)) denote the Banach algebra of all bounded continuous V (R)-valued functions b on R+ with the norm b(·, ·) Cb (R+ ,V (R)) = sup b(r, ·) V . r∈R+

As usual, let C0∞ (R+ ) be the set of all infinitely differentiable functions of compact support on R+ . Let dμ(t) = dt/t for t ∈ R+ . Mellin pseudo-differential operators are generalizations of Mellin convolution operators. The following boundedness result for Mellin pseudo-differential operators was obtained in [20, Theorem 6.1] (see also [19, Theorem 3.1]). Theorem 5 If b ∈ Cb (R+ , V (R)), then the Mellin pseudo-differential operator Op(b), defined for functions f ∈ C0∞ (R+ ) by the iterated integral 

 1 Op(b)f (r) = 2π

 iλ r df or r ∈ R+ , dλ b(r, λ) f (-) R R+





extends to a bounded linear operator on every space Lp (R+ , dμ) with p ∈ (1, ∞), and there is a number Cp ∈ (0, ∞) depending only on p such that Op(b) B(Lp (R+ ,dμ)) ≤ Cp b Cb (R+ ,V (R)) . Following [32], a function f ∈ Cb (R+ ) is called slowly oscillating (at 0 and ∞) if for each (equivalently, for some) λ ∈ (0, 1), lim osc(f, [λx, x]) = 0 (s ∈ {0, ∞}),

x→s

, where osc(f, [λx, x]) := sup |f (r) − f (-)| : r, - ∈ [λx, x] is the oscillation of f on the segment [λx, x] ⊂ R+ . Obviously, the set SO(R+ ) of all slowly oscillating (at 0 and ∞) functions in Cb (R+ ) is a unital commutative C ∗ -algebra properly containing C(R+ ), the C ∗ -algebra of all continuous functions on R+ := [0, +∞]. Let SO(R+ , V (R)) denote the Banach subalgebra of Cb (R+ , V (R)) consisting of all V (R)-valued functions b on R+ that slowly oscillate at 0 and ∞, that is, C lim cmC x (b) = lim cmx (b) = 0,

x→0

x→∞

Localization of Muckenhoupt Weights

297

where 1 ,1 1b(r, ·) − b(-, ·)1 ∞ : r, - ∈ [x, 2x] . cmC x (b) = max L (R) Let E(R+ , V (R)) be the Banach algebra of all V (R)-valued functions b belonging to SO(R+ , V (R)) and such that 1 1 lim sup 1b(r, ·) − bh (r, ·)1V = 0,

|h|→0 r∈R+

where bh (r, λ) := b(r, λ + h) for all (r, λ) ∈ R+ × R. The following result on compactness of commutators of Mellin pseudodifferential operators was obtained in [21, Theorem 3.5] (see also [19, Corollary 8.4]). Theorem 6 If a, b ∈ E(R+ , V (R)), then the commutator [Op(a), Op(b)] is a compact operator on every space Lp (R+ , dμ) with p ∈ (1, ∞).

5.2 Symbols of Mellin Pseudo-Differential Operators Lemma 4 If a function σ ∈ SO(R+ ) satisfies the condition ν+ < 1, 0 0 in view of (38) and (41). Hence, we conclude from (40) and (42) that, for all x ∈ [3 ν− − 1/p,3 ν+ − 1/p] and all λ ∈ R, |E(x, λ)| ≤ M1 := 2/(1 − d) < ∞.

(43)

Since coth2 z = 1 + sinh−2 z, we deduce from (40) and (43) that

 

|b(r, λ)| = coth π λ + π i(1/p + σ (r)) < M2 for all (r, λ) ∈ R+ × R,

(44)

where M2 := (1 + M12 )1/2 < ∞. Further, by (40) and (42), |E(x, λ)| ≤

1 sinh (π λ) + 2−1 (1 − d) 2

(45)

,

which implies that 

 sup

1/p+x∈[3 ν− ,3 ν+ ] R

|E(x, λ)|dλ ≤ M :=

R

dλ sinh (π λ) + 2−1 (1 − d) 2

< ∞. (46)

Since π ∂b (r, λ) = −   = −π E(σ (r), λ), 2 ∂λ sinh π λ + π i(1/p + σ (r))

(47)

we infer from (46) that 



∂b V (b(r, ·)) =

(r, λ) dλ ≤ π M. R ∂λ

(48)

Hence, we see from (44) and (48) that sup b(r, ·) V ≤ M2 + π M.

(49)

r∈R+

Further, we obtain     b(r, λ) − b(-, λ) = coth π λ + π i(1/p + σ (r)) − coth π λ + π i(1/p + σ (-))  σ (r) dx = πi (50)  . 2 π λ + π i(1/p + x) sinh σ (-) Hence, we deduce from (50), (40) and (43) that |b(r, λ) − b(-, λ)| ≤ π M1 |σ (r) − σ (-)|.

(51)

Localization of Muckenhoupt Weights

299

In virtue of (40), for all λ ∈ R we have  E(σ (r), λ) − E(σ (-), λ) = −2π i

σ (r) σ (-)

  coth π λ + π i(1/p + x)   dx, sinh2 π λ + π i(1/p + x)

which implies in view of (47) that

∂b



∂b



(-, λ) = π E(σ (r), λ) − E(σ (-), λ)

(r, λ) − ∂λ ∂λ  

 σ (r)

coth π λ + π i(1/p + x)

2

≤ 2π  

dx

2 σ (-) sinh π λ + π i(1/p + x) ≤ 2π 2 M2

|σ (r) − σ (-)| .  sinh π λ) + 2−1 (1 − d) 2

(52)

Further, applying (52), for all r ∈ R+ we get





∂b

(r, λ) − ∂b (-, λ) dλ

∂λ ∂λ R  dλ ≤ 2π 2 M2 |σ (r) − σ (-)|  2 −1 R sinh π λ) + 2 (1 − d)

V (b(r, ·) − b(-, ·)) =

≤ 2π 2 M2 M|σ (r) − σ (-)|.

(53)

Combining (51) and (53), we deduce that sup b(r, ·) − b(-, ·) V ≤ (π M1 + 2π 2 M2 M)|σ (r) − σ (-)|. r,-∈R+

By (49) and (54), b ∈ Cb (R+ , V (R)). Moreover, by (51), cmC x (b) ≤ π M1 osc(σ, [x, 2x]), x ∈ R+ . Since σ ∈ SO(R+ ), we infer from the latter estimate that lim cmC x (b) = lim osc(σ, [x, 2x]) = 0 (s ∈ {0, ∞}).

x→s

x→s

Thus, b ∈ SO(R+ , V (R)). It remains to prove that b ∈ E(R+ , V (R)). It follows from (47) that   ∂E ∂ 2b 2 coth π λ + π i(1/p + σ (r)) (σ (r), λ) = 2π (r, λ) = −π  . ∂λ ∂λ2 sinh2 π λ + π i(1/p + σ (r))

(54)

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Yu. I. Karlovich

Hence, we infer from (44) and (42) that

2

∂ b

1 2



∂λ2 (r, λ) ≤ 2π M2 sinh2 (π λ) + 2−1 (1 − d) . For all (r, h) ∈ R+ × R, we infer from (47) and (43) that

 λ+h



∂b (r, y)dy

≤ π M1 |h|. b(r, ·) − bh (r, ·) L∞ (R) = sup

∂y λ λ∈R On the other hand, for r ∈ R+ and h > 0, we deduce from (55) that





∂b ∂b h



V (b(r, ·) − b (r, ·)) =

∂λ (r, λ + h) − ∂λ (r, λ) dλ R

  λ+h 2



∂ b

= (r, y)dy

dλ

2 ∂y R λ

  λ+h 2

∂ b



≤ dλ

∂y 2 (r, y) dydλ R λ  λ+h  dy 2 ≤ 2π M2 dλ . 2 sinh (πy) + 2−1 (1 − d) R λ

(55)

(56)

(57)

Changing the order of integration and taking into account (46), we get for h > 0,  λ+h  y   dy dy dλ dλ = 2 −1 sinh2 (πy) + 2−1 (1 − d) R R sinh (πy) + 2 (1 − d) y−h λ ≤ Mh.

(58)

Combining (57) and (58), we see that V (b(r, ·) − bh (r, ·)) ≤ 2π 2 M2 Mh (h > 0).

(59)

Analogously it can be shown that V (b(r, ·) − bh (r, ·)) ≤ 2π 2 M2 M(−h) (h < 0).

(60)

Thus, by (59) and (60), we conclude that sup V (b(r, ·) − bh (r, ·)) ≤ 2π 2 M2 M|h| for all h ∈ R.

(61)

r∈R+

Combining (56) and (61), we arrive at the equality lim sup b(r, ·) − bh (r, ·) V = 0,

|h|→0 r∈R+

which means that b ∈ E(R+ , V (R)).

 

Localization of Muckenhoupt Weights

301

Lemma 5 If functions σ1 , σ2 ∈ SO(R+ ) satisfy the condition ν+ < 1 (k = 1, 2), 0 0 there exists an x0 > 0 such that |0 (t) − 0 (τ )| < ε for all t, τ ∈ [x/2, x] and all x > x0 .

(81)

Then, for such t and τ , it follows that 

1  t

1 τ



0 (s)ds − 0 (s)ds

|1 (t) − 1 (τ )| =

t 0 τ 0 

1  x

1 x



=

0 (yt/x)dy − 0 (yτ/x)dy

x 0 x 0 

1 x

≤ 0 (yt/x) − 0 (yτ/x) dy, x 0

(82)

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Yu. I. Karlovich



where, by (81), yt/x, yτ/x ∈ [y/2, y] and therefore 0 (yt/x) − 0 (yτ/x) < ε if y > x0 . Hence, taking M := supx∈R+ |0 (x)|, we infer from (80) that sup |k (x)| ≤ sup |k−1 (x)| ≤ M x∈R+

(83)

x∈R+

and therefore   x 



1  x0 1 x

0 (yt/x) − 0 (yτ/x) dy

0 (yt/x) − 0 (yτ/x) dy ≤ + x 0 x 0 x0 ≤

2x0 M x − x0 + ε. x x

(84)

Thus, by (82) and (84), lim

sup

x→∞ t,τ ∈[x/2,x]

|1 (t) − 1 (τ )| = 0,

and therefore the function 1 slowly oscillates at ∞. Moreover,    (x) = 0 for all s ∈ {0, ∞}. lim xVk (x) − xVk−1

x→s

(85)

Clearly, it suffices to show (85) for k = 1. For s = 0, this follows from (80) and [22, Remark 6.3]. If s = ∞, then we infer for x ∈ (2n−1 x0 , 2n x0 ] in view of (80), (81) and (83) that  x







xV (x) − xV  (x) ≤ 1

tV (t) − xV  (x) dt 1 x 0  x0 n  s



 1 1 2 x0



tV (t) − xV  (x) dt + ≤ n−1 tV (t) − xV  (x) dt n−1 2 x0 0 2 x0 2s−1 x0 s=1

≤

1 2n−1 x0



x0

2Mdt +

0

1 2n−1 x0

n 

2s x

0

s−1 s=1 2 x0

(n − s + 1)εdt

s 2M + 2ε , n−1 2s 2 n

=

s=1

 s which is small along with ε if n is sufficiently large, because ∞ s=1 2s < ∞. This proves (85) for s = ∞. Letting (DV )(x) := xV  (x), we deduce from (80) that the functions D3 = 2 − 3 , D 2 3 = 1 − 22 + 3 , D 3 3 = 0 − 31 + 32 − 3

Localization of Muckenhoupt Weights

305

belong to SO(R+ ) as well. Finally, by (80) and (83), inequalities (77) remain valid for all xVk (x) (k = 1, 2, 3) in place of xV  (x). Then the weights eVk are in Ap (R+ ) for all k = 1, 2, 3. It is easily seen (cf. [21, Theorem 5.3]) that the function b3 given by   b3 (r, λ) := coth π λ + π i(1/p + rV3 (r)) for all (r, λ) ∈ R+ × R

(86)

belongs to E(R+ , V (R)) along with the function b given by (79). Furthermore, it follows from [21, Theorems 5.2, 5.3] that T(eV3 SR+ e−V3 I ) = OP (b3 ) + K,

(87)

where K ∈ K(Lp (R+ , dμ)). As the function eV −V3 belongs to SO(R+ ) and therefore the commutator [eV −V3 , SR+ ] is a compact operator on the space Lp (R+ ), we deduce from (87) that T(eV SR+ e−V I ) − OP (b3 ) ∈ K(Lp (R+ , dμ).

(88)

Since the function b3 − b ∈ E(R+ , V (R)) vanishes on the boundary of the set R+ × R in view of (79) and (86), we infer from [21, Theorem 3.4] that OP (b3 ) − OP (b) ∈ K(Lp (R+ , dμ)). Finally, by (88) and (89), we obtain (78).

(89)  

Note that equality (78) is proved in Theorem 7 under essentially less restrictive conditions in comparison with papers [6] and [7].

6 Localization of Muckenhoupt Weights Satisfying Condition (A) In what follows, we assume that p ∈ (1, ∞),  = [0, δ] ⊂ R and a weight w = ev ∈ Ap () satisfies condition (A). Following Sect. 4.4, we associate with the weight 3 = eV3 ∈ w ∈ Ap () the locally equivalent (at a neighborhood γ of zero) weight W Ap (γ ) such that the function σ , given by 3 (x) for all x ∈ γ \ {0}, σ (x) = x V

(90)

belongs to the set SO0 (γ ). This allows us to define the numbers 3 (ξ ) for every ξ ∈ M0 (SO0 (γ )), δξ := ξ(σ ) := ξ V

(91)

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where M0 (SO0 (γ )) denotes the fiber over the point 0 of the maximal ideal space M(SO0 (γ )) of the C ∗ -algebra SO0 (γ ). By (23), δξ ∈ (−1/p, 1/q). Given p ∈ (1, ∞), a weight w = ev ∈ Ap () satisfying condition (A), and the Ap,w (), where the Banach algebra 3 Ap,w () is defined operator A := wS w −1 I ∈ 3 by (4), let us study the cosets Aπp,ξ, := Aπ + Jπp,ξ, ∈ πp,ξ () for all ξ ∈ M00 (QC()). Theorem 8 Let p ∈ (1, ∞),  = [0, δ] and let w = ev ∈ Ap () be a weight satisfying condition (A). If ξ ∈ M00 (QC()), then [wS w −1 I ]πp,ξ, = [wξ S wξ−1 I ]πp,ξ, ,

(92)

where wξ (x) = x δξ for all x ∈ , and δξ ∈ (−1/p, 1/q) is given by (90) and (91). 3 Proof Along with v = ln w ∈ BMO(), we consider the functions V and V defined on  \ {0} by formulas (19) and (35), respectively. Then it follows that 3 ∈ BMO(γ ), the function : x → xV  (x) is in QC(γ ) by condition (A), v, V , V 3 (x) belongs to SO0 (γ ), where γ ⊂  is a closed and the function x → x V 3 ∈ QC(γ ) neighborhood of zero on . Then we infer from (34) and (36) that v − V 3(x) = xV  (x) + x V 3 (x) belongs to QC(γ ). because the function x → v(x) − V 3 = eV3 are locally equivalent on . Moreover, Then the weights w = ev and W taking the characteristic function χγ of γ , we obtain 3 S W 3 −1 χγ I ]π [χγ wS w −1 χγ I ]π = [χγ W 3

3

(93)

3

because ev−V ∈ QC(γ ) and hence χγ ev−V S eV −v χγ I ∈ K(Lp ()). 3 from γ to a continuous function on (0, +∞) that Extending the function V 3 for this extension, vanishes at a neighborhood of +∞ and saving the notation V 3 V we conclude that e ∈ Ap (R+ ). Hence, we deduce from Theorem 4 that       3 (x) ≤ 1/p + lim sup x V 3 (x) < 1. 0 < 1/p + lim inf x V x→0

(94)

x→0 3



Moreover, setting R+ = [0, +∞] and replacing eV by an equivalent weight eV 3  3(x) − V (x)) = 0 for s ∈ {0, ∞}, and therefore such that eV −V ∈ C(R+ ), limx→s (V 3  V − V p I, SR+ ] ∈ K(L (R+ )), we can attain the relations [e        (x) ≤ 1/p + sup x V  (x) < 1 0 < 1/p + inf x V x∈R+

x∈R+

instead of (94), and the equality  π  π 3  3  χγ eV SR+ e−V χγ I = χγ eV SR+ e−V χγ I .

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Thus, we may without loss of generality assume that       3 (x) ≤ 1/p + sup x V 3 (x) < 1. 0 < 1/p + inf x V x∈R+

(95)

x∈R+

Then it follows from Theorem 7 that 3 3 T(eV SR+ e−V I ) = Op(3 b) + K,

(96)

where T is given by (75)–(76), the function 3 b ∈ E(R+ , V (R)) is given by   3 3 (r)) , b(r, λ) := coth π λ + π i(1/p + r V

(97)

and K ∈ K(Lp (R+ , dμ)). Let Tγ : Lp (γ ) → Lp (γ , dμ) be a restriction of T. By (96), we obtain   V3   3 π  π 3 π 3 Tγ e Sγ e−V I = χγ T eV SR+ e−V χγ I = χγ Op(3 b)χγ I .

(98)

Since the operators aI for all functions a ∈ QC(γ ) commute with the operator 3 3 eV Sγ e−V I to within compact operators in K(Lp (γ )), we deduce that the commu tators aI, χγ Op(3 b)χγ I belong to K(Lp (γ , dμ)) for all a ∈ QC(γ ). 3 (x) A small in the norm of SO(R+ ) perturbation of the function σ : x → x V leads to a function θ ∈ SO(R + ), for which the function b given by b(r, λ) :=  coth π λ+π i(1/p+θ (r)) belongs to E(R+ , V (R)), and the operator χγ Op(b)χγ I also commutes to within compact operators with all operators aI for a ∈ QC(γ ). Indeed, taking the function V˘ (x) = −



δ x

θ (τ ) dτ + V˘ (δ) for all x ∈ (0, δ], τ

we infer that x V˘  (x) = θ (x) for all x ∈ (0, δ]. Since we can achieve the inequalities 0 < 1/p + inf θ (x) ≤ 1/p + sup θ (x) < 1 x∈R+

x∈R+

˘ along with (95), we conclude that the weight W˘ = eV belongs to Ap (). Then, similarly to (98),



χγ Op(b)χγ I

π

  ˘ π   ˘ ˘ ˘ π = χγ T eV SR+ e−V χγ I = Tγ eV Sγ e−V I ,

  which means due to Corollary 1 that the commutators aI, χγ Op(b)χγ I belong to K(Lp (γ , dμ)) for all a ∈ QC(γ ). Given ξ ∈ M00 (QC(γ )), we take the associated point  ξ ∈ M0 (SO(R+ )) (see 3 (η) for all η in Sect. 2.2) and choose a function θ ∈ SO(R+ ) such that θ (η) = ηV

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an open neighborhood of a point  ξ ∈ M0 (SO(R+ )) and the norm σ − θ L∞ (R+ ) 3 (η) = η(σ ) and  3 ( 3 (ξ ). is sufficiently small. Note that ηV ξV ξ) = ξV Thus, b(η, λ) = 3 b(η, λ) = 3 b(ξ, λ) for all η in an open neighborhood of a point  ξ ∈ M0 (SO(R+ )) and all λ ∈ [−∞. + ∞]. Hence there exists a function d ∈ SO(R+ ), such that d(ξ ) = 0 and d(3 b − b) = 3 b − b.

(99)

By Lemma 5, 3 b − b Cb (R+ ,V (R)) ≤ C σ − θ L∞ (R+ ) . b − b)χγ I commutes with all operators aI (a ∈ QC(γ )) Then the operator χγ Op(3 to within compact operators, and hence χγ T−1 (Op(3 b − b))χγ I ∈ p (γ ). On the other hand, by (99) and Theorem 6, χγ Op(3 b − b)χγ I = χγ Op(d(3 b − b))χγ I = dχγ Op(3 b − b)χγ I ) χγ Op(3 b − b)χγ dI,

where A ) B means that A − B is a compact operator. Consequently, the coset [χγ T−1 (Op(3 b − b))χγ I ]π belongs to the ideal Jπp,ξ,γ for given ξ ∈ M00 (QC(γ )). Finally, since the function 3 b −3 b(ξ, ·) can be approximated in the 3 functions of the form b − b, we deduce that the coset norm ofCb (R+ , V (R)) by  b −3 b(ξ, ·)) χγ I ]π also belongs to the ideal Jπp,ξ,γ . [χγ T−1 Op(3 Hence, we infer from (98) that  π  π  V3 3 π b))χγ I p,ξ,γ = χγ T−1 (Op(3 b(ξ, ·)))χγ I p,ξ,γ . e t Sγ e−Vt I p,ξ,γ = χγ T−1 (Op(3

(100) 3 = eV3 ∈ Ap (γ ), we consider the function σ ∈ SO(γ ) Taking the weight W 3 (ξ ) for all ξ ∈ M 0 (QC(γ )). Given defined by (90) with values δξ = ξ(σ ) = ξ V 0 0 ξ ∈ M0 (QC(γ )), we consider the power weight wξ given by wξ (x) = x δξ for all x ∈ γ . Similarly to (98), we obtain π  π   = χγ Op(3 b(ξ, ·))χγ I , Tγ χγ wξ SR+ wξ−1 χγ I

(101)

where the function 3 b(ξ, ·) ∈ E(R+ , V (R)) is given in view of (97) by   3 3t (ξ )) . b(ξ, λ) := coth π λ + π i(1/p + ξ V Hence, by (101), π  π  π  b(ξ, ·)))χγ I p,ξ,γ , wξ Sγ wξ−1 I p,ξ,γ = χγ wξ SR+ wξ−1 χγ I p,ξ,γ = χγ T−1 (Op(3

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which implies in view of (100) that  π  V3 3 π 3ξ Sγ w 3ξ−1 I p,ξ,γ . e Sγ e−V I p,ξ,γ = w

(102)

It follows from (93) and (102) that  π  3  π 3 π wSγ w −1 I p,ξ,γ = eV Sγ e−V I p,ξ,γ = wξ Sγ wξ−1 I p,ξ,γ .

(103)

Identifying the points ξ in M00 (QC()) and M00 (QC(γ )), and taking into account the equality (χγ I )π Jπp,ξ, (χγ I )π = Jπp,ξ,γ , we infer (92) from (103).  

References 1. M.A. Bastos, C.A. Fernandes, and Yu.I. Karlovich, C ∗ -algebras of integral operators with piecewise slowly oscillating coefficients and shifts acting freely. Integral Equations and Operator Theory 55 (2006), 19–67. 2. M.A. Bastos, Yu.I. Karlovich, and B. Silbermann, Toeplitz operators with symbols generated by slowly oscillating and semi-almost periodic matrix functions. Proc. London Math. Soc. (3) 89 (2004), 697–737. 3. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction. Springer, Berlin, 1976. 4. A. Böttcher, I. Gohberg, Yu. Karlovich, N. Krupnik, S. Roch, B. Silbermann, and I. Spitkovsky, Banach algebras generated by N idempotents and applications. Operator Theory: Advances and Applications 90 (1996), 19–54. 5. A. Böttcher and Yu.I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics 154, Birkhäuser, Basel, 1997. 6. A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, Mellin pseudo-differential operators with slowly varying symbols and singular integral on Carleson curves with Muckenhoupt weights. Manuscripta Math. 95 (1998), 363–376. 7. A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, The method of limit operators for onedimensional singular integrals with slowly oscillating data. J. Operator Theory 43 (2000), 171–198. 8. A. Böttcher, S. Roch, B. Silbermann, and I.M. Spitkovsky, A Gohberg-Krupnik-Sarason symbol calculus for algebras of Toeplitz, Hankel, Cauchy, and Carleman operators. Operator Theory: Advances and Applications 48 (1990), 189–234. 9. A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators. 2nd edition, Springer, Berlin, 2006. 10. A. Böttcher and I.M. Spitkovsky, Toeplitz Operators with PQC Symbols on Weighted Hardy Spaces. J. Funct. Anal. 97 (1991), 194–214. 11. A. Böttcher and I.M. Spitkovsky, The factorization problem: Some known results and open questions. Operator Theory: Advances and Applications 229 (2013), 101–122. 12. F. Cobos, T. Kühn, and T. Schonbek, One-sided compactness results for Aronszajn-Gagliardo functors. J. Funct. Anal. 106 (1992), 274–313. 13. M. Cwikel, Real and complex interpolation and extrapolation of compact operators. Duke Math. J. 65 (1992), no. 2, 333–343. 14. R.G. Douglas, Banach Algebra Techniques in Operator Theory. Academic Press, New York, 1972. 15. J.B. Garnett, Bounded Analytic Functions. Academic Press, New York, 1981.

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16. I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations. Vols. I and II. Birkhäuser, Basel, 1992. 17. P. Hartman, On completely continuous Hankel matrices. Proc. Amer. Math. Soc. 9 (1958), 862–866. 18. A.Yu. Karlovich, Yu.I. Karlovich, and A.B. Lebre, Invertibility of functional operators with slowly oscillating non-Carleman shifts. Operator Theory: Advances and Applications 142 (2003), 147–174. 19. Yu.I. Karlovich, An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. London Math. Soc. (3) 92 (2006), 713–761. 20. Yu.I. Karlovich, Pseudodifferential operators with compound slowly oscillating symbols. Operator Theory: Advances and Applications 171 (2007), 189–224. 21. Yu.I. Karlovich, Nonlocal singular integral operators with slowly oscillating data. Operator Theory: Advances and Applications 181 (2008), 229–261. 22. Yu.I. Karlovich and I. Loreto Hernández, Algebras of convolution type operators with piecewise slowly oscillating data. I: Local and structural study. Integral Equations and Operator Theory 74 (2012), 377–415. 23. Yu.I. Karlovich and I. Loreto Hernández, Algebras of convolution type operators with piecewise slowly oscillating data. II: Local spectra and Fredholmness. Integral Equations and Operator Theory 75 (2013), 49–86. 24. Yu.I. Karlovich and I. Loreto Hernández, On convolution type operators with piecewise slowly oscillating data. Operator Theory: Advances and Applications 228 (2013), 185–207. 25. Yu.I. Karlovich and E. Ramírez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces. Integral Equations and Operator Theory 48 (2004), 331–363. 26. M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik, and P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions. Nauka, Moscow, 1966 (Russian); English transl.: Noordhoff I.P., Leyden, 1976. 27. M.A. Naimark, Normed Algebras. Wolters-Noordhoff, Groningen, 1972. 28. S.C. Power, Hankel Operators on Hilbert Space. Bull. London Math. Soc. 12 (1980), 422–442. 29. S.C. Power, Hankel Operators on Hilbert Spaces. Pitman Research Notes in Math. 64. Pitman, Boston, 1982. 30. S. Roch, P.A. Santos, and B. Silbermann, Non-commutative Gelfand Theories. A Tool-kit for Operator Theorists and Numerical Analysts. Springer, London, 2011. 31. D. Sarason, Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207 (1975), 391–405. 32. D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols. Indiana Univ. Math. J. 26 (1977), 817–838.

Carleman Regularization and Hyperfunctions Otto Liess

Dedicated to Luigi Rodino on the occasion of his 70th birthday.

Abstract Carleman regularization is a method to give a meaning to Fouriertype integrals which are highly divergent in a classical sense. We use it to give a local representation of hyperfunctions in terms of such integrals. While such representations are not unique, uniqueness can be achieved in terms of Dolbeault type cohomology with coefficients in L2 spaces with weights. Keywords Carleman regularization · Hyperfunctions · Fourier transform

1 Introduction Originally Carleman regularization was about (formal) integrals of type u(x) =  exp [ixξ ]μ(ξ )dξ where μ is measurable on the real line R and has at most R polynomial growth at infinity. (See [2].) Since μ is then a temperate distribution, u can be understood in terms of the (more general) theory of the Fourier transform in temperate distributions of L.Schwartz, but such distributions were not available at the time of [2]. Very similar to this is the case when one looks into integrals of form  u(x) = exp [ix, ξ ]μ(ξ ) dξ, (1) X

with X some unbounded set in Rn . The main obstacle in the study of such integrals is that it is in general not possible to give a pointwise meaning for every x to them.

O. Liess () Department of Mathematics, University of Bologna, Bologna, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_18

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Both T.Carleman and L.Schwartz circumvented this obstacle by directly defining averages over various x. (See Remark 3.) While the procedures of Carleman and Schwartz differ significantly, the results which they obtain are for the cases studied by Carleman the same. On the other hand Carleman’s approach offers some advantages when one wants to work with the Fourier transform in hyperfunctions and is interested in local results. It has been observed by a number of authors that it is then natural to admit functions μ which satisfy an estimate of form |μ(ξ )| ≤ exp [ϕ(ξ )] for some sublinear function ϕ. We will say that ϕ is “sublinear” if limx→∞ ϕ(ξ )/|ξ | = 0. The method of Carleman is general enough to give a meaning for Fourier transforms of finite order distributions which satisfy growth restrictions of the type just imposed on μ. For some results in this direction see e.g., [8, 11]. While such extensions are also interesting for what we have in mind, we will defer, to keep this paper in size, these extensions to another paper in which also applications to microlocal analysis will be given. In this paper we are mainly interested to discuss an extension of (1) to integrals of a related form, but when integration is on sets in Cn , i.e., we look at formal integrals of type  exp [ix, ζ ]μ(ζ ) dλ(ζ ),

u(x) =

(2)

X

where dλ is the Lebesgue measure times (2π )−n on Cn . (The factor (2π )−n is related to the Fourier inversion formula.) Such Fourier representations in terms of integrals on Cn are essential in results referring to the so-called fundamental principle: see [3, 8, 13], but only A.Kaneko in [8] treated them in the spirit of Carleman. In this paper, ξ will always denote a variable in Rn , ζ one in Cn .) The idea of Carleman regularization to give a meaning to (1), or (2), is now to look at first for the case when the domain of integration is included in a set of form {ξ ∈ Γ }, or {Re ζ ∈ Γ }, where Γ is a sharp open convex cone in Rn . The integrals on such X are still not converging for real x, but the associated integrals 

 exp [iz, ξ ]μ(ξ ) dξ, h(z) =

h(z) = X

exp [iz, ζ ]μ(ζ ) dλ(ζ ),

(3)

X

will converge, under suitable growth restrictions on the function μ, for z in a complex wedge W = {z ∈ Cn , z ∈ Ω, I m ζ ∈ Γ ⊥ }, and will define a holomorphic function there. Here Γ ⊥ is the polar of Γ and Ω is an open set in Cn , which depends on the estimates for μ. Since h is a holomorphic function defined on a wedge, it has a “boundary value” at Ω ∩ Rn . In the most general case, this boundary value will be a hyperfunction, but if by chance the initial integral is convergent for real x, then the boundary value of μ will be (but this is a somewhat vague statement) the one given by the integral. In the general case, when X is not related to a sharp convex cone in the variables Re ζ , we will split X into a finite union of such sets, associate to each of these sets a meaning as explained just before. The “value” of (3) is then the sum of the values of the various contributions. The fact that this sum does not depend on the

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way we have written X as a union of simpler domains, is an “edge-of-the-wedge” phenomenon. If we regard (2) as a representation of u in terms of μ, then it is clear that representations by integrals over complex sets give us greater flexibility when we want to find some μ when u is given. In particular we have in Theorem 3 a case where we have an almost computational method to find μ. This is important when u will depend on additional parameters and is needed in our applications. On the other hand it is only in the complex setting where we can give a characterization of the non-uniqueness in such representations: see Theorem 2, c), but we will prove part c) there only in a forthcoming paper. Many ideas on Carleman regularization have been present implicitly or explicitly in [12] and in [8, 9] (among others). The method itself has been discussed in its historical context and compared with the theory of temperate distributions of L.Schwartz by T.Kato-C.D.Struppa in [10] and by C.O.Kiselman in [11]. Also see A.Kaneko, [9].

2 Notations and Review of Hyperfunctions Lebesgue spaces with respect to the Lebesgue measure on Rn or Cn will be denoted by Lp . The space of locally L2 functions will be denoted by L2loc . Holomorphic functions on some open set Ω in Cn will be denoted by O(Ω). We will from the very beginning state the growth type conditions at infinity for μ in (2) in terms of L2 type estimates. Specifically, we will assume that  |μ|X,ϕ,−ε < ∞, where |μ|X,ϕ,−ε is the quasinorm ( X |μ(ζ )|2 exp [−2ϕ(ζ ) + 2ε|I m ζ |]dλ(ζ ))1/2 . Here ϕ is sublinear and ε > 0. We will write L2 (Cn , ϕ, −ε) for the space {μ; |μ|Cn ,ϕ,−ε < ∞} with |μ|Cn ,ϕ,−ε as a norm. The reasons why we prefer L2 estimates instead of pointwise estimates, are manifold. The main one however is that the proof of part c) in Theorem 2 depends at some moment on Hörmander’s theory of weighted ∂¯ estimates for the ∂¯ operator, and that theory works best in weighted L2 estimates (see [7]). Another reason for preferring L2 type estimates is that in some instances they lead to (slightly) sharper results. Thus e.g., if  → κ is a sequence of C 1 (Cn ) functions which have uniformly bounded derivatives, with κ (ζ ) = 1 for |ζ | ≤  , κ (ζ ) = 0 for |ζ | ≥  + 1, and if μ ∈ L2 (Cn , ϕ, −ε) then κ μ → μ and (∂¯j κ )μ → 0, both in L2 (Cn , ϕ, −ε), but the same is not true in the related (quasi)-norm supCn |μ(ζ )|/ exp [ϕ(ζ ) − ε|I m ζ |]. Hyperfunctions on an open set U ⊂ Rn will be denoted by B(U ), germs of hyperfunctions at 0 ∈ Rn by B0  . We recall that hyperfunctions on U are equivalence s classes of formal sums S = j =1 hj of holomorphic functions hj defined on infinitesimal wedges over U . If U is open in Rn then an infinitesimal wedge over U is a set of form W = {z ∈ Cn ; Re z ∈ U, I m z ∈ G}∩Ω, where G ⊂ Rn is an open ˙ n = Rn \ {0} and Ω is an open set in Cn which intersects Rn on convex cone in R U . In this situation we denote the infinitesimal wedge by U + ioG. The equivalence  relation is defined as follows. (See e.g., [9] for a variant of this.) If S  = sj =1 hj

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s   and S  = j =s  +1 hj are two such formal sums, then we will say that S is    equivalent to S and write S ∼ S , if (with the notation Gij = Gi + Gj if Gi and Gj are two open cones) we can find hij ∈ O(U + ioGij ), i, j ∈ {1, . . . , s  } with     hij = −hj i , hi (z) = sj =1 hij (z) for i ≤ s  , respectively hi (z) = − sj =1 hij (z) for s  < i ≤ s  and for every z in the common domain of definition of the hi , hij .    Note that while the sums sj =1 hj , sj =s  +1 hj are formal, the relation S  ∼ S  has a non-formal meaning. It follows that if G ⊂ G and h ∈ O(U + ioG), then the restriction of h to U + ioG is equivalent to the initial h. We denote by “b” the map which sends a formal sum S to its equivalence class and call it the boundary value map. As a justification for this terminology we recall that when |h(z)| ≤ c|I m z|−b for some holomorphic function h defined on an infinitesimal wedge W for some c and b ∈ R, then the limit limz∈W,I m z→0 h(z) exists in distributions (see [16]). It has been shown by A.Martineau, [12], that this distribution is also the hyperfunctional boundary value in the natural embedding of distributions into hyperfunctions. Taking equivalence classes commutes with taking restrictions to open subsets U  ⊂ U . If h is a function defined and holomorphic in a complex neighborhood of U , then h also defines a hyperfunction on U . We conclude the section with the definition of the “analytic wave front set”, introduced by M.Sato under the name “singular spectrum”: see [14, 15]. (For distributions a related notion, both in the C ∞ and the analytic category, was defined by L.Hörmander [5, 6]. For various relations between them, see [1, 10].) ˙ n is not in the analytic Definition 1 ([1, 9, 10]) We shall say that (x 0 , ξ 0 ) ∈ U × R 0 0 wave front set of u ∈ B(U ), and write (x , ξ ) ∈ / W FA u, if we can find open ˙ n , j = 1, . . . , s, ε > 0, d > 0, and holomorphic functions convex cones Gj ⊂ R hj definedon {x + iy ∈ Cn ; x ∈ Rn , y ∈ Rn , |x − x 0 | < ε, y ∈ Gj , |y| < d} so that u = j b(hj ) near x 0 and so that ξ 0 is not in the polar G⊥ j of Gj . It is easy 0 0 to see that if u coincides with v near 0, then (x , ξ ) ∈ / W FA u is equivalent with (x 0 , ξ 0 ) ∈ / W FA v. Remark 1 Let h ∈ O(U × iG⊥ d ). Then, by the bipolar theorem, W FA b(h) ⊂ U × co G, with “co” denoting closed convex hulls.

3 Carleman Regularization: Preparations For a generic cone Γ ⊂ Rn and a generic constant C ≥ 0 we denote by A(Γ, C) = {ξ ∈ Γ ; |ξ | ≥ C}. We begin with: Lemma 1 a) Let Γj , j = 1, . . . , s, be open cones, consider for some  C ≥ 0 L∞ functions n gj on R with supports in A(Γj , C), and assume that sj =1 gj (ξ ) ≡ 0 on

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∞ functions g with supports in A(Γ ∩ Γ , C), A(∪Γj , C). Then we can find L ij i j such that gij = −gj i and gi = sj =1 gij for |ξ | > C.    b) We use this to compare two partitions of unity, sj =1 gj ≡ 1, sj =s  +1 gj ≡ 1, on A(∪j Γj , C) associated with some cones Γj as before. We obtain a set of   functions gj , j = 1, . . . , s  , with sj gj ≡ 0 by setting gj = gj for j ≤ s  , gj = −gj for j ≥ s  + 1. We conclude that we can find gij , i, j = 1, . . . , s  ,  with gij = −gj i on A(Γi ∩ Γj , C) such that gi = sj =1 gij for j = 1, . . . , s    on |ξ | > C and gi = sj =s  +1 gj i |ξ | > C for j = s  + 1, . . . , s  .

Proof The proof is by induction in s, the case s = 2 being trivial. Assume that the statement is proved for some s. To prove it for s + 1 we use that gs+1 = − s=1 g . Also let χ be the characteristic function of the support of gs+1 calculated in distributions. We set gj,s+1 = −χgj = −g . . . , s and gs+1,s+1 = 0. s+1,j for j = 1, We have then that gs+1 = χgs+1 = − sj =1 gj,s+1 = s=1 gs+1,j . Further set  gj = gj + gs+1,j for j ≤ s, so sj =1 gj ≡ 0. By induction we can find then s gij , i, j = 1, . . . , s, with gij = −gj i and gi = i=1 gij , i = 1, . . . , s. We consider together with gij , i, j ≤ s, the functions gij where at least one of i or j is s + 1 obtained above and obtain a collection of functions gij with the required properties.   We now return to the integrals in (3). Let K be a compact in Rn or Cn . We denote by HK , respectively HK , the supporting functions of K. In the real case, HK (ξ ) = supx∈K x, ξ  and if K ⊂ Cn , then HK (ζ ) = supz∈K (Re z, I m ζ  + I m z, Re ζ ). However, also more specific notations will be used later, e.g., “HG,d ”, to be introduced in the sequel. As for compacts in Cn they will often have the form K1 + iK2 where K1 , K2 , are compacts in Rn . In this case we will have that HK (ζ ) = HK1 (I m ζ ) + HK2 (Re ζ ). Remark 2 There will be two closely related situations in which we will obtain integrability in the variables Re ζ in Carleman regularization. One is when the support of the integrand lies in a set {ζ ∈ Cn ; Re ζ ∈ Γ } for Γ a sharp convex ˙ n . We denote by G = Γ ⊥ , choose some cone G ⊂⊂ G, and can use that cone in R y, ξ  ≥ c|y| |ξ | when y ∈ G and ξ ∈ Γ . We obtain from this that | exp [iz, Re ζ ]| ≤ exp [−c|I m z| |Re ζ |] when I m z ∈ G , Re ζ ∈ Γ.

(4)

˙ n is contained in G.) (G ⊂⊂ G means that the closure of G in R A second situation is when we start from sharp open convex cones G ⊂⊂ G ⊂ n ˙ and consider for d > 0 the supporting function HG,d (ξ ) = supy∈G,|y|≤d y, ξ  R of the set {y ∈ G; |y| ≤ d}. To estimate HG,d (ξ ) we observe that HG,d (ξ ) ≥ 0 for all ξ since 0 ∈ Gd . Further, supy∈Gd y, ξ  > 0 precisely if ξ is not in −G⊥ . Indeed, at first, if −ξ ∈ G⊥ , then y, −ξ  ≥ 0 for every y ∈ G, so y, ξ  ≤ 0 for all y ∈ G

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and therefore supy∈Gd y, ξ  ≤ 0. In the other direction, assume −ξ ∈ / G⊥ , so there is y 0 ∈ G with y 0 , −ξ  ≤ 0, whence y 0 , −ξ  ≥ 0 so that supy∈Gd y, ξ  ≥ 0. Now choose Γ  ⊂⊂ −(Rn \ G⊥ ). As a consequence of the above, y, ξ  ≥ c|y| ˜ |ξ | for some constant c˜ if y ∈ G and ξ ∈ Γ  . We conclude that HG,d (−ξ ) ≥ c|ξ | if ξ ∈ Γ  , Γ  ⊂⊂ (Rn \ G⊥ ).

(5)

Despite the fact that there are some differences between (4) and (5), the two inequalities will serve a similar purpose. Let us observe that if we start from some ˙ n. open convex cones G ⊂⊂ G and denote Γ = G⊥ , then Γ ∪ (Rn \ (G )⊥ ) = R n Thus for every ξ ∈ R , one of the two inequalities will apply. Proposition 1 We assume ψ(ζ ) = ψ1 (Re ζ ) + ψ2 (I m ζ ) with the ψi sublinear and want to study (3) in the case when X ⊂ Cn . (The case when X ⊂ Rn is similar and actually the argument which follows can be used to study the real situation as ˙ n. a particular case.) Γ will be a sharp open convex cone in R  a) In this part X is {ζ ∈ Cn ; Re ζ ∈ Γ, |ζ | ≥ C} and μ satisfies X |μ(ζ )|2 exp [2ε|I m ζ | − 2ψ(ζ )]dλ(ζ ) < ∞. Then h is well-defined and holomorphic for z in the set Ω = {z ∈ Cn ; |Re z| < ε, I m z ∈ Γ ⊥ }. b) In the special case when ψ1,2 (ξ ) = b ln(1+|ξ |) then h defined in part a) satisfies  |h(z)| ≤ c |I m z|−b for some c , b ≥ 0. The boundary value of h is then accordingly a distribution. ˙ n , and Γ = G⊥ , Γ  = c) In this part, G ⊂⊂ G are open convex cones in R  ⊥ (G ) . The conclusions in the parts a), b), remains valid with Ω replaced by  n n Ω  = {z ∈2 C ; |Re z| < ε, I m z ∈ G, |I m z| < d} if X = C and μ satisfies Cn |μ(ζ )| exp [2HG,d (−Re ζ )+2ε|I m ζ |−2ψ1 (Re ζ )−2ψ2 (I m ζ )]dλ(ζ ) < ∞ for some sublinear functions ψi . The statement corresponding to a) is immediate. It also follows that we can write h = h + h where h is holomorphic on {z; |Re z| < ε, I m z ∈ G} and h is real analytic near 0. Proof The proofs are straightforward and standard. We prove first a) and b). We start from | exp [iz, ζ ]| ≤ exp [−Re z, I m ζ  − I m z, Re ζ ]. When |Re z| < ε for some 0 < ε < ε (as is the case when we assume |Re z| < ε), then −Re z, I m ζ  − ε|I m ζ | + ψ2 (I m ζ ) ≤ −(ε − ε )|I m ζ | + c2 , if we also use the sublinearity of ψ2 . This will lead to exponential decay in the variable I m ζ uniformly in a neighborhood of z, so the integrals which will give h can be estimated well in the variable I m ζ . As for the estimates regarding Re ζ , we observe first that it suffices to estimate the part of the integral which corresponds to |Re ζ | ≥ 2 (say), in that the integral over |Re ζ | < 2 is anyway bounded and gives a holomorphic function. For I m z ∈ G and Re ζ ∈ Γ we use the estimate (4) and the sublinearity of ψ1 . This gives I m z, Re ζ  − ψ1 (Re ζ ) ≤ −c |Re ζ | for some constant c which depends on I m z, but which can be assumed the same for I m z in a neighborhood of some fixed value in G .

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After these preparations, we return to (3). We write the integrand there as exp [iz, ζ ]μ(ζ ) = ν1 (ζ )ν2 (ζ ) with ν1 (ζ ) = exp [iz, ζ  + 2ψ1 (Re ζ ) + 2ψ2 (I m ζ ) − 2ε|I m ζ |], ν2 (ζ ) = μ(ζ ) exp [−2ψ1 (Re ζ ) − 2ψ2 (I m ζ ) + 2ε|I m ζ |]. Then ν1 is exponentially decaying for Re ζ ∈ Γ and I m z ∈ G and ν2 is by assumption L2 . The product ν1 ν2 is then L1 , uniformly in z near some fixed z0 . We can then integrate to give a meaning to the integral in (3). We can also derivate the integral parametrically in z. When we derivate the integral in the parameter z, we obtain the same kind of integral, if we absorb the resulting monomial factors in ζ into new ψ’s. Since the integrand is uniformly integrable for z in a neighborhood of any given z as before and since this is true also for the derivatives in z, it follows that the integral gives a holomorphic function. This gives part a) of the theorem. As for part b), we may again argue for |Re ζ | ≥ 2. We must show that if we fix  ε < ε then there are constants b ≥ 0, c , so that |I m z|b |h(z)| ≤ c if z ∈ Ω  and |Re z| < ε . This is based on the following estimate in which we assume that b = b + n/2 + 1/2 and in which c is the constant from (4): 

|I m z|b exp [−(c/2)|I m z||Re ζ | + b ln(1 + |Re ζ |)] (6) b

b

≤ c1 |I m z| |Re ζ | |Re ζ |

b−b

exp [−(c/2)|I m z||Re ζ |] ≤ c2 |Re ζ |−n/2−1/2 . 

To obtain the last inequality, we have used t b exp [−(c/2)t] ≤ c3 for some c3 if   t > 0. Once we have (6), we can estimate |I m ζ |−2b |Re ζ |≥2 |μ1 (ζ )|2 dλ(ζ ) by  −n−1 exp [−2(ε − ε) |I m ζ |]dλ(ζ ) < ∞. Cn |Re ζ | Part c). Also see Theorem 1. When z0 ∈ Ω  , then Re iz, Re ζ ] − HG,d (−Re ζ ) ≤ −δ|Re ζ | for some δ > 0 for z in a neighborhood of z0 . It follows with the arguments above that h is holomorphic in a neighborhood of z0 . To obtain the remaining statement, we divide the domain of integration into two sets. One is over Γ , the other over its complement. The integral over Γ is treated as before and gives a holomorphic function h on the domain Ω of part a), so we ˙ n \ Γ . Here we use (6) and obtain exponential decay of remain with the one over R the integrand for small z.   It should be observed that when in part a) instead of working for |Re z| < ε we want the function h to be holomorphic on greater sets, we can e.g., replace the estimate for μ by an estimate of form |μ| exp [−ϕ(ζ ) + HK (I m ζ )] ∈ L2 (Cn ), where K ⊂ Rn is a convex compact. h defined in (3) will then be holomorphic for Re z in the interior of K and I m z ∈ Γ ⊥ . Non-convex domains are of course not accessible by inequalities in the Fourier dual variables.

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4 The Regularization We now return to the case of integrals of type (2) when μ is a measurable function on Cn such that |μ(ζ )| exp [−ϕ(ζ ) + ε|I m ζ |] is in L2 (Cn ). We take a finite partition of unity of Rn with positive L∞ functions gj , j = 1, . . . , s  , such that for each j the support of gj as a distribution lies in some  sharp open convex cone Γj . We then define hj , j = 1, . . . , s  , by hj (z) = Cn exp [iz, ζ ]gj (Re ζ )μ(ζ )dλ(ζ ). By proposition 1 the hj are holomorphic on {z; |Re z| < ε, I m z ∈ Γj⊥ } and  we set u = j b(hj ) as a hyperfunction on {x; |x| < ε}. The hyperfunction u does not depend on the choice of the partition gj in view of Lemma 1. Indeed,   n if gj , j = and  s + 1, . . . , s , is another partition of unity  on R as before, if hj = Cn exp [iz, ζ ]gj (Re ζ )μ(ζ )dλ(ζ ) for j = s + 1, . . . , s  , then we s  s  s  have that j =1 b(hj ) = j =s  +1 b(hj ) since we have hi = j =1 hij for   s   i = 1, . . . , s  , respectively hi = j =s  +1 hj i for i = s + 1, . . . , s , with  hij = Cn exp [iz, ζ ]gij (Re ζ )μ(ζ )dλ(ζ ), and gij given by lemma 1 with C = 0. We also observe that we are rather free to choose the gj . In particular, we will take them with additional regularity later. What we do at 0 will be explained then.  We now have given a meaning to (3) in hyperfunctions. If sj =1 b(hj ) = u we will write that u = F −1 (μ). (F −1 should come from “Fourier-inverse transform”.) We will see in Theorem 2 that locally every hyperfunction u ∈ B0 is of the form u = F −1 (μ) for some μ. It should be noted that there is no real need to involve partitions of unity in the above argument, it just suffices to split μ into a sum of functions with the appropriate support conditions. However, using partitions of unity gives a more systematic way to achieve this splitting. Thus forexample if we are given a finite number of μj of type μ and want to define F −1 ( j μ), then it is trivial to see, by using partitions,  that the result is j F −1 (μj ).  We also mention that when one looks at integrals of form Cn exp [iz, ζ ]dμ(ζ ) for some Radon measures μ on Cn , (or similar expressions when μ is a distribution of finite order), then one can still work with partitions of unity, but they must be of greater regularity than those in Lemma 1. Thus, e.g., in the case of Radon measures, they must be at least continuous. Then of course, the constant C in that lemma cannot be taken zero, and we can only achieve partitions of unity in a region of form |Re ζ | ≥ C > 0. (It is to be prepared for such situations that we stated Lemma 1 for a general C.) This will leave us with a region of form |Re ζ | < C which in general will not pose any integrability problems. Remark 3 We explain here how both T.Carleman and L.Schwartz used averaging to give a meaning to (1) for μ with growth. Thus, Schwartz considered  polynomial  instead of (1) the expressions Rn Rn exp [ix, ξ ]g(x)μ(ξ )dx dξ for g in the space S(Rn ) of functions which decay rapidly at infinity together will all their derivatives, i.e., he took the “averages” of the formal expression giving u with densities g ∈ S(Rn ). The meaning of the parametric integral in (1) with x as a

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parameter is then in the dual space of S(Rn ) by functional analysis. In Carleman’s method the “average” is built in the process of taking boundary values of the holomorphic functions which he associates with the integrals in (1) as described in Proposition 1,b). To give an example, consider a holomorphic function h which is e.g., defined in the complex upper half plane {z ∈ C; I m ζ > 0} and satisfies locally in Re z an estimate of form |h(z)| ≤ c|I m z|−k for some constants c, k. It is then known from the theory of boundary values of holomorphic functions mentioned above that the limit limy→o+ h(x + iy) = u(x) ˜ exists in the sense of distributions for some distribution u. ˜ What this means is that if g ∈ C0∞ (R) then the  limit limy→0,y∈G Rn g(x)h(x + iy)dx exists and is u(g). ˜ In this case, we thus take an average of x → h(x + iy) against the density g and then pass to the limit for y → 0+.

5 The Representation Theorems We start with a result on holomorphic functions defined on wedges. ˙ n an open convex cone and Theorem 1 Let U ⊂ Rn be on open convex set G ⊂ R d > 0. Also let h be holomorphic on U + i{y ∈ G; |y| < d}. Then there is a L2loc function μ and a continuous function ρ : Cn → R such that for every compact K with K ⊂⊂ U + i{y ∈ G; |y| < d} it follows that sup HK (ζ )/ exp [ρ(ζ )] < ∞, and we have  |μ(ζ )|2 exp [−2ρ(ζ )]dλ(ζ ) < ∞, (7) Cn

and  h(z) =

Cn

exp [iz, ζ ]μ(ζ )dλ(ζ ) for z ∈ U + i{y ∈ G; |y| < d}.

(8)

The result is a particular case of a very general theorem of Ehrenpreis-Palamodov on the representation of solutions to overdetermined systems of linear constant coefficient partial differential operators, in this case the Cauchy-Riemann system, but the situation here is in fact much simpler. Cf. [1, 3, 7, 8, 13] for the general theory, and [4] for a direct argument. Actually, in these papers, the representation (8) is achieved in terms of Radon measures with appropriate growth restrictions at infinity rather than L2loc -functions. The reason why the representation in terms of Radon measures also gives the representation in the theorem is that in spaces of holomorphic functions locally L2 norms and supremum norms are equivalent. While representations in terms of measures cannot be avoided in the general theorem, we prefer in the case of Theorem 1 representations by functions. It should be said that the proof in [4] is organized in such a way that it implicitly also gives Theorem 1.

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We now study what the conditions on the weight function ρ in Theorem 1 say. This will also show why sublinear functions do appear in the theory. Since we do not insist on results in which h is represented on all of U , the following information about weight functions of type ρ will suffice. We fix an open convex sharp cone ˙ n and two constants 0 ≤ d  < d. We also denote by HG,d  ,d (η) = G ⊂ R supy∈G;d  ≤|y|≤d y, η the supporting function of the set {y ∈ G; d  ≤ |y| ≤ d}. (Of course this set is not convex in the Euclidean norm, but convexity will not be important in what follows since we argue for all d  < d.) We consider then a convex cone G ⊂⊂ G and a function ϕ : Rn → R such that for every d  > 0 we have supη∈Rn HG ,d  ,d (η) − ϕ(η) < ∞. A consequence is that for every j ∈ N there is a constant cj with HG ,1/j,d (η) ≤ ϕ(η) + cj . Note that the closures of the sets {y ∈ G ; d  ≤ |y| ≤ d} form a fundamental system of compacts in G. Since we do not insist on sharp estimates for G, we will even work for {y ∈ G; d  ≤ |y| ≤ d} rather than for {y ∈ G ; d  ≤ |y| ≤ d}. Lemma 2 Let ϕ : Rn → R+ be given. a) assume that for every j there is a constant cj with HG,1/j,d (η) ≤ ϕ(η) + cj . Then there is a sublinear function ψ  for which HG,d (ξ ) ≤ ϕ(ξ ) + ψ  (ξ ). b) Conversely, if there is a sublinear function ψ such that HG,d (ξ ) ≤ ϕ(ξ ) + ψ(ξ ), then for every j we can find a constant cj with HG,1/j,d (ξ ) ≤ ϕ(ξ ) + ψ(ξ ) + cj . Proof Proof of a). (Part b) is trivial.) To prove a), we observe that HG,d (η) = sup0 0 and when ξ ∈ then we can use (9). Combining these informations, we have then − y, ξ  − HG,d (ξ ) ≤ −d2 |y||ξ | − d3 |ξ |, if y ∈ G and |y| < d1 .

(10)

If y ∈ G , |y| < d1 , is fixed, this inequality remains valid with smaller constants, also in a small neighborhood of y. Moreover, if ϕ and ψ  are as in item a) in lemma 2, then it follows since ψ  is sublinear, that − y, ξ  − ϕ(ξ ) ≤ cj ,

(11)

if y ∈ G , 1/j ≤ |y| < d. Let now K ⊂ U × G , be a compact, U as above, and choose compacts K1 ⊂ U, K2 ⊂ G with K ⊂ K1 + iK2 . Then we must have HK (ζ ) ≤ HK1 (I m ζ ) +

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HK2 (Re ζ ). If then K2 = {y ∈ G ; 1/j ≤ |y| ≤ d}, then HK2 = HG ,1/j,d by a previous notation. Assume now that ϕ : Cn → R satisfies HK1 (I m ζ ) + HG ,1/j,d (Re ζ ) ≤ ϕ(ζ ) + cj , ∀j.

(12)

 Then Cn exp [iz, ζ  − ϕ(ζ )]dμ(ζ ) is convergent for z ∈ K1 + iGd . 2 n We now come to  the main2representation result in which L (C , ϕ, −ε) denotes the functions with Cn |μ(ζ )| exp [−2ϕ(Re ζ ) + 2ε|I m ζ |]dλ(ζ ) < ∞. Theorem 2 a) If u ∈ B0 then there are ε > 0, a sublinear ϕ and μ ∈ L2 (Cn , ϕ, −ε) such that u = F −1 (μ) near 0. b) (0, ξ 0 ) ∈ / W FA u precisely ξ 0 and  if we can find a convex cone Γ which contains 2 n 2 μ ∈ L (C , ϕ, −ε) with Re ζ ∈Γ exp [2δ|Re ζ | − 2ε|I m ζ |]|μ(ζ )| dλ(ζ ) < ∞. c) Further, let μ ∈ L2 (Cn , ϕ, −ε) be given such that F −1 (μ) = 0 for |x| < ε   2 n  and fix εn < ε. Then there are ϕ , νj ∈ L (C , ϕ , −ε), j = 1, . . . , n, such that ¯ μ = =1 ∂ ν . s Proof Proof of a). If u = j =1 b(hj ) for some holomorphic functions defined on {z; |Re z| < ε, I m z ∈ Gj , |I m z| < d} then we can write the single hj as F −1 (μj ). Here we have μj exp [−ϕ(ζ )+HGj ,d (Re ζ )+ε|I m ζ |] ∈ L2 (Cn ) for the single j , but then we also have μj exp [−ϕ(ζ ) + ε|I m ζ |] ∈ L2 (Cn ). The function  s j =1 μj has then the property in a). s Proof of b). Assume that (0, ξ 0 ) ∈ / W FA u and let u = j =1 b(hj ) in a neighborhood of 0, with hj ∈ A(U + ioGj ) = {x; ∈ Rn ; |x| < ε} and Gj open convex cones with ξ0 ∈ / G⊥ j . We apply part a) for the hj and can write them  as hj (z) = Cn exp [iz, ζ ]μj (ζ )dλ(ζ ) and Cn |μj (ζ )|2 exp [2HGj ,d (I m ζ ) + 2ε|I m ζ |]dλ(ζ ) < ∞. (See Lemma 2.) We obtain from this exponential decay in a conic neighborhood of ξ 0 if we also use (5). Conversely, if μ ∈ L2 (Cn , ϕ, −ε)  with Re ζ ∈Γ exp [2δ|Re ζ | + 2ε|I m ζ |]|μ(ζ )|2 dλ(ζ ) < ∞, then we pick open 0 sharp convex cones Γj , j = 0, 1, . . . , s, with sΓ0 ⊂⊂ Γ , ξ ∈ Γ0 , Γ0 ∩ Γj = ∅ s n ˙ for j ≥ 1, ∪j =0 Γj = R and write μ as j =0 μj with |μj (ζ )| ≤ |μ(ζ )|, ∀ζ , supp μj ⊂ {ζ ; Re ζ∈ Γj } for j ≥ 1, supp μ0 ⊂ {ζ ∈ Cn ; Re ζ ∈ Γ }. Then s we can write u = j =0 b(hj ) near 0 with hj = Cn exp [iz, ζ ]μj (ζ )dλ(ζ ). Since μ0 is exponentially decreasing, h0 is real-analytic near 0. Furthermore, the μj are for j ≥ 1 holomorphic on {z; |Re z| < ε, I m z ∈ Γ ⊥ }. Thus, by Remark 1, (0, ξ 0 ) ∈ / W FA b(hj ) whatever j is.   While wewill prove part c) only in a forthcoming paper, let us show that 2 n ¯ if μ = k ∂k νk (calculated in distributions) with νk ∈ L (C , ρ, −ε) then −1 F (μ) = 0. The natural way would be to study Fourier inverses of general finite order distributions, but we will argue directly for the case at hand. As s −1 (g (Re ζ )μ(ζ )) for a prescribed by the definitions, we must study F j j =0 suitable partition of unity gj , j = 0, 1, . . . , s. However, for later arguments we

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need to take them C 1 (Rn ), which we are allowed to do (any partition can be used for μ), except that at 0 we must adapt them somewhat. We will then take them with supports in {|ξ | ≥ 1, ξ ∈ Γj }, for some sharp convex cones Γj for j ≥ 1, with bounded derivatives and g0 with support in {|ξ | ≤ 2}. It is immediate to see that this variation does not change F −1 (μ), since the term corresponding to g0 will give a holomorphic function near 0. The problem is now that the ∂¯k νk are not L2 (Cn , ρ, −ε), so F −1 (gj (Reζ )∂¯k νk ) has still to be defined. We know already that the functions hj (z) = Cn exp [iz, ζ ]gj (Re ζ )μ(ζ )dλ are holomorphic on Wj = {z; |Re z| < ε, I m z ∈ Γj⊥ } for j ≥ 1. The greater part of our calculations will in fact be in holomorphic functions on Wj . To define F −1 (gj ∂¯k νk ) we take a sequence of C 1 functions κ as in the beginning of Sect. 2. We set hj k (z) = lim→∞ (κ gj ∂¯k νk )(exp [iz, ζ ]). Here κ gj ∂¯k νk ) has compact support, so (κ gj ∂¯k νk )(exp [iz, ζ ]) makes sense in distributions. It is easy to see, arguing as in the proof of Proposition 1, that the hj k are holomorphic functions on Wj and do not depend on how we choose the κ . Now (calculating in distributions) (κ gj ∂¯k νk )(exp [iz, ζ ]) = −νk (∂¯k κ gj exp [iz, ζ ]) = −νk ((∂¯k κ )gj exp [iz, ζ ]) − νk (κ (∂¯k gj ) exp [iz, ζ ]). Passing to the limit for  → ∞ and taking into account that lim→∞ νk ((∂¯k κ )gj exp [iz, ζ ]) = 0, lim→∞ νk (κ (∂¯k gj ) exp [iz, ζ ]) = νk ((∂¯k gj ) exp [iz, ζ ]), we obtain hj k = (∂¯k gj )νk (exp [iz, ζ ]). At thispoint, the (∂¯k gj )νkare in L2 (Cn , ρ, −ε), so s −1 ¯ ¯ F j (∂k gj )νk = 0 since (∂k j gj )νk = 0 in that j =0 gj ≡ 1. There is an interpretation of Theorem 2 in terms of Dolbeault cohomology. We introduce at first some notations for complex forms, but only give the minimum necessary for our purpose. (See [7] for more information.) We willneed only (0, n− 1) and (0, n) forms. (0, n−1) forms are expressions of type ω = aj1 ,...,jn−1 d ζ¯j1 ∧ d ζ¯j2 ∧ · · · ∧ d ζ¯jn−1 , where 1 ≤ j1 < j2 < · · · < jn−1 ≤ n are n − 1 natural numbers in {1, . . . , n}, the aj1 ,...,jn−1 are L2loc (Cn ) functions and the sum is over all finite collections j1 , j2 , . . . , jn−1 of this type. Practically, j1 , j2 , . . . , jn−1 is thus an ordered subset of {1, . . . , n} in which just one number in {1, . . . , n} is left out. If this number is j , the corresponding index is (1, 2, . . . , j −1, j +1, . . . , n) (with obvious modifications when j = 1 or j = n). We will write then bj for this aj1 ,...,jn−1 . The ∂¯ operator on ω is defined in this notation by ¯ = ∂ω

n

(−1)j +1 (∂bj /∂ ζ¯j )d ζ¯j ∧ d ζ¯1 ∧ d ζ¯2 ∧ · · · ∧ d ζ¯n , j =1

with derivations calculated in distributions. We now assume coefficients both for (0, n − 1) and (0, n) forms to be in Z = ∪L2 (Cn , ϕ, −ε), the union being for all sublinear ϕ and all ε > 0. The (0, n) forms with coefficients in Z can then readily be identified with Z and every germ u of a hyperfunction at 0 can be written as u = F −1 (ω) for some ω ∈ Z. If F −1 (ω) = 0 for some ω then ω is in view of ¯  for some (0, n − 1) form ω . If we denote Z(0,q) Theorem 2 of form ω = ∂ω for q = n − 1, q = n, the complex (0, q) forms with coefficients in Z, we can ¯ (0,n−1) . (Identification is as follows: for given therefore identify B0 with Z(0,n) /∂Z

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323

¯ (0,n−1) of some μ for which u we denote F(u) the equivalence class in Z(0,n) /∂Z u = F −1 (μ).) This is in so far interesting as hyperfunctions in Sato’s definition ˇ were introduced as relative Cech cohomology classes.

6 Representation of Real Analytic Functions 1. For subclasses of hyperfunctions the representation given in Theorem 2 can be improved, in that we can obtain additional information on the representation functions. We only look here at the rather simple case of real analytic functions. Our interest is in functions p(z, ξ ) where p is a symbol of a (possibly infinite order) microdifferential operator and we need to find a representation as a function of z with ξ a parameter. We want this representation to be “robust” in order to be able to study it in relation with its dependence on the parameter. We describe how to do this in this section. We are only interested in analytic functions in a small neighborhood of 0. Let us then assume that u is of form (we use standard multiindex notations) u(z) =

aα zα ,

α∈Nn

where the aα ∈ C satisfy Cauchy’s inequalities: there are constants c, c so that |aα | ≤ cc|α| α!.

(13)

Actually, we want our Fourier representation to be valid in a complex neighborhood of 0. The construction of the representation function for u will then be performed in the following way: we shall choose representation functions σα for zα , i.e., σα will be so that  exp[iz, ζ ]σα (ζ ) dλ(ζ ). (14) zα = Cn

 We can then take α aα σα as a representation function for u. Here the σα are of course not unique, and indeed, we will have to choose them according to the value of c in (13). However, once c in (13) is fixed, we can work with the same σα , independently of u, so that our construction is stable if we have additional parameters. The main step in the argument is the following Lemma 3 There are constants c, c1 , so that for any A > 0 we can find measurable functions σα on Cn with the following properties: a) supp σα ⊂ {ζ ∈ Cn ; A|α| − 1/2 ≤ |ζ | ≤ A|α| + 1/2}, |α| b) ||σα ||L2 (Cn ) ≤ cc1 A−|α| ,

324

O. Liess

 c) for any h ∈ A(Cn ) we have Dζα h(0) = Cn h(ζ )σα (ζ ) dλ(ζ ). In particular, (14) is valid. (Dζα stands for (1/ i)|α| (∂/∂ζ )α .) The statement b) can also be replaced by |α|

b)’ ||σα ||L∞ (Cn ) ≤ cc1 A−|α| . (If α = (α1 , . . . , αn ) is a multiindex, |α| =

n

j =1 αj .)

Before the proof we observe that there are distributions wj ∈ E  (Cn ), j = 1, . . . , n, with supports in D = {z; |z| ≤ 1} and v ∈ C0∞ (Cn ) with support in {z; 1/2 ≤ |z| ≤ 1} such that δ=v+

n

(∂/∂ z¯ j )wj , δ the Dirac distrbution at 0.

(15)

j =1

 Indeed, if Δ is the Laplace operator 4 nj=1 (∂/∂ z¯ j )(∂/∂zj ) in Cn and F a fundamental solution of ΔF = δ, and if G ∈ C ∞ is identically one on {z ∈ Cn ; |z| ≤ 1/2} and vanishes outside D = {z; |z| ≤ 1} then δ = Δ(GF ) + v where v is already  as in the statement. To conclude the proof of (15) we observe that Δ(GF ) = nj=1 (∂/∂ z¯ j )wj with wj = 4(∂/∂zj )(GF ). Proof We now prove Lemma 3. Let A > 0 be given and denote for some multiindex α by Uα = {ζ ∈ Cn ; A|α| − 1/2 ≤ |ζ | ≤ A|α| + 1/2}. We next use that if D = {ζ ∈ Cn ; |ζ | ≤ 1/2} and h is holomorphic in a neighborhood of D, then   by the mean value property |h(0)| ≤ c D |h(ζ )|dλ(ζ ) ≤ c ( D |h(ζ )|2 dλ(ζ ))1/2 for some computable constants c , c . Since for |ζ | = A|α| and η ∈ / Uα it follows that |ζ − η| ≥ 1/2, we obtain from this inequality applied for the disks D(ζ ) = {η ∈ Cn ; |ζ − η| ≤ 1/2} that there is a constant c2 so that sup|ζ |=A|α| |h(ζ )| ≤ c2 [ Uα |h(θ )|2 dλ(θ ) ]1/2 . Also recall that in view of Cauchy’s |α|

inequalities |Dζα h(0)| ≤ c3 A−|α| sup|ζ |=A|α| |h(ζ )| and therefore |α|

|Dζα h(0)| ≤ c2 c3 A−|α| [

 |h(θ )|2 dλ(θ )]1/2 . Uα

The desired conclusion in a) and b) follows thus from the Hahn-Banach (together with the Riesz-Fischer) theorem. Since part c) is clear, it remains to observe that part b)’ follows if we use (15). It follows from there that if h is holomorphic on a  neighborhood of D, that |h(0)| = | Cn v(z)w(z)h(z)dλ(z)| ≤ ||v||s · ||h||−s for any Sobolev norm H −s . Taking s = n + 1 we can now use Sobolev’s embedding theorem and continue as in the case of b). It should be noted that (15) can also be used in the case of b), but “constants” cannot easily be computed then.   2. We can now return to the representation problem for u.

Carleman Regularization

325

 Theorem 3 Let u be of form u(z) = α aα zα , and assume that the coefficients aα satisfy (13). Then we can find a representation function μ for u with the following properties:  i) It is of form μ = α aα σα , with σα as in lemma 3, for some suitable A, which depends only on c . ii) There is c > 0 so that exp[c |ζ |]μ ∈ L2 (Cn ). Proof We will have ||aα σα ||L2 (Cn ) ≤ c(c ˜  c1 )|α| A−|α| , so it will be necessary to decide on how to choose A to obtain the estimate in ii). Let us for this purpose calculate for k = 0, 1, 2, . . . ,

aα σα ||L2 (k≤|ζ |≤k+1) . (16) || α

When k ≤ |ζ | ≤ k + 1 and ζ ∈ supp σα it follows that (k − 1/2)/|A| ≤ |α| ≤ (k + 3/2)/A, so there are at most c4 (k/A)n terms in the sum from (16) which have a nontrivial contribution. Thus

aα σα ||L2 (k≤|ζ |≤k+1) ≤ c5 (k/A)n (c c1 A−1 )(k+3/2)/A . || α

We may here choose A with c c1 A−1 = 1/e, so ||

aα σα ||L2 (k≤|ζ |≤k+1) ≤ c6 (1 + |ζ |)n exp[−|ζ |/A].

α

 All in all we obtain || exp[|ζ |/A](1 + |ζ |)−2n−1 aα σα ||L2 (k≤|ζ |≤k+1) ≤ c7 (1 + k)−n−1 since |ζ | ∼ k on the domain under consideration. We now sum in k to get the desired result.   Remark 4 Assume that μ is a representation function for u in the sense of β Theorem 3, then ζ → ζ β μ(ζ ) is a representation function for Dz u. 3. We have stated Theorem 3 in terms of L2 -type estimates. In the applications of the theorem it is sometimes more convenient to work with L∞ -type estimates. For this purpose we consider the following strengthened form of Theorem 3: Theorem 4 Let u be as in the assumptions of Theorem 3. Then the conclusion there remains valid, even if we replace the condition b) by the following stronger condition ii ) There is a constant c > 0 so that exp[c |ζ |]μ ∈ L∞ (Cn ).

326

O. Liess

The only thing which we need to change in the proof of Theorem 3 is that we have to replace condition b) in the conclusion of Lemma 3 (for perhaps different functions σα ) by the stronger condition b)’ there.

References 1. Björk,J.E.: Rings of differential operators, North-Holland Mathematical Library, vol:21, North-Holland Publishing Co., Amsterdam-New York, 2. Carleman,T., L’intégrale de Fourier et questions qui s’y rattachent. Lecons Professéss à l’Insitut Mittag Leffler, Almqvist Wiksells Boktryckeri, 1944, 119 pp. 3. Ehrenpreis,L.: Fourier Analysis in several complex variables. Wiley-Interscience, New York, 1970. pp.xiii+506. 4. Hansen,S.: Localizable analytically uniform spaces and the fundamental principle. Trans. Amer. Math. Soc. 264 (1981), 235–250. 5. Hörmander,L.: Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math., vol.24 (1971), 671–704. 6. Hörmander,L.: Fourier integral .Hörmander, Lars. Fourier integral operators. I. Acta Math. 127 (1971), no. 1–2, 79–183. 7. Hörmander,L.: An introduction to complex analysis in several variables. North-Holland Publ. Comp., Amsterdam-London, 3-rd revised printing, 1990. 8. Kaneko,A.: Fundamental principle and extension of solutions of partial differential equations with constant coefficients. Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., Vol. 287. pp. 122–134. 9. Kaneko,A.: Introduction to hyperfunctions, Mathematics and its Applications (Japanese Series) Mathematics and its Applications (Japanese Series), vol. 3,1988, pp. xiv+458. 10. Kato,G. and Struppa,D.C.: Fundamentals of algebraic microlocal analysis. Monographs and Textbooks in Pure and Applied Mathematics, vo. 217, Marcel Dekker, Inc., New York, 1999, xii+296 pp. 11. Kiselman,C.O.: Generalized Fourier transformations: the work of Bochner and Carleman viewed in the light of the theories of Schwartz and Sato, Microlocal analysis and complex Fourier analysis, World Sci. Publ., River Edge, NJ, 2002, pp 166–185. 12. Martineau,A.: Distributions et valeurs au bord des fonctions holomorphes. Theory of Distributions (Proc. Internat. Summer Inst., Lisbon, 1964), Inst. Gulbenkian Ci., Lisbon, 1964, pp. 193–326. 13. Palamodov,V.P.: Linear differential operators with constant coefficients. (Russian), Moscow, 1967, English version by Springer-Verlag, Grundlehren der Math. Wissenschaften, vol.168. (1970). 14. Sato,M.: Hyperfunctions and differential equations. Proc. Intern. Conf. on Functional Analysis, Tokyo, 1969, Univ. Tokyo Press,1970. 15. Sato,M.: Regularity of hyperfunctions solutions of partial differential equations, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris 1971,pp 785–794. 16. Tillmann, H.G.: Distributionen als Randverteilungen analytischer Funktionen. II. Math. Z. 76 (1961) 5–21.

Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space Daniel Lorenz

Abstract We consider the strictly hyperbolic Cauchy problem ⎧ m−1 ⎪ ⎨D m u −  t

⎪ ⎩



j =0 |γ |+j =m

γ

j

am−j, γ (t, x)Dx Dt u = 0,

Dtk−1 u(0, x) = gk (x), k = 1, . . . , m,

for (t, x) ∈ [0, T ] × Rn with coefficients belonging to the Zygmund class C∗s in x and having a modulus of continuity below Lipschitz in t. Imposing additional conditions to control oscillations, we obtain a global (on [0, T ]) L2 energy estimate 2m0 without loss of derivatives for s ≥ max{1 + ε, 2−m }, where m0 is linked to the 0 modulus of continuity of the coefficients in time. Keywords Cauchy problem · Modulus of continuity · Zygmund space · Low-regular · Strictly hyperbolic · Higher order

1 Introduction In this paper we study the strictly hyperbolic Cauchy problem ⎧ m−1 ⎪ ⎨D m u −  t



j =0 |γ |+j =m

γ

j

am−j, γ (t, x)Dx Dt u = 0,

⎪ ⎩ k−1 Dt u(0, x) = gk (x), k = 1, . . . , m,

(1)

D. Lorenz () TU Bergakademie Freiberg, Faculty of Mathematics and Computer Science, Institute of Applied Analysis, Freiberg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_19

327

328

D. Lorenz

for (t, x) ∈ [0, T ]×Rn . We aim to derive a global (in time) energy estimate for this equation, when the coefficients are not smooth but their regularity with respect to time is below Lipschitz and with respect to x they belong to the Zygmund space C∗s . The derived energy estimate implies in a more or less standard way well-posedness in H m−1 × H m−2 × . . . × L2 . Historically, many papers have been devoted to study the well-posedness of the second order strictly hyperbolic Cauchy problem utt −

n

∂xl (akl (t, x)uxk ) = 0,

(2)

k, l=1

u(0, x) = u0 (x), ut (0, x) = u1 (x), (t, x) ∈ [0, T ] × Rn . It is well-known that, if the coefficients akl are Lipschitz in t and only L∞ in x, then the Cauchy problem (2) is well-posed in H 1 × L2 (see e.g. [12, 14, 19]). If the akl ’s are Lipschitz continuous in time and B ∞ in space (i.e. C ∞ and all derivatives are bounded), then one can prove well-posedness in H s × H s−1 for all s ∈ R. Furthermore, in that case one is able to obtain an energy estimate with no loss of derivatives. If we lower the regularity of the coefficients below Lipschitz in time, then, generally, we have some loss of derivatives. There have been different approaches to describe the non-Lipschitz behavior of the coefficients. For the x-independent case, the authors of [6] proposed the so called LogLip-property, that is, the coefficients akl = akl (t) satisfy |akl (t + t0 ) − akl (t)| ≤ Ct0 | log(t0 )|,

(3)

for all t0 > 0, t, t + t0 ∈ [0, T ]. Concerning coefficients that also depend on x, the authors of [9] and [2] considered problem (1) with B ∞ coefficients in x and LogLipregularity in t. They proved well-posedness in Sobolev spaces and C ∞ , respectively, without describing the exact loss of derivatives. For second order equations like (1), the authors of [4] provided a classification, linking the loss of derivatives to the modulus of continuity of the coefficients with respect to time. For coefficients that are low regular in time and space, the authors of [9] considered coefficients isotropic Log-Lipschitz in (t, x). They proved an energy estimate with loss of derivatives which forces us to limit t to a small interval when we want to get an existence result. More recently, in [5] the authors studied strictly hyperbolic Cauchy problems of general order m with coefficients low-regular in time (i.e. Lipschitz or less regular) and smooth in space. Most importantly, all results that follow this first approach have in common that, in general, there appears a loss of derivatives when the coefficients are less regular than Lipschitz in t. Another approach to describe the low-regularity of the coefficients with respect to time goes back to [7]. Generalizing their approach to (1) gives well-posedness if the coefficients satisfy akl = akl (t),

akl ∈ C[0, T ] ∩ C 1 (0, T ],

 |akl (t)| ≤ Ct −1 for t ∈ (0, T ].

Hyperbolic Cauchy Problems with Low-Regular Coefficients

329

In general, also that case comes with a loss of derivatives. However, the C[0, T ] ∩ C 1 (0, T ] regularity in t is not sufficient to give a precise knowledge of the loss of derivatives. If we assume that the coefficients are more regular, for example akl ∈ C[0, T ] ∩ C 2 (0, T ], then we are able to link the loss of derivatives to the control of oscillations [8, 10]. Definition 1 Assume that akl = akl (t),

akl ∈ C[0, T ] ∩ C 2 (0, T ],

   γ q (q) |akl (t)| ≤ C t −1 log t −1 , (4)

for t ∈ (0, T ] and q = 1, 2. We say that the coefficients akl oscillate very slowly, slowly, fast or very fast, if γ = 0, γ ∈ (0, 1), γ = 1, if (4) is not satisfied for γ = 1, respectively. Theorem 1 ([20]) Consider problem (2) with coefficients satisfying (4). Then the energy inequality Eν−ν0 (u)(t) ≤ C(T )Eν (u)(0), holds for all t ∈ (0, T ] and large ν, where Eν (u)(t) := (∇x u(t, ·), ut (t, ·)) H ν . Moreover, we have ν0 = 0 if γ = 0; ν0 is an arbitrarily small positive constant if γ ∈ (0, 1); ν0 is a positive constant if γ = 1; there does not exist a positive constant ν0 if γ > 1, i.e. the loss of derivatives is infinite. The paper [11] answers the question if it is possible to lower the C 2 regularity of the coefficients but still keep the characterization of oscillations and its connection to the loss of derivatives. The authors prove that instead of working with C 2 regularity in (4), it is sufficient to assume a suitable C 1, β−1 regularity, β ∈ (1, 2]. Then, by adjusting Definition 1 suitably it is possible to obtain similar results to those of Theorem 1 and keep the characterization of oscillations and loss of derivatives provided by [20]. Finally, in [16] the authors provide a general C 1, η, μ, - theory that combines a global condition like (3) for a general modulus of continuity, with generalized local conditions like (4). For our purpose, this second approach to describe non-Lipschitz behavior of the coefficients is more helpful. By adding a control of oscillations, it is possible to obtain global (in time) energy estimates with no loss of derivatives in certain cases, when the coefficients are B ∞ in x (see e.g. [16]).

330

D. Lorenz

The starting point of our investigations in this paper is the work [16]; more precisely, we start from Theorem 2.2 in [16] which provides conditions under which the strictly hyperbolic Cauchy problem utt −

n

akl (t, x)uxk xl = 0,

u(0, x) = u0 (x), ut (0, x) = u1 (x),

(5)

k, l=1

is well-posed in L2 (without loss of derivatives) with coefficients smooth in x but less regular than Lipschitz in t. To obtain this result the authors of [16] imposed additional local conditions on the first derivative with respect to time of the coefficients. We aim to generalize the results of [16] to equations of general order m and with coefficients low-regular in x. By this we mean, that we assume that the coefficients belong to the Zygmund space C∗s with respect to x for some s > 0. Our goal is to make s as small as possible. Definition 2 ([23]) Let s > 0 and write s = [s]− + {s}+ , where [s]− denotes the largest integer strictly smaller than s and 0 < {s}+ ≤ 1. The Zygmund space C∗s − consists of all functions u ∈ C [s] such that u C∗s =

Dxα u L∞ +

|α|≤[s]−

|α|=[s]−

sup

x=y

α |Dxα u(x) − 2Dxα u( x+y 2 ) + Dx u(y)|

|x − y|{s}

+

< +∞.

Notably, for s ∈ / N the Zygmund spaces C∗s coincide with the usual Hölder space. The result we obtain reveals that for coefficients that are sufficiently regular in time (see Corollaries 1 and 2), it is possible to lower s to 1 + ε for any ε > 0. The minimum regularity in time that we have to suppose for this to be true is Hölder continuous with exponent 13 . If the coefficients are less regular in time than that, we have to choose larger values for s for our result to be valid. Let us consider the second order equation utt −

n

akl (t, x)uxk xl = 0,

u(0, x) = u0 (x), ut (0, x) = u1 (x),

(6)

k, l=1

with coefficients akl (t, x) ∈ C μ ([0, T ]; C∗s ) ∩ C 2 ((0, T ]; C∗s ), where C μ denotes the space of continuous functions with modulus of continuity μ. Definition 3 Let μ : [0, 1] → [0, 1] be a continuous, concave and increasing function. Then μ is called a modulus of continuity if it satisfies μ(0) = 0. A function

Hyperbolic Cauchy Problems with Low-Regular Coefficients

331

f ∈ C(Rn ) belongs to C μ (Rn ) if and only if |f (x) − f (y)| ≤ Cμ(|x − y|), holds for all x, y ∈ Rn , |x − y| ≤ 1 and some constant C. In this setting, our results imply the following statements. Corollary 1 Consider equation (6) with coefficients akl having the modulus of continuity μ(r) = r log

 1 , r

and satisfying the oscillation conditions 1

∂t akl (t, ·)

C∗s

e 2t ,  t

and 1

∂t2 akl (t, ·) C∗s 

et , t2

for t ∈ (0, T ]. Then the energy inequality (Dx u(t, ·), Dt u(t, ·))T L2 ≤ CT (u0 (·), u1 (·))T L2 , holds for all t ∈ (0, T ] provided that the index s of the Zygmund space C∗s satisfies s ≥ 1 + ε, for any ε > 0. Corollary 2 Consider equation (6) with coefficients akl having the modulus of continuity μ(r) = r α , α ∈ (0, 1), and satisfying the oscillation conditions 2−α

∂t akl (t, ·) C∗s  t − 2(1−α) , and 2−α

∂t2 akl (t, ·) C∗s  t − 1−α ,

332

D. Lorenz

for t ∈ (0, T ]. Then the energy inequality (Dx u(t, ·), Dt u(t, ·))T L2 ≤ CT (u0 (·), u1 (·))T L2 , holds for all t ∈ (0, T ] provided that the index s of the Zygmund space C∗s satisfies s≥

" 1+ε

if α ≥ 13 ;

2 1−α 1+α

if α < 13 .

The paper is organized as follows: Sect. 2 states the main results of this paper. Examples and remarks are discussed in Sect. 3. Section 4 reviews some definitions and provides an introduction to the pseudodifferential calculus used in this paper. Finally, in Sect. 5 we proceed to prove the theorem of Sect. 2. In Sect. 6 we present some concluding remarks.

2 Statement of Results We consider the strictly hyperbolic Cauchy problem ⎧ m−1 ⎪ ⎨D m u −  t



γ

j =0 |γ |+j =m

j

am−j, γ (t, x)Dx Dt u = 0,

⎪ ⎩ k−1 Dt u(0, x) = gk (x), k = 1, . . . , m,

(7)

for (t, x) ∈ [0, T ] × Rn . We assume that (A1) gk ∈ H m−k , k = 1, . . . , m, (A2) the characteristic roots τk = τk (t, x, ξ ) of τm −

m−1



am−j, γ (t, x)ξ γ τ j = 0,

j =0 |γ |+j =m

are real and distinct for all (t, x, ξ ) ∈ [0, T ] × Rnx × Rnξ \ {0}, (A3) there exist auxiliary functions η and - defined on (0, r0 ] (r0 small) that can be continuously extended to r = 0, satisfying η ∈ C ∞ (0, r0 ], η(0) = 0, η > 0, η < 0, |η(k) (r)| ≤ Ck r −(k−1) η (r), k ≥ 1,

and - ∈ C ∞ (0, r0 ], -(0) = 0, - > 0, - ≤ 0, |-(k) (r)| ≤ Ck r −(k−1) - (r), k ≥ 1,

such that

Hyperbolic Cauchy Problems with Low-Regular Coefficients

(A4) the function f (r) :=

r η(r)

333

is increasing on (0, r0 ] and lim f (r) = 0, r→0+

(A5) the coefficients satisfy the global condition 1 m−1 1  1 1



(am−j, γ (t + t0 , ·) − am−j, γ (t,

j =0 |γ |+j =m

sup

γ ·)) |ξ |ξm−j

t0 η(t0 )

t, t0 , t+t0 ∈[0, T ]

1 1 1 1

C∗s

≤ Cs ,

for ξ ∈ Rn \ {0}, (A6) the coefficients satisfy the local condition 1  m−1

1 1∂t 1

j =0 |γ |+j =m

12  1  ξ γ 1 2 d , am−j, γ (t, ·) m−j 1 ≤ −C s 1 s |ξ | dt η−1 (t) C∗

for ξ ∈ Rn \ {0}, t ∈ (0, T ], (A7) the coefficients satisfy the additional local condition 1 m−1 1  1 1 sup



1 1 γ (∂t am−j, γ (τ + t0 , ·) − ∂t am−j, γ (τ, ·)) |ξ |ξm−j 1 1

j =0 |γ |+j =m

C∗s

-(t0 )

t0 >0 τ, τ +t0 ∈[t, T ]

≤ −Cs

 1 d , −1 dt -(η (t))

for ξ ∈ Rn \ {0} and t ∈ (0, T ]. Theorem 2 Consider the Cauchy problem (7) under the assumptions (A1)-(A7) and suppose that for some m0 ∈ (0, 1] we have 1 m0 ∈ S1, 0, η(ξ −1 )  1 d m0 −1 ξ −1 ∈ S1, 0 , for t ≥ η(ξ  ) dt η−1 (t − ξ −1 )  1 d m0 −1 ∈ S1, -(ξ −1 ) 0 , for t ≥ η(ξ  ). dt -(η−1 (t − ξ −1 )) Then the energy inequality (Dx m−1 u(t, ·), . . . , Dtm−1 u(t, ·))T L2 ≤ CT (g1 (·), . . . , gm (·))T L2 ,

(8) (9) (10)

334

D. Lorenz

holds for all t ∈ (0, T ] provided that the index s of the Zygmund space C∗s satisfies  2m0  , s ≥ max 1 + ε, 2 − m0

(11)

for any ε > 0. Remark 1 We note that the conditions (8)–(10) are always satisfied for m0 = 1 (Lemma 5.7. in [16]). This yields the following corollary. Corollary 3 The energy estimate in Theorem 2 holds true even if we just assume (A1)–(A7) (and not (8)–(10)) provided that the index s of the Zygmund space C∗s satisfies s ≥ 2. Remark 2 We note that the terms in condition (11) for s are in “equilibrium”, i.e. 2m0 , if 1 = 2−m 0 2

η(r) = r 3 , that is, if the coefficients are Hölder continuous in time with exponent 1 −

2 3

= 13 .

3 Examples and Remarks Let us begin with some remarks and explanations regarding the assumptions of the previous theorem. Remark 3 To formulate the assumptions (A5)–(A7) we use the two auxiliary functions η and -. Typical examples of η are m] 1 −1 η(r) = (log[3 r) ,

η(r) = (log 1r )−α , α > 0,

η(r) = r β , β ∈ (0, 1).

Typical examples of - are the same as for η but -(r) = r is also admissible. Remark 4 The global condition (A5) states that the coefficients have the modulus r of continuity η(r) in time. Due to the assumptions on η and (A4) this modulus of continuity is always weaker than Lipschitz, i.e. the coefficients are less regular than Lipschitz. Remark 5 The local condition (A6) is basically a control of oscillations. Remark 6 The additional local condition (A7) is there to replace a condition on the second derivatives of the coefficients. If the coefficients are C 2 with respect to time, then we can choose -(r) = r and condition (A7) turns into a condition on the second derivative with respect to time of the coefficients.

Hyperbolic Cauchy Problems with Low-Regular Coefficients

335

If the coefficients are not C 2 , then we cannot choose -(r) = r since the supremum might not exist. In these cases, condition (A7) states that the coefficients are more regular than just C 1 and - describes how much more regular (than C 1 ) the coefficients are. We note that the right hand side of this condition becomes more restrictive (for small t) the closer we come to C 1 regularity of the coefficients, i.e. the larger -(r) gets for small r. Let us try to get a feeling for these conditions by looking at some examples. r First, we note that the choice of the modulus of continuity, i.e. η(r) , and the choice of -(r) are completely independent of each other. Moreover, their choice does not influence the result of the theorem. Choosing a less regular modulus of continuity automatically yields more restrictive conditions (A6) and (A7). Similarly, choosing - such that we are closer to C 1 regularity gives a more restrictive condition (A7). We start by considering the local condition (A6) for different moduli of continuity. The results are shown in : Table 1. As expected the right hand side of (A6) gets more restrictive, i.e. the term − dtd η−11(t) is smaller for small t, the further away we get from Lipschitz regularity, i.e. η(r) = 1. Next, we look at condition (A7) for different η and -. The examples are shown in Table 2, where the resulting therms for the right hand side of (A7) are written in the respective table cells below η(r) and to the right of -(r). As expected, we see that condition (A7) gets more restrictive, i.e. the terms in the cells of Table 2 are growing slower for t → 0+, the further away we are from C 2 regularity, i.e. -(r) = r. Table 1 Some examples of the local condition (A6) for different moduli of continuity : Modulus of continuity η(r) − dtd η−11(t) 1

r   r log

1 r

r α , α ∈ (0, 1)

Case excluded by assumptions on η

 log

 −1 1 r

1

e 2t t 2−α

(1 − α)− 2 t − 2(1−α) 1

r 1−α

Table 2 Examples of the additional local condition (A7) for different η and -. The resulting terms d 1 for − are shown in the respective table cells dt -(η−1 (t))   −1 -(r)\η(r) η(r) = log 1r η(r) = s 1−α   −1 -(r) = log 1r -(r) = r β -(r) = r

1 t2 β β et t2 1 1 et t2

1 −1 t 1−α β − 1−α+β t 1−α 1−α 1 − 2−α t 1−α 1−α

336

D. Lorenz

Let us now consider the conditions (8)–(10). These three conditions are used in the proof to apply sharp Gårding’s inequality for possibly smaller values of s. As stated in Corollary 3, it is possible to ignore these conditions but then one may only work with s ≥ 2. Table 3 reviews some examples of the weight 1 , η(ξ −1 ) i.e. condition (8), for different moduli of continuity. Similarly, Tables 4 and 5 present some examples for condition (9) and condition (10), respectively, for different moduli of continuity and different -.

Table 3 Some examples of the weight

1 , i.e. condition (8) for different moduli of continuity η(ξ −1 )

1 η(ξ −1 )

Modulus of continuity   r log 1r

log(ξ )

r α , α ∈ (0, 1)   −α log 1r ,

ξ 1−α  −α ξ  log(ξ )

Symbol class  ε S1, 0

ε>0 1−α S1, 0



ε>0

1−ε S1, 0

m0 Any m0 > 0 m0 = 1 − α m0 = 1

α ∈ (0, ∞)    1 Table 4 Some examples of the weight ξ −1 − dtd η−1 (t−ξ , i.e. condition (9), for different −1 ) moduli of continuity    1 Modulus of continuity ξ −1 − dtd η−1 (t−ξ Symbol class m0 −1 )    ε r log 1r (log(ξ ))2 S1, 0 Any m0 > 0 r α , α ∈ (0, 1)

ε>0 1−α S1, 0

ξ 1−α

m0 = 1 − α

   1 , i.e. condition (10), for different η and -. Table 5 Examples of -(ξ −1 ) − dtd -(η−1 (t−ξ −1 ))     d 1 −1 The resulting terms for -(ξ  ) − dt -(η−1 (t−ξ −1 )) are shown in the respective table cells   −1 -(r) \ η(r) η(r) = log 1r η(r) = r 1−α   −1 -(r) = log 1r

log(ξ )

-(r) = r β

β(log(ξ ))2

-(r) = r

(log(ξ ))2

1 ξ 1−α 1 − α log(ξ ) β ξ 1−α 1−α 1 ξ 1−α 1−α

Hyperbolic Cauchy Problems with Low-Regular Coefficients

337

Table 6 Some examples for the index s of C∗s for different moduli of continuity Modulus of continuity   r log 1r

Possible m0

Possible s

Any m0 > 0

s ≥1+ε

r α , α ∈ (0, 1)   −α log 1r ,

m0 = 1 − α

s ≥ max{1 + ε, 2 1−α 1+α }

m0 = 1

s≥2

α ∈ (0, ∞)

Looking at the examples in Table 5 we note that the weight   1 d -(ξ −1 ) − dt -(η−1 (t − ξ −1 )) changes slightly if we change -. However, the weight mainly depends on our choice of η. For fixed η, we can choose one m0 such that the weight   1 d -(ξ −1 ) − dt -(η−1 (t − ξ −1 )) m0 belongs to S1, 0 for any of the given examples for -. We conclude from this observation, that the possible values for s are mainly determined by our choice of the modulus of continuity. Finally, Table 6 summarizes the discussed examples for the different weights and conditions. For a given modulus of continuity Tabel 6 shows possible choices of m0 and the resulting possible values for s. We also refer to Corollaries 1 and 2 in Sect. 1 as examples of our result for ρ(r) = r.

4 Definitions and Tools Let x = (x1 , . . . , xn ) be the variables in the n-dimensional Euclidean space Rn and by ξ = (ξ1 , . . . , ξn ) we denote the dual variables. Furthermore, we set ξ 2 = 1 + |ξ |2 . We use the standard multi-index notation. Precisely, let Z be the set of all integers and Z+ the set of all non-negative integers. Then Zn+ is the set of all n-tuples α = (α1 , . . . , αn ) with ak ∈ Z+ for each k = 1, . . . , n. The length of α ∈ Zn+ is given by |α| = α1 + . . . + αn . Let u = u(t, x) be a differentiable function, we then write ut (t, x) = ∂t u(t, x) =

∂ u(t, x), ∂t

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D. Lorenz

and  ∂xα u(t,

x) =

∂ ∂x1

α1



∂ ... ∂xn

αn u(t, x).

Using the notation Dxj = −i ∂x∂ j , where i is the imaginary unit, we write also Dxα = Dxα11 · · · Dxαnn . Similarly, for x ∈ Rn we set x α = x1α1 · · · xnαn . Let f be a continuous function in an open set  ⊂ Rn . By supp f we denote the support of f , i.e. the closure in  of {x ∈  | f (x) = 0}. By C k (), 0 ≤ k ≤ ∞, we denote the set of all functions f defined on , whose derivatives ∂xα f exist and are continuous for |α| ≤ k. By C0∞ () we denote the set of all functions f ∈ C ∞ () having compact support in . The Sobolev space H k,p () consists of all functions that are k times differentiable in Sobolev sense and have (all) derivatives in Lp (). We use C as a generic positive constant which may be different even in the same line. An import tool in our approach is the division of the extended phase space into zones. For this purpose we define tξ by tξ = Nη(|ξ |−1 ), N ≥ 2, |ξ | ≥ M. The pseudodifferential zone Zpd (N, M) is then given by Zpd (N, M) = {(t, x, ξ ) ∈ [0, T ] × Rnx × Rnξ : t ≤ tξ , |ξ | ≥ M}. The hyperbolic zone Zhyp (N, M) is defined by Zhyp (N, M) = {(t, x, ξ ) ∈ [0, T ] × Rnx × Rnξ : t ≥ tξ , |ξ | ≥ M}. m and related Let us recall some results and definitions for the symbol space C∗s S1, 0 pseudodifferential operators.

4.1 Pseudodifferential Operators with Limited Smoothness m as in [13] and the space C s S m of We introduce the standard symbol space Sρ, ∗ 1, δ δ symbols having limited smoothness in x.

Hyperbolic Cauchy Problems with Low-Regular Coefficients

339

Definition 4 Let m, ρ, δ ∈ R with 0 < ρ ≤ 1 and 0 ≤ δ < 1. A function m if for all α, β ∈ Nn we have the estimate p = p(x, ξ ) ∈ C ∞ (R2n ) belongs to Sρ, δ |∂ξα Dxβ p(x, ξ )| ≤ Cα, β ξ m−|α|ρ+|β|δ , for all ξ ∈ Rn . The space of the associated pseudodifferential operators is denoted m . by ρ, δ m Definition 5 Let s, m, δ ∈ R with s ≥ 0, 0 ≤ δ < 1. Then we denote by C∗s S1, δ the set of all functions p = p(x, ξ ) which are smooth in ξ and belong to the Zygmund space C∗s (see Definition 2) with respect to x such that for all α, β ∈ Nn with |β| < [s]− , we have the estimates

|∂ξα Dxβ p(x, ξ )| ≤ Cα, β ξ m−|α|+|β|δ , for |β| ≤ [s]− , and ∂ξα p(·, ξ ) C∗s ≤ Cα, s ξ m−|α|+δs , where [s]− denotes the largest integer strictly smaller than s. The space of the m . associated pseudodifferential operators is denoted by C∗s 1, δ 4.1.1

Mapping Properties

The following mapping results as well as the results for composition, adjoint and sharp Gårding’s inequality for pseudodifferential operators with limited smoothness can be proved using a technique called symbol smoothing. We refer the reader to Section 1.3 in [22] for information about that technique. m and Let us now briefly review some mapping properties of operators from 1, δ m s C∗ 1, δ . m with 0 ≤ Proposition 1 (Chapter 3, Theorem 2.7. in [17]) Let p(x, Dx ) ∈ ρ, δ δ < ρ ≤ 1, then

p(x, Dx ) : H r+m → H r , continuously, for all r ∈ R. The following results are valid for the more general nonhomogeneous Besov s . spaces Bp, q Definition 6 ([3]) Let χ0 ∈ C0∞ (Rn ) be a radial function with " χ0 (ξ ) =

1,

|ξ | ≤ 1,

0,

|ξ | ≥ 2,

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D. Lorenz

and assume that the mapping r → χ0 (re) is non-increasing over R+ for all unitary vectors e ∈ R n . We set ϕ(ξ ) = χ0 (ξ ) − χ0 (2ξ ), and define for j ≥ 0 χj (ξ ) := χ0 (2−j ξ ),

ϕj (ξ ) := ϕ(2−j ξ ),

and j := 0 if j ≤ −1,

0 := χ0 (Dx ),

j := ϕj (Dx ) if j ≥ 1.

s Let s ∈ R, 1 ≤ p, q ≤ ∞. The nonhomogeneous Besov space Bp, q consists of all tempered distributions u such that

1  1 s := 1 2j s j u Lp u Bp, q

1 1 1q

j ∈Z l (Z)

< ∞.

s s Corollary 4 We note that for s > 0 we have B∞, ∞ = C∗ ; for s ∈ R we have s = H s. B2, 2

Proposition 2 (Lemma 3.4 in [1]) Let s > 0, 0 ≤ δ ≤ 1, 1 ≤ l, q ≤ ∞, m ∈ R, m . Then and p(x, Dx ) ∈ C∗s 1, δ r+m r p(x, Dx ) : Bq, l → Bq, l ,

if −(1 − δ)s < r < s. Proposition 3 (Lemma 3.5 in [1]) Let s1 , s2 > 0, 1 ≤ l, q ≤ ∞, m ∈ R. Let m ∩ C s2  m−ϑ for some ϑ ∈ (0, s ). Then p(x, Dx ) ∈ C∗s1 1, 1 ∗ 1, 0 0 r+m−ϑ r → Bq, p(x, Dx ) : Bq, l, l

if −s1 + ϑ < r < s1 .

4.1.2

Composition, Adjoint and Sharp Gårding’s Inequality

For two symbols p1 and p2 we introduce the notation (p1 #k p2 )(x, ξ ) :=

1 ∂ α p1 (x, ξ )Dxα p2 (x, ξ ), α! ξ

|α|≤k

for k ∈ N. Consequently, we write (p1 #k p2 )(x, Dx ) = Op(p1 #k p2 )(x, Dx ).

Hyperbolic Cauchy Problems with Low-Regular Coefficients

341

Proposition 4 (Theorem 3.6 in [1]) Let 1 ≤ p, q ≤ ∞, m1 , m2 ∈ R, s1 , s2 > 0, m1 choose ϑ ∈ (0, s2 ) and set s = min{s1 , s2 −[ϑ]− }. Let p1 = p1 (x, Dx ) ∈ C∗s1 1, 0 s2 m2 and p2 = p2 (x, Dx ) ∈ C∗ 1, 0 . For every r such that |r| < s,

r > −(s2 − ϑ),

−s2 + ϑ < r + m1 < s2 ,

we have that Rϑ = Rϑ (x, Dx ) := p1 (x, Dx )p2 (x, Dx ) − (p1 #[ϑ]− p2 )(x, Dx ), is a bounded operator from r+m1 +m2 −ϑ r → Bp, Bp, q q.

The analogous result holds for Bessel potential spaces if 1 < p < ∞ and ϑ ∈ / N. Remark 7 We cite [1] not because it is the first result of this kind but because the technique and notations used there are similar to ours. An earlier version of this result can also be found in [18]. Let us briefly consider the cases of Proposition 4 that are vital for our approach. Corollary 5 Let p = q = 2 and take 0 , then • p1 , p2 ∈ C∗1+ε 1, 0

Rϑ=1 : H r−1 → H r , if − ε < r < ε; 1 , p ∈ C 1+ε  0 , then • p1 ∈ C∗1+ε 1, 2 ∗ 0 1, 0

Rϑ=1 : H r → H r , if − ε < r < ε; 0 , p ∈ C 1+ε  1 , then • p1 ∈ C∗1+ε 1, 2 ∗ 0 1, 0

Rϑ=1 : H r → H r , if − ε < r < ε. The following proposition states a result about the adjoint of an operator from 1 . C∗s 1, 0 1 with s > Proposition 5 (Proposition 2.3.A in [22]) Let p = p(x, Dx ) ∈ C∗s 1, 0 1. Then we have that

R = R(x, Dx ) := p(x, Dx )∗ − q(x, Dx ), is a bounded operator from Hr → Hr,

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D. Lorenz

if 1 − s < r < s, where σ (q(x, Dx ))(x, ξ ) = p(x, ξ ). Remark 8 In the book [22] the author has the condition −s < r < s. However, from the short comments about the proof, the author of this paper can only obtain 1 − s < r < s. Lastly, the following proposition gives a sharp Gårding inequality for operators m . with symbols in C∗s S1, 0 Proposition 6 (Corollary II.5 in [21]) Consider the N × N symbol p(x, ξ ) ∈ m with p(x, ξ ) ≥ 0. Then for all u ∈ C ∞ , we have C∗s S1, 0 0 6(p(x, Dx )u, u) ≥ −C1 u 2L2 , provided that 0 ≤ s ≤ 2 and m ≤

2s s+2 .

1 , then the condition Remark 9 If we have p(x, ξ ) ∈ C∗s S1, 0

1≤

2s , s+2

yields s ≥ 2.

5 Proof The steps of the proof are basically the same as in [16]. We just have to pay attention to the fact that the used pseudodifferential operators and symbols are not smooth with respect to x. In Sect. 5.1 we introduce regularized coefficients and characteristic roots which are smooth with respect to time. After deriving some estimates for the regularized roots, we introduce a suitable symbol class in Sect. 5.2 which takes account of the behavior of the regularized roots in each zone of the extended phase space. We continue by transforming the original Cauchy problem to a Cauchy problem for a first order system in Sect. 5.3 and perform two steps of diagonalization in Sect. 5.4. Finally, after another change of variables to deal with some lower order terms in Sect. 5.5, we conclude the proof in Sect. 5.6. As written above, we divide the extended phase space into two zones. The basic idea is that in the pseudodifferential zone Zpd (N, M) we use the global condition on the coefficients, whereas in the hyperbolic zone Zhyp (N, M) we use the local conditions on the coefficients. The separating line of these zones is given by tξ = Nη(|ξ |−1 ), N ≥ 2, |ξ | ≥ M.

(12)

Hyperbolic Cauchy Problems with Low-Regular Coefficients

343

The pseudodifferential zone Zpd (N, M) is then given by Zpd (N, M) = {(t, x, ξ ) ∈ [0, T ] × Rnx × Rnξ : t ≤ tξ , |ξ | ≥ M}, consequently, the hyperbolic zone Zhyp (N, M) is defined by Zhyp (N, M) = {(t, x, ξ ) ∈ [0, T ] × Rnx × Rnξ : t ≥ tξ , |ξ | ≥ M}.

5.1 Regularization Since the coefficients are not smooth with respect to time in t = 0, it is helpful to regularize them.  Definition 7 Let ψ ∈ C0∞ (R) be a given function satisfying R ψ(x)dx = 1 and 0 ≤ψ(x)  ≤ 1 for any x ∈ R with supp ψ ⊂ [−1, 1]. Let ε > 0 and set ψε (x) = 1 x ψ ε ε . Then we define aε, m−j, γ (t, x) := (am−j, γ ∗t ψε )(t, x), for j = 0, . . . , m − 1. For convenience, we write a(t, x, ξ ) :=

m−1



am−j, γ (t, x)

j =0 |γ |+j =m

ξγ , |ξ |m−j

and aε (t, x, ξ ) :=

m−1



aε, m−j, γ (t, x)

j =0 |γ |+j =m

ξγ . |ξ |m−j

Proposition 7 ([5, 16]) We choose ε = ξ −1 . Then the regularized coefficients satisfy the following estimates for all α ∈ Nn . (i) For (t, ξ ) ∈ [0, T ] × {|ξ | ≥ M}, we have 1 1 1 α 1 1∂ξ aε (t, ·, ξ )1 s ≤ Cα, s ξ −|α| . C∗

(ii) For (t, ξ ) ∈ [0, T ] × {|ξ | ≥ M}, we have 1  1 1 α 1 1∂ξ aε (t, ·, ξ ) − a(t, ·, ξ ) 1

C∗s

≤ Cα, s ξ −1−|α|

1 . η(|ξ |−1 )

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D. Lorenz

(iii) For (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}, we have 1  1 1 α 1 1∂ξ aε (t, ·, ξ ) − a(t, ·, ξ ) 1

C∗s

≤ −Cα, s ξ −1−|α| -(|ξ |−1 )

 1 d . dt -(η−1 (t − |ξ |−1 ))

(iv) For (t, ξ ) ∈ [0, T ] × {|ξ | ≥ M}, we have 1 1 1 α 1 1∂ξ ∂t aε (t, ·, ξ )1

C∗s

≤ Cα, s ξ −|α|

1 . η(|ξ |−1 )

(v) For (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}, we have 1 1 1 α 1 1∂ξ ∂t aε (t, ·, ξ )1

C∗s

 1  1 d 2 ≤ Cα, s ξ −|α| − . dt η−1 (t − |ξ |−1 )

(vi) For (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}, we have 1 1 1 α 2 1 1∂ξ ∂t aε (t, ·, ξ )1

C∗s

≤ −Cα, s ξ 1−|α| -(|ξ |−1 )

 1 d . dt -(η−1 (t − |ξ |−1 ))

Proof The estimates for (t, ξ ) ∈ [0, T ] × {|ξ | ≥ M} follow from Proposition 4.3 in [5]. The estimates for (t, ξ ) ∈ [tξ ] × {|ξ | ≥ M} are derived following the proofs of Lemma 4.1 and Lemma 5.1 in [16].   We introduce λk = λk (t, x, ξ ) to be the solutions to λm −

m−1



aε, m−j, γ (t, x)ξ γ λj = 0.

j =0 |γ |+j =m

We renumber these regularized roots such that λ1 < . . . < λm . Proposition 8 ([5, 16]) We choose ε = ξ −1 . Then the regularized roots satisfy the following relations for all α ∈ Nn and all k = 1, . . . , m. 1 ). (i) We have λk ∈ C([0, T ]; C∗s S1, 0 (ii) For (t, ξ ) ∈ [0, T ] × {|ξ | ≥ M}, we have

1  1 1 α 1 1∂ξ λk (t, ·, ξ ) − τk (t, ·, ξ ) 1

C∗s

≤ Cα, s ξ −|α|

1 . η(|ξ |−1 )

(iii) For (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}, we have 1  1 1 1 α 1∂ξ λk (t, ·, ξ ) − τk (t, ·, ξ ) 1

C∗s

≤ −Cα, s ξ −|α| -(|ξ |−1 )

 d 1 . dt -(η−1 (t − |ξ |−1 ))

Hyperbolic Cauchy Problems with Low-Regular Coefficients

345

(iv) For (t, ξ ) ∈ [0, T ] × {|ξ | ≥ M}, we have 1 1 1 α 1 1∂ξ ∂t λk (t, ·, ξ )1

C∗s

≤ Cα, s ξ 1−|α|

1 . η(|ξ |−1 )

(v) For (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}, we have 1 1 1 α 1 1∂ξ ∂t λk (t, ·, ξ )1

C∗s

 1  1 d 2 ≤ Cα, s ξ 1−|α| − . dt η−1 (t − |ξ |−1 )

(vi) For (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}, we have 1 1 1 α 2 1 1∂ξ ∂t λk (t, ·, ξ )1

C∗s

≤ −Cα, s ξ 2−|α| -(|ξ |−1 )

 1 d . dt -(η−1 (t − |ξ |−1 ))

5.2 Symbol Space We introduce the symbol space P m which takes account of the estimates we observe in Proposition 8. m+1 Definition 8 Let m ∈ R, b = b(t, x, ξ ) ∈ L∞ ([0, T ]; C∗s S1, 0 ). Then b belongs m to the symbol class P , if it satisfies 1 1 1 1 1 α , 1∂ξ b(t, ·, ξ )1 s ≤ Cα, s |ξ |m−|α| C∗ η(|ξ |−1 )

for (t, ξ ) ∈ [0, tξ ] × {|ξ | ≥ M}, and 1 1 1 1 α 1∂ξ b(t, ·, ξ )1

C∗s

  1 d ≤ Cα, s ξ m−|α| -(|ξ |−1 ) − −1 −1 dt -(η (t − |ξ | ))   1 d +Cα, s ξ m−|α|−1 − , dt η−1 (t − |ξ |−1 )

for (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}. m+1 Corollary 6 (Lemma 4.3 in [16]) If b ∈ P m , then b ∈ L∞ ([0, T ]; C∗s S1, 0 ). ∞ n Furthermore, if ψ ∈ C0 (R ) with supp ψ ⊂ {|ξ | ≥ M}, then 1 ψ(ξ )λk (t, x, ξ ) ∈ L∞ ([0, T ]; C∗s S1, 0 ),

and 2 ψ(ξ )Dt λk (t, x, ξ ) ∈ L∞ ([0, T ]; C∗s S1, 0 ).

With these preparations complete, let us now tend to the original Cauchy problem.

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D. Lorenz

5.3 Transformation to a First-Order System We consider the Cauchy problem (7) and set U = (Dx m−1 ψ(Dx )u, Dx m−2 ψ(Dx )Dt u, . . . , ψ(Dx )Dtm−1 u)T , where ψ localizes to large frequencies. This yields Dt U = A(t, x, Dx )U + R(t, x, Dx )U, with ⎞ 0 ξ  0 ··· 0 ⎟ ⎜ .. .. .. ⎟ ⎜ . . . ⎟, A(t, x, ξ ) = σ (A(t, x, Dx )) = ⎜ ⎟ ⎜ ⎠ ⎝ 0 ··· 0 ξ  0 am, γ (t, x, ξ ) · · · am−j, γ (t, x, ξ ) · · · a1, γ (t, x, ξ ) ⎛

where am−j, γ (t, x, ξ ) =

|γ |=m−j

1 am−j, γ (t, x)ξ γ ξ −(m−1−j ) ∈ L∞ ([0, T ]; C∗s S1, 0 ),

(13) −∞ and R(t, x, Dx ) ∈ L∞ ([0, T ]; 1, 0 ).

5.4 Diagonalization We perform two steps of diagonalization. The first step is done in both zones, whereas the second step of diagonalization is only done in the hyperbolic zone.

5.4.1

First Step of Diagonalization

We introduce the diagonalizer M1 = M1 (t, x, Dx ) with symbol ⎛ ⎜ ⎜ ⎜ M1 (t, x, ξ ) = σ (M1 ) = ⎜ ⎜ ⎝

1 λ1 ξ 

λ1 ξ 

.. . m−1

··· ··· ···

1

⎟ ⎟ ⎟ .. ⎟, ⎟ . m−1 ⎠

λm ξ 



λm ξ 

⎞

Hyperbolic Cauchy Problems with Low-Regular Coefficients

347

31 = M 31 (t, x, Dx ) with its as well as the matrix pseudodifferential operator M symbol 31 ) = σ (M1 )−1 = (cp, q (t, x, ξ ))1≤p, q≤m , σ (M given by

cp, q = (−1)q−1 ξ q−1

λi1 · . . . · λim−q

m G

(λi − λp )−1 ,

i=1 i=p

(m)

S{p} (m−q)

for 1 ≤ q ≤ m − 1 and by cp, m = (−1)

m−1

ξ 

m−1

m G

(λi − λp )−1 ,

i=1 i=p

where , SB(m) (k) := (i1 , . . . , ik ) ∈ Nk : 1 ≤ i1 < . . . < ik ≤ m, il ∈ / B, l = 1, . . . , k . 31 defined Proposition 9 For the matrix pseudodifferential operators M1 and M above, we have the following properties. 31 ) ∈ L∞ ([0, T ]; C∗s S 0 ) for |ξ | ≥ M. (i) We have σ (M1 ), σ (M 1, 0 (ii) In Zhyp (N, M) ∪ Zpd (N, M), we have for s = 1 + ε > 1. 31 (t, x, Dx )M1 (t, x, Dx ) = I + R1 (t, x, Dx ), M with R1 (t, x, Dx ) : H r−1 → H r continuously for |r| < ε and (t, x) ∈ [0, T ] × Rn . Proof Both assertions follow essentially from the composition result for operators m (see Proposition 4 and Corollary 5) and the observation that σ (M 31 ) is from C∗s 1, 0 the inverse matrix of σ (M1 ).   We set U = M1 (t, x, Dx )U1 and obtain that Dt U = (Dt M1 )U1 + M1 Dt U1 = AM1 U1 + RU, which gives 31 AM1 U1 − M 31 (Dt M1 )U1 − R1 (Dt U1 ) + M 31 RM1 U1 , D t U1 = M

(14)

where R ∈ L∞ ([0, T ];  −∞ ) and R1 (t, x, Dx ) : H r−1 → H r continuously for |r| < ε and (t, x) ∈ [0, T ] × Rn .

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D. Lorenz

Proposition 10 The Cauchy problem (14) is for s = 1 + ε > 1 equivalent to Dt U1 = (A1 + B1 − C1 + R2 + R∞ )U1 − R1 (Dt U1 ),

(15)

where ⎛ ⎜ σ (A1 ) = ⎝

⎞

λ1 ..

⎟ ⎠,

.

σ (B1 ) ∈ P 0 ,

λm and σ (C1 ) = (ep, q )1≤p, q≤m with

ep, q =

⎧ m  ⎪ ⎪ −D λ ⎪ t p ⎪ ⎪ i=1 ⎪ ⎪ i=p ⎪ ⎪ ⎪ ⎨ m C

p = q,

1 λi −λp ,

(λi −λq )

⎪ i=1 ⎪ ⎪ i=p, q ⎪ ⎪ −Dt λq C , m ⎪ ⎪ ⎪ (λi −λp ) ⎪ ⎪ ⎩ i=1

p = q,

i=p

31 RM1 : and R1 as above, R2 (t, x, Dx ) : H r → H r for |r| < ε and R∞ = M H r → H q for |q|, |r| < s and all (t, x) ∈ [0, T ] × Rn . 31 AM1 and M 31 (Dt M1 ) and analyze Proof To obtain (15) from (14), we compute M the respective symbols. m (see Proposition 4 Following our composition result for operators from C∗s 1, 0 and Corollary 5), we obtain that 31 )σ (A)σ (M1 ) + R, 3 31 AM1 ) = σ (M σ (M 3 : H r → H r for |r| < ε. We write with R ⎛ ⎜ ⎜ σ (A) = ⎜ ⎝ ⎛ ⎜ +⎜ ⎝

0 .. .

ξ 

0 .. .

···

⎞

0 .. .

⎟ ⎟ ⎟ ⎠

0 ··· 0 ξ  0 aε, m, γ · · · aε, m−j, γ · · · aε, 1, γ ⎞

⎟ ⎟, ⎠ (am, γ − aε, m, γ ) · · · (am−j, γ − aε, m−j, γ ) · · · (a1, γ − aε, 1, γ ),

Hyperbolic Cauchy Problems with Low-Regular Coefficients

where aε, m−j, γ = aε, m−j, γ (t, x, ξ ) =

349

aε, m−j, γ (t, x)ξ γ ξ −(m−1−j ) ,

|γ |=m−j

and am−j, γ are given by (13). Noting that (am−j, γ − aε, m−j, γ ) ∈ P 0 , (due to Proposition 7) we have ⎛ 31 )σ (A)σ (M1 ) = ⎜ σ (M ⎝

⎞

λ1 ..

⎟ ⎠ + B1 ,

. λm

with σ (B1 ) ∈ P 0 . 31 (Dt M1 ), we first look at (Dt M1 ). Due to Proposition 8, we have Concerning M for (t, ξ ) ∈ [0, T ] × {|ξ | ≥ M} that 1 1 1 1 1 α , (16) 1∂ξ ∂t λk (t, ·, ξ )ξ −1 1 s ≤ Cα, s ξ −|α| C∗ η(|ξ |−1 ) and for (t, ξ ) ∈ [tξ , T ] × {|ξ | ≥ M}, we have 1 1 1 α 1 1∂ξ ∂t λk (t, ·, ξ )ξ −1 1

C∗s

 1  1 d 2 ≤ Cα, s ξ −|α| − . dt η−1 (t − |ξ |−1 )

(17)

Thus σ (Dt M1 ) does not belong to P 0 (due to the exponent 12 ) but it clearly belongs 1 ). Application of the composition result (see Proposition 4 to L∞ ([0, T ]; C∗s S1, 0 and Corollary 5), yields ¯ 31 )σ (Dt M1 ) + R, 31 (Dt M1 )) = σ (M σ (M 31 )σ (Dt M1 ) = where R¯ : H r → H r for |r| < ε. Finally, computing σ (M (ep, q )1≤p, q≤m yields

ep, q =

⎧ m  ⎪ ⎪ −Dt λp ⎪ ⎪ ⎪ i=1 ⎪ ⎪ i=p ⎪ ⎪ ⎪ ⎨ m C

1 λi −λp ,

p = q,

(λi −λq )

⎪ i=1 ⎪ ⎪ i=p, q ⎪ ⎪ −Dt λq C , m ⎪ ⎪ ⎪ (λi −λp ) ⎪ ⎪ ⎩ i=1

p = q,

i=p

3 concludes this part of the proof. and writing R2 = R¯ + R

350

D. Lorenz

31 RM1 and observing that Finally, setting R∞ = M M1 : H r → H r , for |r| < s, R : H r → H q , for any r, q, 31 : H q → H q , for |q| < s, M  

concludes the proof.

Remark 10 Looking at (15), we note that the operator A1 is diagonal and σ (B1 ) ∈ P 0 . If σ (C1 ) belonged to P 0 , there would be no need for a second step of diagonalization and we could continue with the next step of the proof. We note that the behavior of σ (C1 ) is characterized by (16) and (17). Its behavior in the pseudodifferential zone, i.e. for t ≤ tξ is fine—there it satisfies the estimate required to belong to P 0 . However, its behavior in the hyperbolic zone does not fit into P 0 , which is why we carry out the second step of diagonalization only in the hyperbolic zone.

5.4.2

Second Step of Diagonalization

We consider the Cauchy problem (15) and want to diagonalize C1 . We follow the standard diagonalization procedure and set  e  t p, p σ (M2 ) = (dp, q )1≤p, q≤m = I + χ . Nη(ξ −1 ) λp − λq In other words

dp, q =

⎧ ⎪ ⎪ ⎨1, Dλ − t q ⎪ ⎪ λp −λq

⎩

m C i=1 i=p, q

(λi − λq )

m C

p = q, (λi − λp )−1 , p = q,

i=1 i=p

32 with symbol σ (M 32 ) = σ (M2 )−1 , for (t, x, ξ ) ∈ Zhyp (N, M). We introduce M which exists since all columns are linearly independent of each other. 32 defined Proposition 11 For the matrix pseudodifferential operators M2 and M above, we have the following properties. 32 ) ∈ L∞ ([0, T ]; C∗s S 0 ) for |ξ | ≥ M. (i) We have σ (M2 ), σ (M 1, 0 (ii) In Zhyp (N, M) ∪ Zpd (N, M), we have for s = 1 + ε > 1 32 (t, x, Dx )M2 (t, x, Dx ) = I + R3 (t, x, Dx ), M with R3 (t, x, Dx ) : H r−1 → H r continuously for |r| < ε and (t, x) ∈ [0, T ] × Rn .

Hyperbolic Cauchy Problems with Low-Regular Coefficients

351

Proof Both assertions follow essentially from the composition result for operators m (see Proposition 4 and Corollary 5). from C∗s 1,   0 We perform the second step of diagonalization by setting U1 = M2 U2 and obtain Dt U1 = (Dt M2 )U2 + M2 Dt U2 = (A1 + B1 − C1 + R2 + R∞ )M2 U2 − R1 (Dt U1 ). (18)

Proposition 12 The Cauchy problem (18) is for s = 1 + ε > 1 equivalent to Dt U2 = (A1 + A2 + B2 + R2 + R∞ )U2 − R3 (Dt U2 ) − R1 (Dt U1 ), where ⎞

⎛ λ1 ⎜ .. σ (A1 ) = ⎝ .

⎟ ⎠, λm

⎛ m  1 λi −λ1 ⎜Dt λ1 i=1 ⎜ ⎜ i=1 ⎜ .. ⎜ σ (A2 ) = ⎜ . ⎜ m  ⎜ Dt λm ⎝

i=1 i=m

(19) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ 1 ⎟ λi −λm ⎠

with σ (B2 ) ∈ P 0 and • • • •

R1 (t, x, Dx ) : H r−1 → H r for |r| < ε, R2 (t, x, Dx ) : H r → H r for |r| < ε, R3 (t, x, Dx ) : H r−1 → H r for |r| < ε, and R∞ (t, x, Dx ) : H r → H q for |q|, |r| < s.

Proof The result follows from the standard diagonalization routine (see e.g. [15]) and the composition result in Proposition 4 and Corollary 5.   We carry out one more change of variables to derive the system from which we conclude the desired energy estimate. We define the matrix pseudodifferential operator M3 = M3 (t, x, Dx ) with symbol ⎞ ⎛ w1 ⎟ ⎜ .. σ (M3 ) = ⎝ ⎠, . wm where  t wp = wp (t, x, ξ ) = exp

Ds λp (s, x, ξ ) m 

0 i=1 i=p

(λi − λp )(s, x, ξ )

ds .

352

D. Lorenz

33 = M 33 (t, x, Dx ) with We introduce the matrix pseudodifferential operator M symbol ⎛ 33 ) = σ (M3 )−1 = ⎜ σ (M ⎝

w1−1

⎞ ..

⎟ ⎠.

. −1 wm

33 defined Proposition 13 For the matrix pseudodifferential operators M3 and M above, we have the following properties. 33 ) ∈ L∞ ([0, T ]; C∗s S 0 ) for |ξ | ≥ M. (i) We have σ (M3 ), σ (M 1, 0 (ii) In Zhyp (N, M) ∪ Zpd (N, M), we have for s = 1 + ε > 1 33 (t, x, Dx )M3 (t, x, Dx ) = I + R4 (t, x, Dx ), M with R4 (t, x, Dx ) : H r−1 → H r continuously for |r| < ε and (t, x) ∈ [0, T ] × Rn . Proof Both assertions follow from the observation that t

Ds λp (s, x, ξ ) m 

0

(λi − λp )(s, x, ξ )

0 ds ∈ L∞ ([0, T ]; C∗s S1, 0 ),

(20)

i=1 i=p

for all (t, x, ξ ) ∈ [0, T ] × Rn × {|ξ | ≥ M} and from the composition result for m (see Proposition 4 and Corollary 5). Relation (20) can be operators from C∗s 1, 0 proved by splitting the integral into two integrals. One integral from 0 to tξ and one integral from tξ to T . Using the properties of λk , k = 1, . . . , m from Proposition 8 and the definition of the zones (12) then gives (20).   Setting U2 = M3 U3 then yields Dt U2 = (Dt M3 )U3 +M3 Dt U3 = (A1 +A2 +B2 +R2 +R∞ )U2 −R3 (Dt U2 )−R1 (Dt U1 ).

(21) Proposition 14 The Cauchy problem (21) is for s = 1 + ε > 1 equivalent to Dt U3 = (A1 + B2 + R2 + R∞ )U3 − R4 (Dt U3 ) − R3 (Dt U2 ) − R1 (Dt U1 ), where ⎛ ⎜ σ (A1 ) = ⎝

⎞

λ1 ..

⎟ ⎠,

. λm

σ (B2 ) ∈ P 0 ,

Hyperbolic Cauchy Problems with Low-Regular Coefficients

353

and • • • • •

R1 (t, x, Dx ) : H r−1 → H r for |r| < ε, R2 (t, x, Dx ) : H r → H r for |r| < ε, R3 (t, x, Dx ) : H r−1 → H r for |r| < ε, R4 (t, x, Dx ) : H r−1 → H r for |r| < ε, and R∞ (t, x, Dx ) : H r → H q for |q|, |r| < s.

Proof The statement of the proposition follows from the composition result for m (see Proposition 4 and Corollary 5) and the fact that operators from C∗s 1, 0 33 )σ (Dt M3 ) = σ (A2 ). σ (M

 

5.5 Conjugation We want to apply Duhamel’s principle to "

Dt U3 − (A1 + B2 + R2 + R∞ )U3 = −R4 (Dt U3 ) − R3 (Dt U2 ) − R1 (Dt U1 ), U3 (0, x) = G(x),

where we consider the terms −R4 (Dt U3 ) − R3 (Dt U2 ) − R1 (Dt U1 ) as an inhomogeneity and where G(x) denotes the vector containing the transformed initial data. In that way, we can argue that it is sufficient to consider the homogeneous Cauchy problem Dt W − (A1 + B2 + R2 + R∞ )W = 0, with initial data W (s, s, x) = −R4 (Dt U3 )(s, x) − R3 (Dt U2 )(s, x) − R1 (Dt U1 )(s, x). We use this approach to obtain an L2 − L2 energy estimate for W without loss of derivatives. We note that it is important for the application of Duhamel’s principle that • R1 (t, x, Dx ) : H r−1 → H r for |r| < ε, • R3 (t, x, Dx ) : H r−1 → H r for |r| < ε, and • R4 (t, x, Dx ) : H r−1 → H r for |r| < ε, which ensures that the initial data −R4 (Dt U3 )(s, x) − R3 (Dt U2 )(s, x) − R1 (Dt U1 )(s, x) is in L2 with respect to x. Let us now consider the homogeneous system Dt W − (A1 + B2 + R2 + R∞ )W = 0,

354

D. Lorenz

and define  V := exp

t −

ϑ(s, Dx )ds W,

0

with ϑ(t, ξ ) = K(2 + ϑ0 (t, ξ )), where  1 t t + χ 2N η(ξ −1 ) η(ξ −1 ) N η(ξ −1 )    1 1 d −1 d − ξ  , × − -(ξ −1 ) dt -(η−1 (t − ξ −1 )) dt η−1 (t − ξ −1 )

  ϑ0 (t, ξ ) = 1 − χ

and K > 0 is a constant. Proposition 15 (Lemmas 5.6 and 5.7. in [16]) We have m0 1 ∞ (i) ϑ0 (t, ξ ) ∈ L∞ ([0, T ]; S1, 0 ) ⊂ L ([0, T ]; S1, 0 ), and t  0 ). (ii) ϑ0 (s, ξ )ds ∈ L∞ ([0, T ]; S1, 0 0

Applying this transformation yields Dt V = iϑ(t, Dx )I V  + exp



t −

 t

ϑ(s, Dx )ds (A1 + B2 + R2 + R∞ ) exp 0

ϑ(s, Dx )ds V ,

0

which gives ∂t V + ϑ(t, Dx )I V − i(A1 + B1 + R2 + R∞ )V )  −i exp −



t

*

 t

ϑ(s, Dx )ds , A1 + B2 + R2 + R∞ exp 0

ϑ(s, Dx )ds V = 0.

0

We define Q0 := K(1 + ϑ0 (t, Dx ))I − i(A1 + B2 + R2 + R∞ ), ) Q1 := K I − i exp



t − 0

*  t ϑ(s, Dx )ds , A1 + B2 + R2 + R∞ exp ϑ(s, Dx )ds . 0

Hyperbolic Cauchy Problems with Low-Regular Coefficients

355

Proposition 16 (Lemma 4.6 in [16]) For s > 1, we have that m0 ∞ s 1 (i) ψ(Dx )Q0 ∈ L∞ ([0, T ]; C∗s 1, 0 ) ⊂ L ([0, T ]; C∗ 1, 0 ),   ∗ Q +Q (ii) σ ψ(Dx ) 0 2 0 ≥ ϑ0 (t, ξ )I .

Proof The first statement follows from the structure of Q0 and Proposition 15. For the second statement we employ Proposition 5 to deal with adjoints of operators 1 . We obtain that from C∗s 1, 0 σ (A∗1 ) = σ (A1 ) + r1 , σ (B2∗ ) = σ (B2 ) + r2 , with r1 , r2 : L2 → L2 continuously. Since the problem is strictly hyperbolic, we conclude that iA1 = −(iA1 )∗ − r1 . Furthermore, since σ (B2 ) ∈ P 0 , we know that |σ (B2 )| ≤ C(1 + ϑ0 (t, ξ )), which gives |σ (iB2 ) + σ ((iB2 )∗ )| ≤ C(1 + ϑ0 (t, ξ )). Thus, we may conclude that we can estimate |σ (iψ(Dx )(A1 + B2 + R2 + R∞ ) + iψ(Dx )(A1 + B2 + R6 )∗ )| ≤ C(1 + ϑ0 (t, ξ )).  

Choosing K sufficiently large then gives the second statement. Proposition 17 (Lemma 4.7 in [16]) For s > 1, we have that 0 ), (i) ψ(Dx )Q1 ∈ L∞ ([0, T ]; C∗s 1, 0   ∗ Q1 +Q1 ≥ C I. (ii) σ ψ(Dx ) 2

Proof Let us denote ) Z := exp



t − 0

*  t ϑ(s, Dx )ds , A1 + B2 + R2 + R∞ exp ϑ(s, Dx )ds , 0

356

D. Lorenz

3 = A1 + B2 + R2 + R∞ to write and use to abbreviation P )



Z = exp

t −

*  t ϑ(s, Dx )ds , P3 exp ϑ(s, Dx )ds .

0

0

Applying Proposition 4 and Corollary 5 yields that σ (Z) =

) ∂ξα σ

 exp

t −

|α|≤1



*  t α 3 Dx exp ϑ(s, Dx )ds , P ϑ(s, ξ )ds + r

0

=

0

 ∂ξα

exp

t −

|α|=1

 t α3 ϑ(s, ξ )ds (Dx P ) exp ϑ(s, ξ )ds + r.

0

0

Analyzing the symbols and applying Proposition 15 allows us to conclude that Z ∈ 0 ). L∞ ([0, T ]; C∗s 1, 0 Similarly to the argument in the proof of Proposition 16, the second result follows for sufficiently large K from the rules for computing the adjoint and the fact that 0 ). Z ∈ L∞ ([0, T ]; C∗s 1,   0

5.6 Conclusion We consider the Cauchy problem ∂t V + Q0 V + Q1 V = 0,

V (0, x) = V0 (x).

We have ∂t V (t, ·) 2L2 = −26(Q0 V , V ) − 26(Q1 V , V ). According to Propositions 16 and 17 we have  Q0 + Q∗0  ≥ 0, σ ψ(Dx ) 2

 Q1 + Q∗1  σ ψ(Dx ) ≥ 0, 2

which allows us to apply sharp Gårding’s inequality (see Proposition 6). Since 0 ψ(Dx )Q1 ∈ L∞ ([0, T ]; C∗s 1, 0 ),

Hyperbolic Cauchy Problems with Low-Regular Coefficients

357

and m0 ψ(Dx )Q0 ∈ L∞ ([0, T ]; C∗s 1, 0 ),

application of our result for sharp Gårding’s inequal is only possible if s≥

2m0 . 2 − m0

This yields −26(Q0 V , V ) − 26(Q1 V , V ) ≤ C V (t, ·) 2L2 , for s ≥

2m0 2−m0 .

Application of Gronwall’s lemma gives V (t, ·) L2 ≤ CT V (t, ·) L2 .

Using the definition of U we obtain (Dx m−1 u(t, ·), . . . , Dtm−1 u(t, ·))T L2 ≤ CT (g1 (·), . . . , gm (·))T L2 ,  where we used that M1 , M2 , M3 and exp

−

t

ϑ(s, Dx )ds

belong to

0

0 ), i.e. they are operators of order zero. L∞ ([0, T ]; C∗s 1, 0 We conclude the proof by noting that condition (11), i.e.

 2m0  , s ≥ max 1 + ε, 2 − m0

(11)

2m0 is due to the restriction s ≥ 2−m coming from sharp Gårding’s inequality and due 0 to the restriction s > 1 coming from our results for composition and adjoints in C∗s  m .

6 Concluding Remarks In this paper we consider the strictly hyperbolic Cauchy problem ⎧ m−1 ⎪ ⎨D m u −  t

⎪ ⎩



j =0 |γ |+j =m

γ

j

am−j, γ (t, x)Dx Dt u = 0,

Dtk−1 u(0, x) = gk (x), k = 1, . . . , m,

358

D. Lorenz

for (t, x) ∈ [0, T ] × Rn , with coefficients low regular in time and space. By this we mean that the coefficients are less regular than Lipschitz in time and belong to the Zygmund space C∗s in x. Under suitable assumptions we prove a global (on [0, T ]) well-posedness result without loss of derivatives if  s ≥ max 1 + ε,

2m0  , 2 − m0

(22)

where ε > 0 and m0 ∈ (0, 1] is related to the regular of the coefficients in time. The number m0 is closer to 1 for less regular (in time) coefficients and closer to 0 if the coefficients are close to Lipschitz in time. Of course one would expect that a higher regular of the coefficients in time would allow us to lower the regularity of the coefficients with respect to x. Condition (22) however tells us that the index of the Zygmund spaces s always has to be strictly larger than 1. At some point, additional regularity in time of the coefficients does not improve the situation for s any further. Of course it would be interesting to study whether it is possible to lower the bound s ≥ 1 + ε to s ≥ 1. A possible approach could be to use paradifferential methods to attack the problem. Acknowledgements The author wants to express his gratitude to Michael Reissig for many fruitful discussions and suggestions. Furthermore, he wants to thank Daniele Del Santo for his hospitality and the suggested improvements during the authors stay at Trieste University.

References 1. Abels, H.: Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients. Comm. Partial Differential Equations 30.10 (2005), 1463–1503, doi: https://doi.org/10.1080/ 03605300500299554. 2. Agliardi, R., Cicognani, M.: Operators of p-evolution with nonregular coefficients in the time variable. J. Differential Equations 202.1 (2004), 143–157, doi: https://doi.org/10.1016/j.jde. 2004.03.028. 3. Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations., Springer, Berlin and Heidelberg (2011). 4. Cicognani, M., Colombini, F.: Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem. J. Differential Equations 221.1 (2006), 143–157, doi: https://doi.org/10.1016/j.jde.2005.06.019. 5. Cicognani, M., Lorenz, D.: Strictly hyperbolic equations with coefficients low-regular in time and smooth in space. J. Pseudo-Differ. Oper. Appl. (2017), online first, doi: https://doi.org/10. 1007/s11868-017-0203-2. 6. Colombini, F., De Giorgi E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola. Norm.-Sci. 6.3 (1979), 511–559. 7. Colombini, F., Del Santo, D., Kinoshita, T.: Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1.2 (2002), 327–358.

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8. Colombini, F., Del Santo, D., Reissig, M.: On the optimal regularity of coefficients in hyperbolic Cauchy problems. Bull. Sci. Math. 127.4 (2003), 328–347, doi: https://doi.org/10. 1016/S0007-4497(03)00025-3. 9. Colombini, F., Lerner, N.: Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77.3 (1995), 657–698, doi: https://doi.org/10.1215/S0012-7094-95-07721-7. 10. Hirosawa, F.: On the Cauchy problem for second order strictly hyperbolic equations with non-regular coefficients. Math. Nachr. 256 (2003), 29–47, doi: https://doi.org/10.1002/mana. 200310068. 11. Hirosawa, F., Reissig, M.: Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients. Ann. Global Anal. Geom. 25.2 (2004), 99–119, doi: https://doi.org/10.1023/B:AGAG.0000018553.77318.81. 12. Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin, Heidelberg (1963). 13. Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators., Reprint of the 1994 ed., Springer, Berlin (2007). 14. Hurd, A. E., Sattinger, D. H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Amer. Math. Soc. 132 (1968), 159–174, doi: https://doi. org/10.2307/1994888. 15. Jachmann, K., Wirth, J.: Diagonalisation schemes and applications. Ann. Mat. Pura Appl (4) 189.4 (2010), 571–590. 16. Kinoshita, T., Reissig, M.: About the loss of derivatives for strictly hyperbolic equations with non-Lipschitz coefficients. Adv. differential Equations 10.2 (2005), 191–222. 17. Kumano-go, H.: Pseudo-differential operators., English-language ed. MIT Press, Cambridge, Massachusetts 1982. 18. Marschall, J., Löfström, J., Marschall, J.: Pseudo-differential operators with nonregular symbols of the class Sm p p. Comm. Partial Differential Equations 12.8 (1987), 921–965, doi: https://doi.org/10.1080/03605308708820514. 19. Mizohata, S.: The theory of partial differential equations., Cambridge University Press, New York 1973. 20. Reissig, M.: Hyperbolic equations with non-Lipschitz coefficients. Rend. Sem. Mat. Univ. Politec. Torino 61.2 (2003), 135–182. 21. Tataru, D.: On the Fefferman-Phong inequality and related problems. Comm. Partial Differential Equations 27.11-12 (2002), 2101–2138, doi: https://doi.org/10.1081/PDE-120016155. 22. Taylor, M.: Pseudodifferential Operators and Nonlinear PDE., Vol. 100. Progress in Mathematics. Birkäuser, Boston 1973. 23. Triebel, H.: Theory of function spaces., Modern Birkhäuser Classics. Springer, Basel 2010.

Quantization and Coorbit Spaces for Nilpotent Groups M. M˘antoiu

Abstract We reconsider the quantization of symbols defined on the product between a nilpotent Lie algebra and its dual. To keep track of the non-commutative group background, the Lie algebra is endowed with the Baker-Campbell-Hausdorff product, making it via the exponential diffeomorphism a copy of its unique connected simply connected nilpotent Lie group. Using harmonic analysis tools, we emphasize the role of a Weyl system, of the associated Fourier-Wigner transformation and, at the level of symbols, of an important family of exponential functions. Such notions also serve to introduce a family of phase-space shifts. These are used to define and briefly study a new class of coorbit spaces of symbols and its relationship with coorbit spaces of vectors, defined via the Fourier-Wigner transform. Keywords Nilpotent lie group · Pseudo-differential operator · Coorbit space

1 Introduction We discuss quantization on connected, simply connected nilpotent Lie groups G involving scalar-valued symbols. The main reason for which this is (at least formally) straightforward is the fact that the exponential exp : g → G is a diffeomorphism and, under it, the Haar measures on G are proportional with the Lebesgue measure on the Lie algebra g . Denoting by g1 the dual of the Lie algebra, the symbols are complex functions defined on G × g1 or, equivalently, on g × g1 . Of course, both these spaces can be seen as cotangent spaces, but we insist on the fact that the quantization is expected to be “global” (choosing charts for constructing the calculus is not needed and would be harmful) and that the group structure of G should play an important role. Another way to see the usefulness of nilpotence is to note that it allows a well-behaved Fourier transformation from functions or

M. M˘antoiu () Facultad de Ciencias, Universidad de Chile, Santiago, Chile e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_20

361

362

M. M˘antoiu

distributions on G to functions or distributions on g1 and this Fourier transformation plays an important role in the pseudo-differential calculus. Basically, if a is a function on G × g1 and ϕ a function on G , under favorable circumstances and with suitable interpretations, one is interested in   Op(a)ϕ (x) :=

  G

g1

eilog(xy

−1 )|ξ 

a(x, ξ )ϕ(y) dydξ

(1)

where, by definition, log := exp−1 : G → g . After suitable isomorphic compositions, this yields the equivalent form (9), in which • is the BackerCampbell-Hausdorff composition on the Lie algebra, leading (via the exponential map) to a group isomorphism (G, ·) ∼ = (g, •) . Although quite natural, the prescription (9) (or (1)) has not been considered in such a general setting until recently, and the problem of defining and studying good Hörmander-type symbol classes is a non-trivial challenge. Important articles have been dedicated to particular cases. Here “particular” very often means restricting to invariant (convolution) operators (formally, the function a in (1) only depends on ξ ). It could also mean that the nilpotent group is two-step, or graded. Since we are not concerned here with the difficult problem of a Hörmander-type calculus, we only cite [2, 7, 14–16, 27, 28, 34–36] without details. Let us mention, however, that in the cited articles of P. Głowacki and D. Manchon, the invariant calculus is studied in depth, partly relying on the important previous work of R. Howe [25, 26]. In [33] the case of nilpotent groups with (generic) flat coadjoint orbit is treated, making a precise connection with the well-developed [11, 12, 32, 37] operator-valued pseudo , where G  is the unitary dual of G , i.e. the family of differential calculus on G × G all equivalence classes of irreducible unitary Hilbert space representations of G . In the present paper, we mainly rely on Harmonic Analysis tools. We give a short description of its content. Section 2 Section 2 contains basic notions and notations concerning Hilbert space operators and nilpotent groups. A remarkable family of exponential functions on 2 := g × g1 is introduced; it will play an important role subsequently. , Section 3 The basic object is the Weyl system E(X, ξ ) | X ∈ g , ξ ∈ g1 , a family of unitary operators in L2 (g) mixing left translations associated to the group (g, •) with multiplication by imaginary exponentials with phase given by the duality between g and g1 . This family  is very far from being a projective representation of the product group (g, •) × g1 , + (that we denote by 2 and call phase space). This is why most of the technics developed in the literature do not apply automatically. It can be used to shift bounded operators A → E(X, ξ )AE(Y, η)∗ in a useful way. The “matrix coefficients” of the Weyl system defines the Fourier-Wigner transformation. It satisfies orthogonal relations and serves to introduce rigorously Berezin-type operators, cf. [30]. Section 4 We introduce the quantization Op and state its connection with the Weyl system, the Fourier-Wigner transformation and the Berezin quantization. Then

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we indicate the ∗ -algebraic laws on symbols that correspond to composition and adjunction of pseudo-differential operators. On the Schwartz space S(2) one gets a Hilbert algebra structure, that can be used to perform various extensions of the laws by duality techniques, that will be useful below. In particular, one gets a (rather large, but not so explicit) Moyal algebra of symbols, quantized by operators that are continuous on the Schwartz space S(g) and extend continuously on the dual S (g) . Section 5 Since the elements of the Weyl system are obtained by quantizing the special exponential function introduced in the first section, for pseudo-differential operators the shift A → E(X, ξ )AE(Y, η)∗ is emulated by a similar one at the level of symbols, involving these exponentials and the intrinsic algebraic laws. We give the definition and the basic properties, that are interesting in their own right, are used in Sect. 6 for constructing coorbit spaces of symbols, and will reappear in our subsequent study of a Beals-Bony commutator criterion on nilpotent groups. Section 6 Section 6 is dedicated to a tentative definition of coorbit spaces at two levels: (i) coorbit spaces of vectors, contained in S (g) , and (ii) coorbit spaces of symbols, contained in S (2) . We declare from the very beginning that the treatment is incomplete from many points of view. Although there is a lot of group theory around, one does not follow the orbits of some (usual or projective) group representation. At both levels, one uses isometric linear mappings, labeled by fixed functions (windows), sending functions (or distributions) on the space we are interested in (here g , respectively 2), to functions or distributions on larger spaces (2 or 2×2, respectively). Then one selects for the coorbit space elements that have a certain behavior under the isometry (belonging to a given subspace, or having a finite given norm). For the first level we use the Fourier-Wigner transform, naturally depending on two vectors, fixing one of them as a window and measuring the dependence of the other one. For the particular case of the Heisenberg group, it would be interesting to compare the outcome with the constructions of [13], in which, a priori, the point of view is different. Doubling the number of variables, such a procedure would probably also work well for (ii), and this is roughly what is done in the Abelian case G = Rn to define coorbit spaces of symbols. However, we adopt another strategy. Adapting some abstract ideas from [29], the isometry we use can be found in Definition 9 and it relies on the previously defined phase-space shifts of the symbol, coupled by duality with the chosen window. To advocate this choice, we put into evidence two properties of our isometry that can be obtained quite easily: 1. For a self-adjoint idempotent window, it is a ∗ -algebra monomorphism (cf. Proposition 6). The consequence (Corollary 2) is the fact that starting with an algebra of kernels defined on 2 × 2 , stable under the natural kernel adjoint, the corresponding coorbit space is a ∗ -algebra of symbols with respect to the intrinsic symbol composition and adjoint (so, by quantization, one gets ∗ -algebras of pseudo-differential operators).

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2. For a good correlation of the chosen windows, there is a formula (25) relying the two types of isometries, the Op-calculus and the calculus of integral operators. This allows in Theorem 1 to state roughly that pseudo-differential operators with symbols in certain coorbit spaces are bounded between certain coorbit spaces of vectors. Actually, this is merely stated in a weaker form making use of coorbit norms on Schwartz spaces. It is obvious that much more effort is needed to transform this sketchy treatment into a theory. Remarks 11 and 12 can be read as a self-criticism. To be brief, let us say that the abstract part is still incomplete, while the concrete part misses completely. Hopefully, there will be some progress in a future publication. It is not yet clear to us how far one can go, since the notions, although quite elementary, have complicated explicit expressions. Probably particular classes of nilpotent groups should be considered first. It is clear that the coorbit theory part of this paper relies on many previous contributions of many authors. The number of interesting articles belonging to Time Frequency Analysis and treating modulation or other coorbit spaces on various mathematical structures and from various points of view is huge. Even if one restricts to classics and to those papers involving pseudo-differential operators, it is not the place here to sketch a history or at least to cite “most” of the references. Not forgetting to mention the central role played by H. Feichtinger and K. Gröchenig, we make a selection of references [3–6, 8–10, 17–24, 38–40] that inspired or are related to the last section of the present article.

2 Framework Conventions The scalar products in a Hilbert space are linear in the first variable. For a given (complex, separable) Hilbert space H , one denotes by B(H) the C ∗ algebra of all linear bounded operators in H, by K(H) the closed bi-sided ∗ -ideal of all compact operators and by B2 (H) the bi-sided ∗ -ideal of all Hilbert-Schmidt operators. The group of unitary operators in H is denoted by U(H) . If F, G are locally convex spaces, one sets L(F, G) for the space of linear continuous operators T : F → G . We admit the abbreviation L(F, F) =: L(F) . Let G be a connected simply connected nilpotent Lie group with unit e , center  . Let g be the Lie algebra of G Z , bi-invariant Haar measure dx and unitary dual G with center z = Lie(Z) and g1 its dual. If X ∈ g and ξ ∈ g1 we set X | ξ  := ξ(X) . We also denote by exp : g → G the exponential map, which is a diffeomorphism. Its inverse is denoted by log : G → g . Under these diffeomorphisms the Haar measure on G corresponds to a Haar measure dX on g (normalized accordingly). It then follows that Lp (G) is isomorphic to Lp (g) . The Schwartz spaces S(G) and S(g) are defined as in [1, A.2]; they are isomorphic Fréchet spaces.

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Remark 1 For X, Y ∈ g we set X • Y := log[exp(X) exp(Y )] . It is a group composition law on g , given by a polynomial expression in X, Y (the Baker-Campbel-Hausdorff formula). The unit element is 0 and X• ≡ −X is the inverse of X with respect to • . One has in fact 1 1 1 X•Y = X+Y + [X, Y ]+ [X, [X, Y ]]+ [Y, [Y, X]]+· · · ≡ X+Y +R(X, Y ) , 2 12 12 (2) where, by nilpotency, the sum is finite. It seemed to us easier to work on the group (g, •) , but transferring all the formalism to its isomorphic version (G, ·) is an obvious task. The adjoint action [1] is Ad : G × g → g ,

Adx (Y ) :=

 d

 x exp(tY )x −1 )

dt t=0

and the coadjoint action of G is Ad1 : G × g1 → g1 ,

(x, η) → Ad1x (η) := η ◦ Adx −1 .

Translating to the Lie algebra, one gets O 1 : g × g1 → g1 , Ad

O 1 (η) := Ad1 (X, η) → Ad X exp X (η).

(3)

  One has the left and the right unitary representations L, R : (g, •) → U L2 (g) , defined by     LZ (u) (X) := u [−Z]•X ,

  RZ (u) (X) := u(X•Z) .

We call (somehow inappropriately)  phase  space the direct product noncommutative group (2, ◦) := (g, •) × g1 , + . Definition 1 For every (Z, ζ ) ∈ 2 we define ε(Z,ζ ) : 2 → T , εZ : g1 → T , εζ : g → T by ε(Z,ζ ) (X, ξ ) := eiX|ζ  e−iZ|ξ  ,

εZ := ε(Z,0) ,

εζ := ε(0,ζ ) .

These functions will play an important role in quantization. The proof of the following lemma consists in simple calculations; for (4) one needs (2).

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Lemma 1 For all the values of the parameters one has ε(Z,ζ ) = εZ εζ ≡ εζ ⊗ εZ , ε(Z1 +Z2 ,ζ1 +ζ2 ) = ε(Z1 ,ζ1 ) ε(Z2 ,ζ2 ) , ε(Z,ζ ) (X•Y, ξ + ζ ) = ε(Z,ζ ) (X, ξ )ε(Z,ζ ) (Y, η)eiR(X,Y )|ζ  .

(4)

One has a linear topological isomorphism F : S(g) → S(g1 ) , the Fourier transformation, given by 

 Fu (ξ ) =

 g

e−iX|ξ  u(X)dX =

 g

εξ (X) u(X)dX .

For a unique good choice of the Haar measure dξ on g1 , the inverse is  −1  F u (X) =

 g1

eiX|ξ  u(ξ )dξ =

 g1

εξ (X) u(ξ )dξ ,

the transformation F is unitary from L2 (g; dX) to L2 (g1 ; dξ ) and extends to a topological isomorphism F : S (g) → S (g1 ) . We are also going to use the total Fourier transformation f → f3 given by f3(Z, ζ ) :=

  g g1

e−iY |ζ  eiZ|η f (Y, η)dY dη .

(5)

Lemma 2 For every f, g ∈ S(2) one has  2

f, εX (2) εX , g(2) dX = f, g(2) .

(6)

Proof First one notes that f, εX (2) = f3(X) and then invokes Plancherel’s Theorem.  

3 Weyl Systems, the Fourier-Wigner Transform Definition 2 For any (Z, ζ ) ∈ g × g1 = 2 one defines a unitary operator E(Z, ζ ) in L2 (g) by   E(Z, ζ )u (X) := eiX|ζ  u([−Z]•X) ,

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with adjoint 

 E(Z, ζ )∗ u (Y ) = e−iZ•Y |ζ  u(Z•Y ) .

This extends the notion of Weyl system (or time-frequency shifts) from the case G = Rn . Since the composition law • is polynomial, these operators also act as isomorphisms of the Schwartz space S(g) and can be extended to isomorphisms of the space S (g) of tempered distributions. We are going to see below how they fit in the pseudo-differential calculus. Lemma 3 Denote by Mult(φ) the operator of multiplication by the function φ . For (Z, ζ ), (Y, η) ∈ 2 one has    E(Z, ζ ) E(Y, η) = Mult ϒ (Z, ζ ), (Y, η); · E(Z•Y, ζ + η) , where   , ϒ (Z, ζ ), (Y, η); X = exp i [−Z]•X − X) | η  . This follows from a direct calculation. The map E is not even a projective representation of the group 2 , so standard tools in coorbit theory relying on group representations will not be available. One also sets EZ := E(Z, 0) ≡ LZ ,

Eζ := E(0, ζ ) ≡ Mζ = Mult(εζ ) .

Note the “multiplication relations” LY LZ = LY •Z ,

Mη Mζ = Mη+ζ ,

LZ Mζ = eiZ

−1 •(·)−(·)|ζ 

Mζ LZ ,

that follow from Lemma 3 or are shown directly. So one has (strongly continuous) unitary representation   M : (g1 , +) → U L2 (g) and

  L : (g, •) → U L2 (g)

that do not commute to each other. Definition 3 For any Y := (Y, η), Z := (Z, ζ ) ∈ 2 := g × g1 we define the linear contraction $Y,Z : B(H) → B(H) ,

$Y,Z (A) := E(Y)A E(Z)∗ = Mη LY A L−Z M−ζ .

In particular, $Y,Y is an automorphism of the C ∗ -algebra B(H) . There are no simple group properties of the family.

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Remark 2 As said above, besides being unitary operators in L2 (g) , the elements E(Z) of the Weyl system can also be seen as isomorphisms of S(g) or of its dual. Therefore $Y,Z also acts on L[S(g)] , L[S (g)] , L[S(g), S (g)] and L[S (g), S(g)] . Definition 4 For u, v ∈ H := L2 (g) one sets Eu,v ≡ Eu⊗v : g × g1 → C by  Eu,v (Z, ζ ) := E(Z, ζ )u, vH =

g

eiY |ζ  u([−Z]•Y )v(Y )dY.

and call it the Fourier-Wigner transform. Lemma 4 The Fourier-Wigner transform extends to a unitary map E: H ⊗ H ∼ = L2 (g × g) → L2 (2) . It also defines isomorphisms E : S(g) ⊗ S(g) ∼ = S(g × g) → S(2) ,

E : S (g) ⊗ S (g) ∼ = S (g × g) → S (2) .

Proof It is composed of a partial Fourier transformation and a unitary change of variables, that is also S-compatible. We denoted by ⊗ the completed projective tensor product, but we recall that S(g) is nuclear.   In particular, one has the orthogonality relations: Eu,v , Eu,v 

! L2 (2)

= u, u H v  , vH .

(7)

From now on, we are going to use the notation ·, ·(2) both for the scalar product in L2 (2) and for the duality between the Schwartz space on 2 and the space of temperate distributions. Similarly for ·, ·(g) . Remark 3 In [30], the Berezin-Toeplitz calculus for suitable symbols h : G × g1 → C has been introduced and studied. As in the present article, G was a connected, simply connected nilpotent Lie group with Lie algebra g1 , but the Berezin operators act in L2 (G) or S(G) . By suitably composing with the diffeomorphism exp : g → G (both at the level of vectors and symbols) this may be recast in the present setting. For convenience of the reader, we indicate the basic definition, making use of the objects E and E introduced above. The full treatment in [30] can easily transported on g × g1 . Let w ∈ S(g) be a normalized vector (it may also be chosen in L2 (g)). We define in L2 (g)   Berw (h) := h(X, ξ ) ·, E(X, ξ )∗ wE(X, ξ )∗ w dXdξ . (8) g g1

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The rigorous definition is in weak sense: for any u, v ∈ L2 (g) one has Berw (h)u, v

 

! (g)

:=

g g1

! ! h(X, ξ ) u, E(X, ξ )∗ w E(X, ξ )∗ w, v dXdξ

  =

g g1

h(X, ξ ) Eu,w (X, ξ ) Ev,w (X, ξ ) dXdξ

= h, Eu,w Ev,w

! (2)

.

This last expression and the properties of the Fourier-Wigner transform allow various interpretations of this formula, under various conditions on u, v, w, h .

4 Pseudo-Differential Operators One has the quantizations of the “phase space” g × g1 - (X, ξ )   Op : L2 (g × g1 ) → B2 L2 (g) ,     Op(f )u (X) = eiX•(−Y )|ξ  f (X, ξ ) u(Y ) dY dξ .

(9)

g g1

2 Remark  2 4 Examining the kernel of Op(f ) , one easily sees that Op : L (2) → 2 B L (g) is indeed an isomorphism. For similar reasons, by restriction or extension, one also has topological linear isomorphisms

  ∼ Op : S(2) −→ L S (g), S(g) ,

  ∼ Op : S (2) −→ L S(g), S (g) .

One may justify (at least heuristically) formula (9) in various ways: • One could start with a canonical dynamical system, built over the left action of (g, •) on itself and then raised to a C ∗ -action on function defined on g . To such a data, there is a canonical construction of a C ∗ -algebra (the crossed product) and of a “Schrödinger representation” in H := L2 (g) . The calculus Op is then obtained from this Schrödinger representation by composing with a partial Fourier transformation. For , details we refer to [32]. • In terms of the Weyl system E(Z, ζ ) | (Z, ζ ) ∈ 2 and the total Fourier transformation f → f3, one can write   Op(f ) =

g g1

f3(Z, ζ )E(Z, ζ ) dZdζ .

• In the simple Abelian case G ≡ g = Rn one has X•(−Y ) = X − Y and (9) boils down to the Kohn-Nirenberg quantization.

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Actually, in terms of the Fourier-Wigner transform, one can write ! Op(f )u, v(g) = f3, Eu,v (2) ,

(10)

allowing various types of ingredients u, v, f , having in view the properties of the transformation E and of the dualities. By using Plancherel’s Theorem one could rewrite (10) as ! Op(f )u, v(g) = f, Wu,v (2) , (11) and (u, v) → Wu,v could be called the Wigner transformation. It is easy to prove the next result: Proposition 1 (i) One has   E(Z, ζ ) = Op ε(Z,ζ ) ,

∀ (Z, ζ ) ∈ 2 .

In particular LZ = Op(εZ ) and Mζ = Op(εζ ) . (ii) One has Op(φ ⊗ ψ) = Mult(φ)ConvL (F−1 ψ) , the product between a multiplication operator and a left convolution operator (that is right invariant). Particular cases: f (X, ξ ) := φ(X) ⇒ Op(f )u = φu , f (X, ξ ) := ψ(ξ ) ⇒ Op(f )u = (F−1 ψ)  u .   (iii) The rank one operator ·, vu coincides with Op Eu,v . Remark 5 In [30, Sect. 6], a connection has been established between pseudo differential and Berezin-type operators with symbols defined on G×g1 . By properly composing with the exponential diffeomorphism, one lands in our framework and finds that the Berezin operator Berw (h) given in (8) is an operator of the form (9), with    f (X, ξ ) := e−iY |ξ  eilog(Z•[−Y ]•X)−Z•X|ζ  g g g1

h(Z, ζ ) w(Z • X) w(Z • [−Y ] • X) dY dZdζ . We treat now the intrinsic algebraic structure on symbols. The pseudo-differential operator (9) with symbol f is an integral operator with kernel Kerf : g × g → C given by      Kerf (X, Y ) = eiX•[−Y ]|ξ  f (X, ξ )dξ = id ⊗ F−1 f X, X•[−Y ] . (12) g1

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Inverting, the symbol may be recuperated from the kernel by means of the formula  f (X, ξ ) =

g

  e−iY |ξ  Kerf X, [−Y ]•X dY.

(13)

Proposition 2 (i) The symbol f #g of the product Op(f )Op(g) is     (f #g)(X, ξ ) =

g g g1 g1

εξ (Y ) εη (X•[−Z]) εζ (Z•[−X]•Y ) f (X, η)g(Z, ζ )dY dZdηdζ .

(ii) The symbol f # of the adjoint Op(f )∗ is   f # (X, ξ ) =

g g1

eiY |η−ξ  f ([−Y ]•X, η) dY dη .

(14)

In particular (φ ⊗ 1)# = φ ⊗ 1 and (1 ⊗ ψ)# = 1 ⊗ ψ . Proof (i) One computes    (f #g)(X, ξ ) = Ker−1 Kerf ◦ Kerg (X, ξ )     = e−iY |ξ  Kerf ◦ Kerg X, [−Y ]•X dY g

  =

g g

  e−iY |ξ  Kerf (X, Z)Kerg Z, [−Y ]•X dY dZ

    =

g g g1 g1

e−iY |ξ  eiX•[−Z]|η eiZ•[−X]•Y |ζ  f (X, η)g(Z, ζ )dY dZdηdζ .

(ii) If K is the kernel of an integral operator, the kernel of the adjoint is given by K  (X, Y ) := K(Y, X) . Hence, by (12) and (13)  f (X, ξ ) = #

G

 =

G

 =

G

  e−iY |ξ  Kerf # X, [−Y ]•X dY   e−iY |ξ  Ker f X, [−Y ]•X dY   e−iY |ξ  Kerf [−Y ]•X, X dY

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 =

e−iY |ξ 

G

  =

G g1

 g1

e−iY |η f ([−Y ]•X, η)dη dY

eiY |η−ξ  f ([−Y ]•X, η)dη dY.  

 # Corollary 1 For every Z ∈ 2 one has Op(εZ )∗ = Op εZ , with # iZ|ξ  −iZ•X|ζ  ε(Z,ζ e . ) (X, ξ ) = e

 # = E(Z)∗ for Proof This follows from (14), or by checking directly that Op εZ every Z ∈ 2 .   One already has the linear topological isomorphism of Gelfand triples -

?

-

-

?

-

?

The horizontal arrows are linear continuous dense embeddings. The first vertical arrow of ∗ -algebras. Taking into account the fact that  2 is also an isomorphism 2 ∗ B L (g) is a H -algebra (i.e. a complete Hilbert algebra) with respect to the operator product, the usual adjoint and the scalar product associated to the trace, one gets easily Lemma 5   (i) L2 (2), # ,# , ·, ·L2 (2) is a H ∗ -algebra.   (ii) S(2), #,# ·, ·L2 (2) is a Hilbert algebra. In particular, this means that for every f, g, h ∈ L2 (2) one has f #g, h(2) = f, h#g #

! (2)

! ! = g, f # #h (2) , f, g(2) = g #, f # (2) .

(15)

This allows a series of extensions by duality. By capital letters we denote distributions. We are going to skip the easy justifications and refer to [31] for an abstract approach. First one extends #

S (2) × S(2) −→ S (2) , #

S(2) × S (2) −→ S (2) ,

F #g, h(2) := F, h#g #

! (2)

,

! g#F, h(2) := F, g # #h (2) .

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Definition 5 The Moyal algebra is , M(2) := F ∈ S (2) | F #S(2) ⊂ S(2) , S(2)#F ⊂ S(2) . It is clear that this is a unital ∗ -algebra with F #G, h(2) := F, h#G#

! (2)

F #, h(2) := h#, F

and

! (2)

,

∀ h ∈ S(2) ;

the unit is the constant function 1. Actually, by construction, it is the largest ∗ algebra in which S(2) is an essential bi-sided self-adjoint ideal. Anyhow, one has M(2)#S(2)#M(2) ⊂ S(2) . We do not intend to discuss its natural topological structure. There is also an obvious way to get extensions #

S (2) × M(2) −→ S (2) ,

#

M(2) × S (2) −→ S (2) .

(16)

By inspecting the definitions one realizes that Proposition 3 The pseudo-differential calculus extends to an isomorphism Op : M(2) → L[S(g)] ∩ L[S (g)] . Finally, let us introduce a symbol version of the C ∗ -algebra of all the bounded linear operators in H = L2 (g) , by pulling back structure through Op . Definition 6 The symbol C ∗ -algebra is A(2) := {F ∈ S (2) | Op(F ) ∈ B(H)} ,

F A(2) := Op(F ) B(H) .

Remark 6 Obviously S(2) ⊂ L2 (2) ⊂ A(2) ⊂ S (2) , and all the inclusions are strict. The exponential functions {εZ | Z ∈ 2} and the constant functions are all in [A(2) ∩ M(2)]\L2 (2) . There is no inclusion between A(2) and M(2) . Since by quantizing symbols only depending on X ∈ g one gets ∞ (g) ⊂ M(2) . multiplication operators, it is clear that L∞ (g) ⊂ A(2) and Cpol Thinking of symbols only depending on ξ ∈ g1 , yielding convolution operators by the inverse Fourier transform of the symbol, one gets other results. Since both A(2) and M(2) are ∗ -algebras, one can generate new examples by performing #-products.

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5 Phase-Space Shifts We introduce at the symbol level the analog of Definition 3. # . Definition 7 For every Y, Z ∈ 2 and f ∈ S (2) we set θY,Z (f ) := εY #f #εZ

In fact, having in view the properties of the functions εZ , we see that the mapping θY,Z acts in any of the spaces S(2), L2 (2), A(2), M(2), S (2) . We also refer to Remark 2. Proposition 4 One has   $Y,Z [Op(f )] = Op θY,Z (f ) . The C ∗ -algebra A(2) is invariant under all the mappings θY,Z . Proof We have # $Y,Z [Op(f )] = E(Y)Op(f )E(Z)∗ = Op(εY )Op(f )Op(εZ )     # = Op εY #f #εZ = Op θY,Z (f ) .

Invariance follows from this, since at an operator level A(2) is B(H) , which is left invariant by multiplying to the left and to the right with elements of the Weyl system.   We are going to need below the following result: Lemma 6 For every u, v ∈ L2 (2) and Y, Z ∈ 2 one has   θY,Z Eu,v = EE(Y)u,E(Z)v .

(17)

Proof One can write         Op θY,Z Eu,v = $Y,Z Op Eu,v = E(Y) ·, vu E(Z)∗   = ·, E(Z)vE(Y)u = Op EE(Y)u,E(Z)v ,  

which implies (17).

Remark 7 The explicit form of θY,Z (f ) is less important than the way it has been constructed, and will not be used here. However, for convenience, we are going to record the diagonal case θZ,Z ≡ θZ (forming a family of automorphisms). One of the reasons is that it leads to the covariant symbol of the operators Op(f ) ; see [30]. By a direct computation one gets   θ(Z,ζ ) (f ) (X, ξ ) =

  g g1

  O 1 (η) dY dη eiX•[−Y ]|η−ξ  eiX−Y |ζ  f [−Z]•X, Ad −Z (18)

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in terms of the coadjoint action (3). If G = Rn the coadjoint action is trivial and X•[−Y ] = X − Y , so one gets   θ(Z,ζ ) (f ) (X, ξ ) = f (X − Z, ξ − ζ ) , implying that the phase-space translations of the symbols are implemented, at the level of the quantization, by conjugations with the Weyl system. , Remark 8 One can use the automorphism family θZ | Z ∈ 2 to define a sophisticated form of convolution, that we intend to use in a future publication. For suitable functions ϕ, f on 2 we write  ϕ ‡θ f = 2

 ϕ(Z)θZ (f ) dZ =

2

# ϕ(Z) εZ #f #εZ dZ .

(19)

For G = Rn this boils down to the usual additive convolution, since θZ reduces to a translation. One gets easily an explicit formula:   ϕ ‡θ f (X) =

 2

φX (Z)f (Z)dZ ,

where   φ(X,ξ ) (Z, ζ ) :=

g g1

e−iX•[−N ]•Z•[−X]|ξ  eiX−X•[−Z]•N |μ eiZ•[−N ]|ζ  ϕ(X•[−Z], μ)dNdμ .

It is easy to verify that, for G = Rn , one gets φ(X,ξ ) (Z, ζ ) = ϕ(X − Z, ξ − ζ ) . As with the usual convolution, setting   ϕt (Z) := t −2n ϕ t −1 Z ,

t > 0 , Z ∈ 2 , ϕ ∈ S(2).

one gets ϕt ‡θ f −→ f pointwise if f is bounded and continuous. t→0

6 Coorbit Spaces—A Short Overview Let us pick a normalized “window” (or “atom”) w belonging to the Fréchet space S(g) → L2 (g) . In terms of the Fourier-Wigner transform, the linear mapping Ew : S (g) → S (g × g1 ) ,

Ew (u) := Eu,w

will be used to pull back algebraic and topological structures. It is isometric from L2 (g) to L2 (g × g1 ) and this has standard consequences (inversion and reproduction

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formulae). Let us record the explicit form of its adjoint  E†w (h) =

h(X)E(X)∗ w dX.

(20)

2

Definition 8 Let (B, · B ) be a normed space continuously embedded in S (g × g1 ) . Its coorbit space (associated to the window w) is , cow (B) := u ∈ S (g) | Ew (u) ∈ B

(21)

with the norm u cow (B) := Ew (u) B . If B is just a vector subspace, we still use (21) to define a subspace of S (g) . The case of a locally convex space B is also important. Remark 9 Developing the abstact part of the theory of the spaces cow (B) is quite standard, relying on the good properties of the isometry Ew , and it will not be done here. Let us just state that if B → S (g × g1 ) is Banach, then cow (B) is a Banach space continuously embedded in S (g) . Simple arguments based on the inversion formula and the mapping properties of E†w show that   cow L2 (g × g1 ) = H ,

  cow S(g × g1 ) = S(g) ,

  cow S (g × g1 ) = S (g) .

Weighted mixed Lpq -spaces of functions defined on g × g1 are nice examples of spaces B to start with. For the case of the Heisenberg group, a comparison with [13] would be interesting. To define coorbit spaces of functions on 2 = g × g1 , that could play the role of symbols of the Op-calculus, one needs a good map (at least an isometry) transforming functions/distributions on 2 into functions/distributions on 2 × 2. One solution is to proceed by analogy, defining in 2 × 2 ∼ = (g × g) × g1 × g1 a Weyl system and a Fourier-Wigner transform, doubling the number of variables. We did not check, but a starting point could be the Weyl system       E (Y, η), (Z, ζ ) h (X, ξ )) := eiZ,ξ  e−iX,ζ  h [−Y ]•X, ξ − η , attaching unitary operators in L2 (2) to points in 2 × 2 . The non-commutative group structure of 2 = g × g1 has been taken into account. It seems that nobody has done this for the case of a nilpotent Lie algebra g , but is is likely that this can be done, and connecting the coorbit spaces on g and on 2 , respectively, via the pseudo-differential calculus, would be successful. We will sketch a different approach, relying on the previously defined phase-space shifts and having some connections with ideas from [29]. Definition 9 Let h ∈ S(2) \ {0} (most often h (2) = 1) . One defines for f ∈ S (2) and Y, Z ∈ 2  ! !  # , h (2) . Eh (f ) (Y, Z) := θY,Z (f ), h (2) = εY #f #εZ

(22)

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Note that, by (16), one has # ∈ M(2)#S (2)#M(2) ⊂ S (2) , εY #f #εZ

so by our choice h ∈ S(2) the expression (22) makes sense. If f ∈ L2 (2) , one has # ∈ A(2)#L2 (2)#A(2) ⊂ L2 (2) εY #f #εZ

(L2 (2) is an ideal in A(2) , since it corresponds to Hilbert-Schmidt operators) and an L2 -window is directly available. Other situations can be accommodated. Proposition 5 For every h, f ∈ L2 (2) one has 1 1 1 Eh (f ) 1

(2×2)

= h (2) f (2) .

(23)

, Proof The family Eu,v | u, v ∈ L2 (G) is total in L2 (2) , since by the Opquantization it yields  all the  rank one operators (see Proposition 1, (iii)), forming a total family in B2 L2 (g) . One has  !  # EEu,v(Eu ,v  ) (X, Y) = εX #Eu ,v  #εY , Eu,v (2) (17)

= EE(X)u ,E(Y)v  , Eu,v

! (2)

(7)

= E(X)u , u(g) v, E(Y)v  (g)

= Eu ,u (X)Ev  ,v (Y) , meaning that EEu,v(Eu ,v  ) = Eu ,u ⊗ Ev  ,v . Then using once again the orthogonal relations (7) leads easily to the result (work with scalar products and then take diagonal values to get (23)).   Definition 10 Let (B, · B ) be a normed space continuously embedded in S (2 × 2) . Its coorbit space associated to the normalized window h ∈ S(2) is , Coh (B) := f ∈ S (2) | Eh (f ) ∈ B

(24)

with the norm u Coh (B) := Eh (f ) B . If B is just a subspace, we still define the vector space (24), but without norm. Recall the algebraic rules of (suitable) integral kernels K, L : 2 × 2 → C :  (K ◦ L)(Y, Z) :=

K(Y, X)L(X, Z)dX ,

K ◦ (Y, Z) := K(Z, Y) .

2

Of course, they are ment to emulate the multiplication and the adjoint of integral operators.

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Proposition 6 If h, k ∈ S(2) , then for every f, g ∈ L2 (2) we have Eh#k (f #g) = Eh (f ) ◦ Ek (g)

and

  Eh# f # = Eh (f )◦ .

Proof One uses the rules (15) of a Hilbert algebra and the relation (6), getting   Eh (f ) ◦ Ek (g) (Y, Z) =

 Eh (f )(Y, X) Ek (g)(X, Z) dX 2

 =

2

 =

2

! ! # # εY #f #εX , h (2) εX #g#εZ , k (2) dX ! ! h# #εY #f, εX (2) εX , k#εZ #g # (2) dX

! ! = h# #εY #f, k#εZ #g # (2) = h#h#εY #f #g, εZ (2) ! # = εY #f #g#εZ , h#k (2) = Eh#k (f #g)(Y, Z) . and ! # ,h Eh (f )◦ (Y, Z) = Eh (f )(Z, Y) = εZ #f #εY (2) ! ! # # # # = h, εZ #f #εY (2) = f #εZ , εY #h# (2) !   # = εY #f # #εZ , h# (2) = Eh# f # (Y, Z) .    ◦ Corollary 2 If the vector space B is an involutive algebra with respect to ◦,   and h = h#h = h# , then Coh (B) is an involutive algebra with respect to #,# . Proof Suppose that f, g ∈ Coh (B) , which means that Eh (f ), Eh (g) ∈ B . Then Eh (f #g) = Eh#h (f #g) = Eh (f ) ◦ Eh (g) ∈ B , implying that f #g ∈ Coh (B) . Invariance under the involution similarly, using the self-adjointness of the window h .

#

is checked     ∗ ◦ Remark 10 In the framework of the Corollary, if · B is a C -norm on B, ◦, ,   then · Coh (B) is a C ∗ -norm on Coh (B), #,# :

1 1 1 1 1 12 g # #g Coh (B) = 1 Eh (g # #g)1B = 1 Eh (g)◦◦Eh (g)1B = 1Eh (g)1B = g 2Coh (B) . We describe now an abstract situation in which pseudo-differential operators with symbols in a coorbit space (of symbols) are well-defined and bounded between two coorbit spaces of vectors. Note that this requires a correlation of the windows; the Wigner transform W has been introduced in (11).

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Theorem 1 Let w1 , w2 ∈ S(g) with w1 (g) = w2 (g) = 1 . Let · B1 and · B2 two norms on S(2) and · B a norm on S(2 × 2) . Suppose that for every    ∈ S(2 ×2) the integral operator Int() is bounded from S(2), · B1 to S(2), · B2 , with operatorial norm less or equal than C  B for some positive absolute constant C. Then, operator Op(f ) is bounded  for every f ∈S(2)  , the pseudo-differential  from S(g), · cow1(B1 ) to S(g), · cow2(B2 ) , with operatorial norm less or equal than C f CoWw ,w (B) . 1

2

Proof We first show that for w1 , w2 ∈ S(g) and f ∈ S(2) , in terms of the Wigner transform (11), one has   Ew2 Op(f )E†w1 = Int EWw1 ,w2 (f ) .

(25)

For this we compute 0 /   Ew2 Op(f )E†w1 g (X) = E(X)Op(f )E†w1 g, w2  0 / (20) = E(X)Op(f ) g(Y)E(Y)∗ dY w1 , w2 2

 = 2

(11)

 !  # w1 , w2 dY g(Y) Op εX #f #εY



=

2

! # εX #f #εY , Ww1 ,w2 (2) g(Y) dY

    = Int EWw1 ,w2(f ) g (X) . Then the norm estimate is easy: for every u ∈ S(g) 1  1  Op(f )u cow2(B2 ) = Ew2 Op(f )u B2 = 1 Int EWw1 ,w2(f ) Ew1 (u) 1B 1 1  1 1 1 ≤ Int EWw1 ,w2(f ) B(B ,B ) Ew1 (u) B1 1 2 1 1 ≤ C 1 EWw1 ,w2(f ) 1B Ew1 (u) B1 1 1 = C 1 f 1Co u cow1(B1 ) . (B) Ww1 ,w2

  Remark 11 Theorem 1 provides a boundedness result for pseudo-differential operators involving coorbit norms both at the level of vectors and at the level of symbols. However, in the statement and the proof of this Theorem, the initial or the induced norms are only defined on (various) Schwartz spaces and the action of the operators are also confined to such spaces. Of course, automatically, there are bounded extensions to the corresponding completions. But, for a really nice result, some technical issues still have to be solved. For example, if B1 denotes

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  the completion of S(2), · B1 , is it true that the (very relevant) completion  of S(g), · cow1(B1 ) may be identified with the coorbit space cow1(B1 ) ? There   is a similar question starting with the completion B of S(2 × 2), · B . In addition, one would like to treat boundedness for coorbit spaces associated to spaces of temperate distributions in which the Schwartz space is not dense. Besides this, many other topics deserves attention, as duality, interpolation, equivalent norms, dependence of windows, decompositions, Schatten-von Neumann behavior, etc. They will be treated systematically in a subsequent publication. Remark 12 Another reason to invest effort in a future article is concreteness. Besides abstract results, valid for general choices, many interesting facts will only occur in particular situations. Even if g = Rn (Abelian), most of the previous work has been dedicated to weighted modulation spaces, having mixed Lp,q -spaces as a starting point. In addition, an important and difficult issue is to compare the modulation (or the coorbit) spaces with other function spaces, defined by different techniques. Frames should also be studied. Hopefully, more specific Lie algebra features will appear at a certain moment. Acknowledgements The author has been supported by the Fondecyt Project 1160359. He is grateful for having the opportunity to participate in the Conference MicroLocal and Time Frequency Analysis 2018 in honor of Luigi Rodino on the occasion of his 70th Birthday.

References 1. L. J. Corwin and F. P. Greenleaf: Representations of Nilpotent Lie Groups and Applications, Cambridge Univ. Press, 1990. 2. M. Christ, D. Geller, P. Glowacki, D. Polin, Pseudodifferential Operators on Groups with Dilations, Duke Math. J., 68, 31–65, (1992). 3. E. Cordero, K. Gröchenig, F. Nicola and L. Rodino: Wiener Algebras of Fourier Integral Operators, J. Math. Pures Appl., 99, 219–233, (2013). 4. E. Cordero, F. Nicola and L. Rodino: Time-Frequency Analysis of Fourier Integral Operators, Comm. Pure Appl. Anal., 9(1), 1–21, (2010). 5. E. Cordero and L. Rodino: Time-Frequency Analysis: Function Spaces and Applications, Note di Matematica, 31, 173–189, (2011). 6. E. Cordero, J. Toft and P. Wahlberg: Sharp Results for the Weyl Product on Modulation Spaces, J. Funct. Anal., 267(8), 3016–3057, (2014). 7. A. S. Dynin: An Algebra of Pseudodifferential Operators on the Heisenberg Group: Symbolic Calculus, Dokl. Akad. Nauk SSSR, 227, 508–512, (1976). 8. H. G. Feichtinger: On a New Segal Algebra, Monatsh. Mat. 92(4), 269–289, (1981). 9. H. G. Feichtinger: Modulation Spaces on Locally Compact Abelian Groups, Proc. of “International Conference on Wavelets and Applications”, Chenai, India, 99–140, 1983. 10. H. G. Feichtinger and K. Gröchenig: Banch Spaces Associated to Integrable Group Representations and Their Atomic Decompositions I, J. Funct. Anal. 86, 307–340, (1989). 11. V. Fischer and M. Ruzhansky: A Pseudo-differential Calculus on Graded Nilpotent Groups, in Fourier Analysis, pp. 107–132, Trends in Mathematics, Birkhäuser, 2014. 12. V. Fischer and M. Ruzhansky: Quantization on Nilpotent Lie Groups, Progress in Mathematics, 314, Birkhäuser, 2016.

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13. V. Fischer, D. Rottensteiner and M. Ruzhansky: Heisenberg-Modulation Spaces at the Crossroads of Coorbit Theory and Decomposition SpaceTheory, Preprint ArXiV. 14. P. Głowacki: A Symbolic Calculus and L2 -Boundedness on Nilpotent Lie Groups, J. Funct. Anal. 206, 233–251, (2004). 15. P. Głowacki: The Melin Calculus for General Homogeneous Groups, Ark. Mat., 45(1), 31–48, (2007). 16. P. Głowacki: Invertibility of Convolution Operators on Homogeneous Groups, Rev. Mat. Iberoam. 28(1), 141–156, (2012). 17. K. Gröchenig: Foundations of Time-Frequency Analysis, Birkhäuser Boston Inc., Boston, MA, 2001. 18. K. Gröchenig: Time-Frequency Analysis of Sjöstrand Class, Revista Mat. Iberoam. 22(2), 703– 724, (2006). 19. K. Gröchenig: Composition and Spectral Invariance of Pseudodifferential Operators on Modulation Spaces, J. Anal. Math. 98, 65–82, (2006). 20. K. Gröchenig: A Pedestrian Approach to Pseudodifferential Operators, In: C. Heil editor, Harmonic Analysis and Applications, Birkhäuser, Boston, 2006. 21. K. Gröchenig and C. Heil: Modulation Spaces and Pseudodifferential Operators, Integral Equations Operator Theory, 34, 439–457, (1999). 22. K. Gröchenig and Z. Rzeszotnik: Banach Algebras of Pseudodifferential Operators and Their Almost Diagonalization, Ann. Inst. Fourier. 58(6), 2279–2314, (2008). 23. K. Gröchenig and T. Strohmer: Pseudodifferential Operators on Locally Compact Abelian Groups and Sjöstrand’s Symbol Class, J. Reine Angew. Math. 613, 121–146, (2007). 24. A. Holst, J. Toft and P. Wahlberg: Weyl Product Algebras and Modulation Spaces, J. Funct. Anal. 251, 463–491, (2007). 25. R. Howe: Quantum Mechanics and Partial Differential Operators, J. Funct. Anal. 38, 188–254, (1980). 26. R. Howe: The Role of the Heisenberg Group in Harmonic Analysis, Bull. Amer. Math. Soc. 3(2), 821–843, (1980). 27. D. Manchon: Formule de Weyl pour les groupes de Lie nilpotente, J. Reine Angew. Mat. 418, 77–129, (1991). 28. D. Manchon: Calcul symbolyque sur les groupes de Lie nilpotentes et applications, J. Funct. Anal. 102 (2), 206–251, (1991). 29. M. M˘antoiu: Coorbit Spaces of Symbols for Square Integrable Families of Operators, Math. Reports, 18(1), 63–83 (2016). 30. M. M˘antoiu: Berezin-Type Operators on the Cotangent Bundle of a Nilpotent Group, submitted. 31. M. M˘antoiu and R. Purice: On Fréchet-Hilbert Algebras, Archiv der Math. 103(2), 157–166, (2014). 32. M. M˘antoiu and M. Ruzhansky: Pseudo-differential Operators, Wigner Transform and Weyl Systems on Type I Locally Compact Groups, Doc. Math., 22 , 1539–1592, (2017). 33. M. M˘antoiu and M. Ruzhansky: Quantizations on Nilpotent Lie Groups and Algebras Having Flat Coadjoint Orbits, J. Geometric Analysis, (2019). 34. A. Melin: Parametrix Constructions for Right Invariant Differential Operators on Nilpotent Groups, Ann. Global Anal. Geom. 1(1), 79–130, (1983). 35. K. Miller: Invariant Pseudodifferential Operators on Two Step Nilpotent Lie Groups, Michigan Math. J. 29, 315–328, (1982). 36. K. Miller: Inverses and Parametrices for Right-Invariant Pseudodifferential Operators on TwoStep Nilpotent Lie Groups, Trans. of the AMS, 280 (2), 721–736, (1983). 37. M. Ruzhansky and V. Turunen: Pseudodifferential Operators and Symmetries, PseudoDifferential Operators: Theory and Applications 2, Birkhäuser Verlag, 2010. 38. J. Sjöstrand: An Algebra of Pseudodifferential Operators, Math. Res. Lett. 1(2), 185–192, (1994).

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39. J. Toft: Continuity Properties for Modulation Spaces, with Applications to Pseudo-differential Calculus. I, J. Funct. Anal. 207(2), 399–426, (2004). 40. J. Toft: Continuity Properties for Modulation Spaces, with Applications to Pseudo-differential Calculus. II, Annals of Global Analysis and Geometry, 26, 73–106, (2004).

On the Measurability of Stochastic Fourier Integral Operators Michael Oberguggenberger and Martin Schwarz

Abstract This work deals with the measurability of Fourier integral operators (FIOs) with random phase and amplitude functions. The key ingredient is the proof that FIOs depend continuously on their phase and amplitude functions, taken from suitable classes. The results will be applied to the solution FIO of the transport equation with spatially random transport speed as well as to FIOs describing waves in random media. Keywords Fourier integral operators · Stochastic Fourier integral operators · Hyperbolic differential equations with random field coefficients

1 Introduction In the theory of hyperbolic partial differential equations (PDEs), Fourier integral operators (FIOs) have become an important tool to examine certain properties of the solution, e.g., the propagation of singularities [1, 6, 12]. When studying waves in random media, the coefficients of the underlying PDEs are random fields. This has become important in seismology [3, 7] and in material science [5, 9]. As the phase and amplitude functions of the FIOs producing a solution or a parametrix are functions of the coefficients of the underlying PDE, one has to ensure that the FIO, respectively its action, stays measurable. The question of measurability of a FIO arises also in its own right, when a deterministic phase or amplitude function is subjected to a stochastic perturbation [8]. This work is dedicated to providing a rigorous proof of various continuity and measurability properties.

M. Oberguggenberger () · M. Schwarz Unit of Engineering Mathematics, University of Innsbruck, Innsbruck, Austria e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_21

383

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Consider a FIO of the form 1 A,a [u](x) = (2π )n

 ei(x,y,ξ ) a(x, y, ξ )u(y) dy dξ .

The first task will be to show that, for ψ ∈ D(Y ), respectively u ∈ E (Y ), the maps (, a) → A,a [ψ],

(, a) → A,a [u]

(1)

are continuous with values in C∞ (X), respectively D (X), on suitable spaces of phase and amplitude functions. (Here X and Y are open subsets of RnX and RnY .) Equipping these spaces with their Borel σ -algebra, we consider random functions ω → (ω (x, y, ξ ), aω (x, y, ξ )) on a probability space (, F, P). The second task will be to infer that the maps ω → Aω ,aω [ψ],

ω → Aω ,aω [u]

are measurable as well. The applicability of these results will be demonstrated in three examples: the transport equation and the half wave equation, both with a spatially random propagation speed, and a random perturbation of the solution operator to the wave equation. The plan of the paper is as follows. We start by recalling required facts from the classical theory of Fourier integral operators. In the subsequent section, we prove the continuity and measurability results. The final section addresses the announced applications. The paper is part of a larger program aiming at studying wave propagation in random media by means of stochastic Fourier integral operators [8, 10]; it provides the probabilistic basis for this program.

2 Classical Theory of Oscillatory Integrals and FIOs 2.1 Oscillatory Integrals Let nY , n2 ∈ N and let Y ⊂ RnY be an open set. The subsequent short exposition follows [11]. Let   I (au) = ei(y,ξ ) a(y, ξ )u(y) dy dξ . (2) Rn2

Y

n2 Here u ∈ D(Y ) is a smooth function

with compact support and  : Y × R is a

phase function, which means that  Y ×(Rn2 \{0}) is smooth, real valued and positively

Stochastic Fourier Integral Operator

385

homogeneous of degree 1 in ξ . Furthermore,  does not have any critical points in Rn2 \{0}, i.e., for all y ∈ Y and ξ ∈ Rn2 \{0} [∂y1 (y, ξ ), . . . , ∂ynY (y, ξ ), ∂ξ1 (y, ξ ), . . . , ∂ξn2 (y, ξ )]T = 0. d (Y × Rn2 ), d ∈ R, The function a : Y × Rn2 is a Hörmander symbol of class S-,δ 0 ≤ δ < 1 and 0 < - ≤ 1. That means, it is smooth and for any given multi-indices k ∈ NnY , l ∈ Nn2 and any compact K ⊂ Y there exists a constant Cl,k,K such that





l k

∂ξ ∂y a(y, ξ ) ≤ Cl,k,K ξ d−-|l|+δ|k| , where y ∈ K and ξ ∈ Rn2 . As usual, we write ξ  = (1 + ξ 2 )1/2 . In general, (ei(y,ξ ) a(y, ξ )u(y)) is not absolutely integrable; the oscillatory integral (2) has to be regularized. We recall the usual procedure, as presented, e.g., in [11]. Let χ ∈ D(Rn2 ) with χ (ξ ) ≡ 1 for ξ < 1 and χ (ξ ) ≡ 0 for ξ > 2. Furthermore, let r(ξ , y) =

 n2

ny

2





∂y (y, ξ ) 2 , ξ 2 ∂ξl (y, ξ ) + k

l=1

(3)

k=1

and αl =

−i (1 − χ ) ξ 2 (∂ξl ), r

βk =

−i (1 − χ )(∂yk ), r

γ = χ.

(4)

Let L be the differential operator Lf = −

n2

∂ξl (α l f ) −

l=1

nY

∂yk (β k f ) + γ f.

k=1

Then, the formal adjoint operator tL t

L=

 n2

α l ∂ξl +

l=1

nY

β k ∂y k + γ ,

k=1

satisfies t

Lei = ei .

0 (Y × Rn2 ), β ∈ S −1 (Y × Rn2 ) and γ ∈ S −1 (Y × Rn2 ). Furthermore, α l ∈ S1,0 1,0 1,0 k

(5)

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Finally, one can choose a κ ∈ N large enough and iteratively apply L to (2), and get a convergent integral by 

 I (au) =

Rn2

ei(y,ξ ) Lκ (a(y, ξ )u(y)) dy dξ . Y

2.2 Classical Theory of FIOs Let nX , nY , n2 ∈ N and X resp. Y be an open subset of RnX resp. RnY . A Fourier integral operator is an operator of the form  A,a [u](x) =



Rn2

ei(x,y,ξ ) a(x, y, ξ )u(y) dy d−ξ ,

(6)

Y

where d−ξ = (2π )−n dξ . Let  : X × Y × Rn2 → R be a phase function on d (X × Y × Rn2 ) an amplitude function. Furthermore, the X × Y × Rn2 and a ∈ S-,δ following conditions for ξ = 0 are assumed: [∂x1 (x, y, ξ ), . . . , ∂xnX (x, y, ξ ), ∂ξ1 (x, y, ξ ), . . . , ∂ξn2 (x, y, ξ )]T = 0, (7) [∂y1 (x, y, ξ ), . . . , ∂ynY (x, y, ξ ), ∂ξ1 (x, y, ξ ), . . . , ∂ξn2 (x, y, ξ )]T = 0, (8) The function  is then called an operator phase function. By classical theory, if condition (8) is satisfied, operator (6) continuously maps D(Y ) into C∞ (X). Under condition (7) operator (6) can be extended to a continuous map from E (Y ) into D (X) by / 0 A[u], φ = u, tA[φ] , where 



ei(x,y,ξ ) a(x, y, ξ )φ(x) dx d−ξ .

t

A,a [φ](y) =

R n2

X

3 Stochastic Fourier Integral Operators The set of operator phase functions and the space of amplitudes are equipped with a natural metrizable topology, which we now describe. The first task in this section will be to prove the continuity of the maps (1). In order to derive the required

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387

inequalities, it will be necessary to replace the conditions on nondegeneracy (7), (8) by strict bounds away from zero. It turns out that this is not a serious restriction in the applications. The resulting space is a closed subset of the space of operator phase functions. To avoid technical difficulties we hereafter exclude 0 from Rn2 and set 2 = n 2 R \ {0}. The sets X and Y are open subsets of RnX and RnY , respectively. If Z is an open subset of Rd , d ∈ {nX , nY }, we define a compact exhaustion KZ,m of Z by KZ,m = {x ∈ Z, x ≤ m, dist(x, Rd \Z) ≥ 1/m}. Definition 1 The space of positively homogeneous functions of degree one is defined by  Mhg (X, Y, 2) =  : X × Y × 2 → R,  smooth and  positively homogeneous of degree 1w.r.t. ξ . Let α > 0. The subspace Mα (X, Y, 2) is given by  Mα (X, Y, 2) =  ∈ Mhg (X, Y, 2) such that ∀x ∈ X, y ∈ Y, ξ ∈ 2 : 12 12 1 1  1 1 1 1 1 1 1 ξ −1 ∇y 1 ξ −1 ∇x (x, y, ξ ) (x, y, ξ ) ≥ α, ≥ α . 1 1 1 1 1 1 1 1 ∇ξ ∇ξ

Let j ∈ NnX , k ∈ NnY , l ∈ Nn2 be multi-indices and let (pm )m∈N be the sequence of seminorms with 



j

pm () := sup ξ l−1 ∂x ∂yk ∂ξl  (x, y, ξ ) , x ∈ KX,m , y ∈ KY,m ,  ξ ∈ 2, |j | + |k| + |l| ≤ m .

(9)

The topology of Mhg (X, Y, 2) is defined by the seminorms (pm )m∈N . Remark 1 Since  is positively homogeneous of degree one with respect to ξ , an equivalent sequence of seminorms is given by 



j p 3m () = sup ∂x ∂yk ∂ηl  (x, y, η)

η=ξ / ξ

, x ∈ KX,m , y ∈ KY,m ,  ξ ∈ 2, |j | + |k| + |l| ≤ m . (10)

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Remark 2 Note that any function in Mα automatically satisfies (7) and (8). However, there are functions satisfying (7) and (8), but there is no α > 0 such that they are in Mα . For X = Y = R consider (x, y, ξ ) = (e−x − y) ξ. 2

Then ∂ξ (x, y, ξ ) = e−x and (|ξ |−1 ∂x (x, y, ξ )) = −2sign (ξ )xe−x , which can be arbitrarily small for |x| large. 2

2

Now let ∂B = {ξ ∈ 2, ξ = 1} be the unit sphere in Rn2 . The space C∞ (X × Y × ∂B) with its usual topology T is a separable Fréchet space and thus metrizable and complete [2, Chapter XVII, Section 2]. We have the following result   1 The space Mhg (X, Y, 2), (pm )m∈N is isomorphic to the space Proposition  C∞ (X × Y × ∂B), T and hence separable, metrizable, and complete. Proof The isomorphism is explicitly given by I : Mhg (X, Y, 2) → C∞ (X × Y × ∂B)     ξ . (x, y, ξ ) → f (x, y, ξ ) → (x, y, ξ ) → f x, y, ξ The bicontinuity of I can be most easily seen by employing local spherical coordinates on ∂B.   Proposition 2 Let α > 0, then Mα (X, Y, 2) is a closed subset of Mhg (X, Y, 2). Proof We show that 1 12 1 ξ −1 ∇x 1 1 (·)1 1 1 ∇ξ continuously maps Mα (X, Y, 2) to R: Fix (x, y, ξ ) ∈ (X × Y × 2), and choose m ≥ 1 large enough, such that x ∈ KX,m and y ∈ KY,m . Then, by (9)





ξ −1 ∂xj (x, y, ξ ) ≤ pm () and



∂ξ (x, y, ξ ) ≤ pm (). l

So in total 12 1 1 1 ξ −1 ∇x 1 ≤ (n2 + nX )(pm ())2 . 1 (x, y, ξ ) 1 1 ∇ξ

Stochastic Fourier Integral Operator

389

Since the 2-norm is continuous as well, the limit of a convergent sequence (n )n∈N can be pulled out 1 1 12 12 1 ξ −1 ∇x 1 ξ −1 ∇x 1 1 1 1 1 lim n (x, y, ξ )1 = lim 1 n (x, y, ξ )1 1 1 ≥ α, n→∞ n→∞ ∇ξ ∇ξ and the continuity is shown. Since the preimage of a closed set is closed, "

2 1 12 1 ξ −1 ∇x 1 1 1  ∈ Mhg (X, Y, 2) : 1 (x, y, ξ )1 ≥ α ∇ξ

1 12 1 ξ −1 ∇y 1 1 is closed. The proof works analogously for 1 (·)1 1 , and the intersection ∇ξ of closed sets is closed again.   Definition 2 The space of amplitude functions on X × Y × 2 is defined by 

 d Sd-,δ (X, Y, 2) = a X×Y ×2 , a ∈ S-,δ (X × Y × Rn2 ) . This space is equipped with the seminorms





j qm (a) = sup ξ −d+-|l|−δ(|j |+|k|) ∂x ∂yk ∂ξl a(x, y, ξ ) , where x ∈ KX,m , y ∈ KY,m , ξ ∈ 2, |j | + |k| + |l| ≤ m .

Note that by [4, Chapter VII, Section 7.8] the space (Sd-,δ , (qm )m∈N ) forms a Fréchet space, and thus it is complete and metrizable. Furthermore, one can check d (X × Y × Rn2 ). that it is separable. Actually, Sd-,δ (X, Y, 2) is isomorphic with S-,δ In any case, both the phase function space and the amplitude function space are closed subsets of separable, metrizable, complete spaces. For any further consideration we will deal with Fdα,-,δ (X, Y, 2) = Mα (X, Y, 2) × Sd-,δ (X, Y, 2), which is equipped with the product topology of the two spaces, induced by the seminorms pm (, a) := pm () + qm (a), m ∈ N. We call Fdα,-,δ (X, Y, 2) the space of FIO operator functions. Lemma 1 Let  ∈ Mα (X, Y, 2) and let α i resp. β i be the coefficients of the regularizing operator L (see (4)). Then for m ∈ N there exists a polynomial Pm such that for any x ∈ KX,m , y ∈ KY,m and ξ ∈ 2



j k l

∂x ∂y ∂ξ α i (x, y, ξ ) ≤ Pm (pm ()) ξ −|l|

(11)

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for i ∈ {1, . . . , n2 } and



j k l

∂x ∂y ∂ξ β i (x, y, ξ ) ≤ Pm (pm ()) ξ −1−|l|

(12)

for i ∈ {1, . . . , nY } and all |j | + |k| + |l| + 1 ≤ m. Proof For this proof we will show the inequalities only for the zeroth and first derivative of (11). Any higher derivative can be treated the same way. Recall the notation from Equation (3): 1 12 1 12 r(x, y, ξ ) = 1 ξ ∇ξ (x, y, ξ )1 + 1∇y (x, y, ξ )1 . Since  ∈ Mα (X, Y, 2) one has that α ξ 2 ≤ r(x, y, ξ ), and therefore,



2





α (x, y, ξ ) =

(1 − χ (ξ )) ξ ∂ξi (x, y, ξ )

i



r(x, y, ξ )



1 − χ (ξ )

≤

∂ξi (x, y, ξ )

α 0 ≤ Cm pm ()=: P0m (pm ()).

We note that for all multi-indices j , k and l with |j | + |k| + |l| + 1 ≤ m there exists a constant Km such that





j k l

∂x ∂y ∂ξ r(x, y, ξ ) ≤ Km ξ 2−|l| (pm ())2 ≤ Km ξ 2−|l| (pm ())2 , for ξ ∈ 2, ξ ≥ 1. Therefore,



∂y α (x, y, ξ )

k i



(1 − χ(ξ )) ξ 2 ∂ (x, y, ξ )

ξi



= ∂yk



r(x, y, ξ )



ξ 2 ∂ξi (x, y, ξ )



≤ (1 − χ(ξ ))∂yk



r(x, y, ξ )



ξ 2 ∂ξi ,yk (x, y, ξ )r(x, y, ξ ) − ξ 2 ∂ξi (x, y, ξ )∂yk r(x, y, ξ )



≤ (1 − χ(ξ ))



r 2 (x, y, ξ )

Stochastic Fourier Integral Operator

391







∂ξi ,yk (x, y, ξ )r(x, y, ξ )



∂ξi (x, y, ξ )∂yk r(x, y, ξ )



≤ (1 − χ(ξ ))

+ (1 − χ(ξ ))

α 2 ξ 2 α 2 ξ 2 1 (pm ())3 =: P1m (pm ()) ≤ Cm

where the last inequality is due to the fact that ∂ξi ∂yk (x, y, ξ ) and ∂ξi (x, y, ξ ) are bounded by a constant times pm () applying (9). Since (1 − χ (ξ )) is nonzero only for ξ > 1, one has no difficulties in the neighborhood of 0. Derivation with respect to ξl yields







(1 − χ (ξ )) ξ 2 ∂ξi (x, y, ξ )

∂ξ α (x, y, ξ ) = ∂ξ

, l i

l

r(x, y, ξ ) which is less or equal to







ξ 2 ∂ξi (x, y, ξ )



2ξl ∂ξi (x, y, ξ )



≤ ∂ξl χ (ξ )

+ (1 − χ (ξ ))



r(x, y, ξ ) r(x, y, ξ )







ξ 2 ∂ξi (x, y, ξ )∂ξl r(x, y, ξ )

ξ 2 ∂ξi ,ξl (x, y, ξ )





+ (1 − χ (ξ ))

+ (1 − χ (ξ ))





r(x, y, ξ ) r 2 (x, y, ξ )







ξ 2 ∂ξi (x, y, ξ )



2ξl ∂ξi (x, y, ξ )



≤ ∂ξl χ (ξ )

+ (1 − χ (ξ ))



α ξ 2 α ξ 2





ξ 2 ∂ξi (x, y, ξ )∂ξl r(x, y, ξ )

ξ 2 ∂ξi ,ξl (x, y, ξ )





+ (1 − χ (ξ ))

+ (1 − χ (ξ ))





α ξ 2 α 2 ξ 4  2  3 ≤ Cm pm () + Cm (pm ())3 ξ −1 =: P2m (pm ()) ξ −1 ,

where we used that ξl / ξ ≤ 1 and the same arguments as before. Having done the estimation for all j , k, l, in the end one can set Pm =

Pim ,

i

since all coefficients of Pim are nonnegative. To prove (12) one can use the same arguments.   Definition 3 The seminorms (πX,m )m∈N for C∞ (X) are defined by





j πX,m (v) = sup ∂x v(x) , x ∈ KX,m , |j | ≤ m ,

v ∈ C∞ (X).

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Proposition 3 Let (n , an )n∈N be a convergent sequence in the product space (Fdα,-,δ (X, Y, 2), (p m )m∈N ) with limit (, a). Furthermore, let Ln be the regularizing operator (cf. Sect. 2.1) for (n , an ). Let m 3 ∈ N and ψ ∈ D(Y ) be fixed. Furthermore, let |j | ≤ m 3. Choose κ ∈ N large enough, such that

 

j in (x,y,ξ ) κ

Ln (an (x, y, ξ )ψ(y)) = O(ξ −(n2 +1) ).

∂x e Choose m large enough such that κ + m 3 + 1 ≤ m and supp(ψ) ⊂ KY,m . Then (a) there exists a constant Cm,pm (n ),qm (an ),ψ , depending on m, pm (n ), qm (an ) and ψ, such that

 

j in (x,y,ξ ) κ

sup Ln (an (x, y, ξ )ψ(y))

∂x e x∈KX,3 m , y∈Y

≤ Cm,pm (n ),qm (an ),ψ ξ −(n2 +1) . (b) the sequence of FIOs An ,an converges in the following sense: lim πX,3 m (An ,an [ψ] − A,a [ψ])) = 0,

n→∞

where   An ,an [ψ](x) =

2 Y

ein (x,y,ξ ) Lκn (an (x, y, ξ )ψ(y)) dy d−ξ ,

and   A,a [ψ](x) =

ei(x,y,ξ ) Lκ (a(x, y, ξ )ψ(y)) dy d−ξ .

2 Y

Proof (a) We will examine only an ψ and Ln (an ψ). The term Lκn (an ψ) can be estimated in the same way, but it is much more tedious. Since ψ has compact support, there exists a constant Cψ,m , depending on m and ψ such that



k

∂y ψ(y) ≤ Cψ,m , for any |k| ≤ m. By assumption, supp(ψ) ⊂ KY,m and therefore sup

x∈KX,3 m , y∈Y

|an (x, y, ξ )ψ(y)| =

sup

x∈KX,3 m , y∈KY,m

|an (x, y, ξ )ψ(y)|

≤ Cψ,m qm (an ) ξ d 0 ξ d . ≤ Cm,p m (n ),qm (an ),ψ

Stochastic Fourier Integral Operator

393

Furthermore, Ln (an (x, y, ξ )ψ(y)) =−

n2

∂ξl (α nl (x, y, ξ )an (x, y, ξ )ψ(y)) −

l=1

nY

∂yk (β nk (x, y, ξ )an (x, y, ξ )ψ(y))

k=1

+ γ n (x, y, ξ )an (x, y, ξ )ψ(y).

(13) Using again that supp ψ ⊂ Ky,m , the first term of (13) can be bounded by sup

x∈KX,3 m , y∈Y

≤



∂ξ (α (x, y, ξ )an (x, y, ξ )ψ(y))

l nl

sup

x∈KX,3 m , y∈KY,m

+

sup



∂ξ α (x, y, ξ )an (x, y, ξ )ψ(y)

l nl

x∈KX,3 m , y∈KY,m



α (x, y, ξ )∂ξ an (x, y, ξ )ψ(y)

l nl

≤ Pm (pm (n )) ξ −1 |an (x, y, ξ )| Cψ,m



+ Pm (pm (n )) ∂ξl an (x, y, ξ ) Cψ,m   ≤ Cψ,m Pm (pm (n )) qm (an ) ξ d−1 + ξ d−- . The second term of (13) can be bounded by sup

x∈KX,3 m , y∈Y

≤

sup







∂yk (β nk (x, y, ξ )an (x, y, ξ )ψ(y))

x∈KX,3 m , y∈KY,m

+ +







∂yk β nk (x, y, ξ )an (x, y, ξ )ψ(y)

sup







β nk (x, y, ξ )∂yk an (x, y, ξ )ψ(y)

sup







β nk (x, y, ξ )an (x, y, ξ )∂yk ψ(y)

x∈KX,3 m , y∈KY,m

x∈KX,3 m , y∈KY,m

≤ P(pm (n )) ξ −1 |an (x, y, ξ )| Cψ,m + Pm (pm (n )) ξ −1 ∂yk |an (x, y, ξ )| Cψ,m + Pm (pm (n )) ξ −1 |an (x, y, ξ )| Cψ,m   ≤ Cψ,m Pm (pm (n ))qm (an ) ξ d−1 + ξ d−1+δ + ξ d−1 .

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Since γ n (x, y, ξ ) ≡ χ (ξ ), which is compactly supported, there exists a constant 3m such that C





3 −(n2 +1) . sup

γ n (x, y, ξ )an (x, y, ξ )ψ(y) ≤ C m Cψ,m qm (an ) ξ  x∈X,y∈Y,ξ ∈2

So for     1 3m qm (an ) Pm (pm (n )) ∨ 1 = Cψ,m n2 + nY + C Cm,p m (n ),qm (an ),ψ it holds that 1 ξ d−((1−δ)∧-) . sup |Ln (an (x, y, ξ )ψ(y))| ≤ Cm,p m (n ),qm (an ),ψ

x∈KX,3 m

Iterative application of L decreases the growth with respect to ξ . For the κth κ such that application of L we get a constant Cm,p m (n ),qm (an ),ψ



κ ξ d−κ((1−δ)∧-) . sup Lκn (an (x, y, ξ )ψ(y)) ≤ Cm,p m (n ),qm (an ),ψ

x∈KX,3 m

If κ is large enough, d − κ((1 − δ) ∧ -) ≤ −(n2 + 1). In the end we set Cm,pm (n ),qm (an ),ψ =

κ

i Cm,p . m (n ),qm (an ),ψ

i=1

(b) To prove this, we would like to use the dominated convergence theorem. So first we note that if (n , an )n∈N converges in (Fdα,-,δ (X, Y, 2), (p m )m∈N ), then  j sup ∂x ein (x,y,ξ ) Lκn (an (x, y, ξ )ψ(y))

x∈K

converges pointwise for all j ∈ NnX , y ∈ Y and ξ ∈ 2 and K ⊂ X compact. Furthermore, since (n , an )n∈N is convergent, , Cm,sup = sup Cm,pm (n ),qm (an ),ψ n∈N

is finite.

Stochastic Fourier Integral Operator

395

So, since ψ is supported in KY,m we see that  

 

j

sup ∂x ein (x,y,ξ ) Lκn (an (x, y, ξ )ψ(y)) dy d−ξ

2 Y x∈K

 

 



j sup ∂x ein (x,y,ξ ) Lκn (an (x, y, ξ )ψ(y)) dy d−ξ

=

2 KY,m x∈K

 

Cm,sup ξ −(n2 +1) dy d−ξ ,

≤ 2 KY,m

by which all requirements of the dominated convergence theorem are satisfied. So lim πX,3 m (An ,an [ψ] − A,a [ψ])

 

j = lim sup

∂x ein (x,y,ξ ) Lκn (an (x, y, ξ )ψ(y)) n→∞

n→∞ x∈K

X,3 m

2 Y

−e



L (a(x, y, ξ )ψ(y)) dy dξ

i(x,y,ξ ) κ

  ≤ lim

−



j sup

∂x ein (x,y,ξ ) Lκn (an (x, y, ξ )ψ(y))

n→∞ 2 Y x∈K X,3 m

−e



L (a(x, y, ξ )ψ(y))

dy d−ξ

i(x,y,ξ ) κ

  =

lim



j sup

∂x ein (x,y,ξ ) Lκn (an (x, y, ξ )ψ(y))

2 Y n→∞ x∈KX,3 m



− ei(x,y,ξ ) Lκ (a(x, y, ξ )ψ(y))

dy d−ξ

= 0.

 

Corollary 1 Let ψ ∈ D(Y ) fixed. The map A,a : Fdα,-,δ (X, Y, 2) → C∞ (X)   ei(x,y,ξ ) a(x, y, ξ )ψ(y) dy d−ξ (, a) → 2 Y

is continuous with respect to the topologies generated by (p m )m∈N and (πX,m )m∈N .

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M. Oberguggenberger and M. Schwarz

Remark 3 Using the same principles of the proof, one can also show that the map Fdα,-,δ (X, Y, 2) × D(Y ) → C∞ (X) (, a, ψ) → A,a [ψ] is continuous. Corollary 2 Let u ∈ E (Y ) be a compactly supported distribution. Then A,a : Fdα,-,δ (X, Y, 2) → D (X) (, a) → A,a [u] is continuous in the weak-∗ topology. As in Sect. 2.2 the operator A is defined by its adjoint 0 ! / A,a [u](x), ψ(x) = u(y), tA,a [ψ](y) Q P   i(x,y,ξ ) − e a(x, y, ξ )ψ(x) dx dξ . = u(y), 2 X

Theorem 1 (Stochastic Fourier Integral Operators) Let (, F, P) be a probability space and d a :  → S-,δ (X × Y × 2)   ω → x, y, ξ ) → a(x, y, ξ , ω)

and  :  → Mα (X × Y × 2)   ω → (x, y, ξ ) → (x, y, ξ , ω) be measurable mappings, where the target spaces are equipped with the Borel σ algebra. Then, for fixed ψ ∈ D(X) resp. u ∈ E (X) the mappings A,a :  → C∞ (X) ω → A,a [ψ]

resp.

A,a :  → D (X) ω → A,a [u]

are measurable with respect to the Borel σ -algebra, induced by the corresponding topology.

Stochastic Fourier Integral Operator

397

4 Applications In this section we apply Theorem 1 in three typical situations. Example 1 Let X = Y = R, 2 = R\ {0}, α > 0 be a real constant and , ∞ C∞ α (X) = f ∈ C (X) : ∀x ∈ X : f (x) ≥ α , and let c :  → C∞ α (X)  ω → x → c(ω, x)) be a measurable map. The transport equation with speed c is then  " ∂t + c(ω, x)∂x u(ω, x, t) = 0 u(ω, x, 0) = u0 (x).

(14)

The characteristic curves γ satisfy ⎧ ⎪ ⎨ d γ (ω, x, t; τ ) = c(ω, γ (ω, x, t; τ )) dτ ⎪ ⎩ γ (ω, x, t; t) = x.

(15)

Then one can check by classical ODE theory that for fixed t ∈ R, ω ∈  the map ∞ S : C∞ α (X) → C (X)   (x → c(ω, x)) → x → γ (ω, x, t; 0)

is continuous in the (πX,3 m )m 3∈N -sense. Therefore, for t ∈ R fixed, ω :  → Mα (X, Y, 2)   ω → (x, y, ξ ) → ξ · (γ (ω, x, t; 0) − y) is measurable again. Furthermore, let a(x, y, ξ ) ≡ 1 and let u0 ∈ D(X) resp. u0 ∈ E (X). Then, for fixed ω, the solution to (14) is given by Aω ,a [u0 ], and the maps  → C∞ (X) ω → Aω ,a [u0 ] are measurable.

resp.

 → D (X) ω → Aω ,a [u0 ]

398

M. Oberguggenberger and M. Schwarz

Example 2 Consider as before X = Y = R, 2 = R\ {0}. The half wave equation in one dimension is  " ∂t + ic(ω, x)P (Dx ) u(ω, x, t) = 0 (16) u(ω, x, 0) = u0 (x), where P is the pseudodifferential operator with symbol P (ξ ) = |ξ | (1 − χ (ξ )), where χ ∈ D(R), χ (ξ ) ≡ 1 for |ξ | < 1/4 and χ (ξ ) ≡ 0 for |ξ | > 1/2. We assume that c :  → C∞ α (X) is measurable and that c(ω, x) and |∂x c(ω, x)| are bounded from above by a constant R for any x ∈ R and ω ∈ . Further, all seminorms πX,m (c(ω, ·)) are assumed to be bounded independently of ω. We fix ω ∈  for now and for notational simplicity we will drop it. A FIO parametrix for (16) can be constructed following [12, Chapter  VIII, §3]. The phase function of the parametrix is of the form (x, y, ξ ) = |ξ | φ x, |ξξ | , t − y · ξ , where φ satisfies the eikonal equation, "

∂t φ(x, t) + c(x)P (∂x φ(x, t)) = 0 φ(x, 0) = x · ξ.

(17)

This is a nonlinear differential equation of the form Q(x, t, φx , φt ) = 0. In [13, Chapter II, Sect 19] one can find an explicit representation of the solution to this problem, which we will write down in the following. Let F (t; x1 , ξ1 ) and G(t; x1 , ξ1 ) satisfy the following equations ⎧ ⎪ ⎨

⎧ ⎪ ⎨

dF = c(F )P  (G) dt

dG = −c (F )P (G) dt

⎪ ⎩ G(0; x , ξ ) = ξ . 1 1 1

⎪ ⎩ F (0; x , ξ ) = x 1 1 1

(18)

There exists a T ∈ R, such that P (G(t; x1 , ξ1 )) ≡ |ξ1 |

and

P  (G(t; x1 , ξ1 )) ≡ sign ξ1

for 0 ≤ t ≤ T and for all ξ1 , x1 ∈ R, |ξ1 | ≥ 1. In that case F does not depend on G and x = F (t; x1 , ξ1 ) is just the flow from (0, x1 ) to (t, x). The inverse flow is then F (−t, x, ξ1 ), which goes from (t, x) to (0, x1 ). Finally, φ is given by  φ(x, t, ξ ) = xξ −

t

c(x)P (G(F (−s; x, ξ ), s, ξ )) ds

(19)

0

for t ∈ [0, T ]. We observe that F depends only on the sign of ξ1 and λG(t; x1 , ξ1 ) = G(t; x1 , λξ1 ) for λ ≥ 1, |ξ1 | > 1 and all x ∈ R and t ∈ [0, T ]. Thus,  ξ φ(x, t, ξ ) − |ξ | φ x, t, |ξ |

Stochastic Fourier Integral Operator

399

is compactly supported with respect to ξ for all x ∈ R and t ∈ [0, T ]. Noting that φ(x, 0, ξ ) = xξ and by the representation (19) one can also see that for a sufficiently small time interval, there is an α such that  ∈ Mα (X, Y, 2). Finally, by classical ODE theory, one can show that F and G continuously depend on c and thus also  (at least for a short time interval). To obtain the amplitude one has to solve a cascade of transport equations of the form   ∂t − c(x)P  (ξ )∂x + H (x, t, ξ ) aj (x, t, ξ ) = 0,

j ≤ 0,

where H is a smooth function, depending on derivatives of c,φ and ak , k > j (for details see again [12, Chapter VIII, §3]). Similarly to the previous example one can show that aj continuously depends  on c. In [11] one can find an explicit construction of a function a such that (a − j ≤0 aj (x, ξ, t)) is smoothing, namely a(x, t, ξ ) =

(1 − χ(ξ/nj ))aj (x, t, ξ ), j ≤0

where χ ∈ D(R), as above, χ (ξ ) ≡ 1 for |ξ | < 1/4 and χ (ξ ) ≡ 0 for |ξ | > 1/2, and nj ∈ N approaches +∞ quickly enough as j goes to −∞. Since the aj depend continuously on c in the C∞ -topology, one can chose nj such that the series converges uniformly together with all derivatives when c is taken from a bounded set in C∞ α (X). Thus, a depends continuously on c as well in the corresponding topologies. Consequently, the phase function  and the amplitude function a of the parametrix A,a depend continuously on c in the corresponding topologies. Recall that c :  → C∞ α (X) is a random function. At fixed ω ∈ , one obtains the parametrix Aω ,aω . By Theorem 1, if u0 ∈ D(R) resp. u0 ∈ E (R), then the map ω → Aω ,aω [u0 ](x, t) is measurable. Example 3 The standard bottom-up approach to modeling waves in random media would place the randomness in the coefficients of the underlying PDEs. However, the solution depends in a strongly nonlinear way on the coefficients of the equation, even if the PDEs are linear. This makes it hard to track or compute the stochastic features of the solution, such as the expectation, the variance or the autocovariance function. Accordingly, a top-down approach has been proposed in [9, 10]. Starting from the mean field equations as constant coefficient PDEs, the deterministic solution can readily be represented by FIOs. The stochastic properties of the medium are then modeled a posteriori by random perturbations of the phase and amplitude functions. In applications to material sciences, e.g., damage detection, these random perturbations can be calibrated to measurement data (just as in the bottom-up approach, where the coefficients are calibrated to measurement data).

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M. Oberguggenberger and M. Schwarz

This program has been carried through in [10] in the case of three-dimensional linear elasticity. The approach leads to stochastic FIOs, whose measurability has to be proven. We present a simplified example using the scalar wave equation. Let Y = Rn , X = Rn × R and 2 = Rn \ {0}. One can solve the deterministic wave equation with constant wave speed c0 ⎧ ⎪ ⎪ (∂tt − c0 )u(x, t) = 0 ⎨ u(x, 0) = u0 (x) ⎪ ⎪ ⎩ ∂t u(x, 0) = 0 using a linear combination of FIOs with phase functions (x, t, ξ , y) = x − y, ξ  ± c ξ t and amplitude function a(x, y, ξ ) ≡

1 . 2

Then, instead of using the deterministic  and a, randomly perturbed version are introduced. One has to make sure that the randomly perturbed phase function is still an operator phase function. Thus fix α > 0 and let n c :  → C∞ α (R )  ω → x → c(ω, x))

be a measurable, smooth random field with expectation value E(c(x)) ≡ c0 . Let ω (x, t, y, ξ ) = x − y, ξ  ± ξ c(ω, x)t. Then, ω ∈ Mα (X, Y, 2). Note that this is guaranteed, since we included the time into the image space X and ξ −1 |∂t ω (x, t, y, ξ )| ≥ α. Furthermore, let a :  → Sd-,δ (X, Y, 2) ω → aω (x, t, y, ξ ) be random with E(aω (x, y, ξ )) = 12 . Then, (ω , aω ) ∈ Fdα,-,δ (X, Y, 2) and, as above, ω → Aω ,aω [u0 ] is measurable for a given u0 ∈ D(X) resp. u0 ∈ E (X) in the corresponding topology. If  and a are independent, the expectation, the variance and the autocovariance function of Aω ,aω [u0 ](x),  resp. Aω ,aω [u0 ], ψ can be computed without much difficulty, observing that E exp(iω (x, t, ξ , y)) is just the characteristic function of the random variable ω (x, t, ξ , y), see [8].

Stochastic Fourier Integral Operator

401

Acknowledgements This work was supported by the grant P-27570-N26 of FWF (The Austrian Science Fund).

References 1. P. Boggiatto, E. Buzano, L. Rodino. Global hypoellipticity and spectral theory. Akademie Verlag, Berlin, 1996. 2. J, Dieudonné. Treatise on Analysis, Volume III. Academic Press, New York, 1972. 3. J.-P. Fouque, J. Garnier, G. Papanicolaou, K. Sølna. Wave propagation and time reversal in randomly layered media. Springer, New York, 2007. 4. L. Hörmander. The Analysis of Liner Partial Differential Operators I. Springer, Berlin Heidelberg, 2003 5. L. Lamplmayr, M. Oberguggenberger, M. Schwarz. Stochastic Fourier Integral Operators for Damage Detection. In: M. Voigt, D. Proske, W. Graf, M. Beer, U. Häußler-Combe, P. Voigt (Eds.), A Proceedings of the 15th International Probabilistic Workshop & 10th Dresdner Probablistik Workshop. TUDpress, Dresden, 2017, 73–84. 6. M. Mascarello, L. Rodino. Partial differential equations with multiple characteristics. Akademie Verlag, Berlin, 1997. 7. B. Nair, B.S. White. High-Frequency Wave Propagation In Random Media—A Unified Approach. SIAM J. Appl. Math. 51 (1991), no. 2, 374–411. 8. M. Oberguggenberger, M. Schwarz. Fourier integral operators in stochastic structural analysis. In: F. Werner, M. Huber, T. Lahmer, T. Most, D. Proske (Eds.), Proceedings of the 12th International Probabilistic Workshop. Bauhaus-Universitätsverlag, Weimar, 2014, 250–257. 9. M. Oberguggenberger, M. Schwarz. Stochastic Methods in Damage Detection. In: M. De Angelis (Ed.), REC2018 - Proceedings of the 8th International Workshop on Reliable Engineering Computing. University of Liverpool, 2018, 1–11. 10. M. Schwarz. Stochastic Fourier Integral Operators and Hyperbolic Differential Equations in Random Media. PhD thesis, University of Innsbruck, Austria, 2019. 11. M. A. Shubin. Pseudodifferential Operators and Spectral Theory. Springer, Berlin Heidelberg, 1987. 12. M. E. Taylor. Pseudodifferential Operators. Princeton University Press, Princeton, NJ, 1981. 13. F. Trèves. Basic Linear Partial Differential Equations. Academic Press, New York, NY, 1975.

Convolution and Anti-Wick Quantisation on Ultradistribution Spaces Stevan Pilipovi´c and Bojan Prangoski

Abstract We present recent advances in convolution theory for the quasi-analytic and non quasi-analytic ultradistribution spaces and generalised Gelfand-Shilov spaces. Additionally, we consider the existence of convolution of non-quasianalytic 2 ultradistribution with the Gaussian kernel es|x| , s ∈ R\{0}, and identify the largest subspace of non-quasi-analytic ultradistributions for which this convolution exists. This gives a way to extend the definition of Anti-Wick quantisation for symbols that are not necessarily tempered ultradistributions. Finally, we discuss convolution in q quasi-analytic classes with es|·| , q > 1, s ∈ R. Keywords Anti-Wick quantisation · Convolution · Ultradistributions

1 Introduction This paper has an expository form up to the last section where we give new results q concerning the convolution with es|·| , q ∈ (1, 2), s > 0, in the quasi-analytic setting. This section is a starting point of our further investigation of convolution in this domain. Convolution is among the most important tools in mathematical analysis. In the classical, distributional, setting this is a much studied topic and most of the central problems have been solved more than half a century ago; however, there are still many interesting recent results: see for example [1, 8, 12, 21, 22]. One of the most important problems is extending the definition of convolution S ∗ T to the case when both S and T are not necessarily test functions (see [32, 33]). The

S. Pilipovi´c Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia e-mail: [email protected] B. Prangoski () Department of Mathematics, Faculty of Mechanical Engineering-Skopje, Ss. Cyril and Methodius University in Skopje, Skopje, Macedonia © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_22

403

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key ingredient for the development of the theory was the study of the topological ˙ d ), DL∞ (Rd ) and D 1 (Rd ) (see [10, 11, 32, 33]). properties of the spaces B(R L This problem in the case of non-quasianalytic ultradistributions of Beurling class was solved a long time ago in [16] (see also [23]). However, the extension of the theory to the non-quasianalytic Roumieu ultradistributions was done only recently in our papers [13, 25]; the difficulty in the Roumieu case comes from the fact ˙ d ) and DL∞ (Rd ) carry much that the ultradistributional variants of the spaces B(R more complicated topologies than their Beurling (and distributional) counterparts. In [29], we have extended this result to the case of quasi-analytic ultradistributions of Beurling and Roumieu class (which have the classical Gelfand-Shilov spaces as a special case); of course, the lack of test functions with compact support makes the development of the theory significantly different although the final results look similar to the non-quasianalytic case. The first part of this survey article (Sect. 3) contains the main results concerning the convolution of both non-quasianalytic and quasianalytic ultradistributions of Beurling and Roumieu type as given in our articles [13, 25, 29]. In Sect. 4 we will consider the existence of convolution of an ultradistribution with the Gaussian 2 kernel es|x| , s ∈ R\{0}. The main result comes from [26] and it basically identifies the largest subspace of ultradistributions which can be convolved with the Gaussian (see [35] for the solution of the same problem in the distributional setting and its applications). Although this is of independent interest, we will present one particular application in the non-quasianalytic case concerning Anti-Wick quantisation (cf. [20, 34]) in Sect. 5. The Anti-Wick quantisation is usually given for symbols that are tempered ultradistributions (generalised Gelfand-Shilov ultradistributions), but one can easily prove that it is equal to the Weyl quantisation of the convolution of the 2 2 symbol with the Gaussian e−|x| −|ξ | . Since the Weyl quantisation can be defined for more general classes of symbols as a mapping from D∗ (Rd ) into D∗ (Rd ) (D∗ (Rd ) stands for the space of compactly supported ultradifferentiable functions of Beurling and Roumieu class with D∗ (Rd ) being its dual) we can extend the AntiWick quantisation to those symbols which can be convolved with the Gaussian by defining it in terms of the Weyl quantisation (the symbols have to satisfy several other technical conditions). Finally, we mention that we have developed a Shubin type calculus of pseudo-differential operators of infinite order in the setting of nonquasianalytic tempered ultradistributions (generalised Gelfand-Shilov spaces) and there is a close connection between the Anti-Wick and Weyl quantisation of symbols of these classes. This calculus has all the essential properties: it is closed under composition, taking adjoint, hypoelliptic symbols have parametrices etc. However, as we will not need it, we will not recall any results from it and just refer the interested reader to [5, 6, 26–28, 31] (see also [3, 4] for a similar calculus). In the last section, independently of the results in Sects. 2 and 3, we present new results related to the convolution within the quasi-analytic class of Roumieu ultradistributions. Actually, we consider a special case, the problem of the existence q of convolution of the function es|·| , s ∈ R, with an ultradistribution of the form  q {p!} e−k· g, g ∈ S 1/q  (Rd ), 1 < q < q  . It is evident that the results of Sect. 5 can {p!

}

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be considered in a more general setting. This will be done in a separate paper. Here we just indicate important steps in this direction.

2 Preliminaries Let {Mp }p∈N and {Ap }p∈N be sequences of positive numbers, M0 = M1 = A0 = A1 = 1. Recall [17], given two weight sequences Mp and Ap , the notation Mp ⊂ Ap (resp. Mp ≺ Ap ) means that there are C, L > 0 (resp. for every L > 0 there is C > 0) such that Mp ≤ CLp Ap , ∀p ∈ N. For a multi-index α ∈ Nd , Mα stands for M|α| , |α| = α1 + . . . + αd . Throughout the article, we impose some of the following conditions on a weight sequence Mp (we will always explicitly state which conditions have to be satisfied by the sequence): (M.1) Mp2 ≤ Mp−1 Mp+1 , p ∈ Z+ ; (M.2) Mp ≤ c0 H p min {Mp−q Mq }, p, q ∈ N, for some c0 , H ≥ 1; 0≤q≤p

(M.3)  (strong non-quasianalyticity) there exists a constant c0 ≥ 1 such that ∞ j =p+1 Mj −1 /Mj ≤ c0 pMp /Mp+1 , ∀p ∈ Z+ ; q (M.5) there exists q > 0 such that Mp is strongly non-quasianalytic, i.e., there ∞ q q q q exists c0 ≥ 1 such that j =p+1 Mj −1 /Mj ≤ c0 pMp /Mp+1 , ∀p ∈ Z+ . (M.6) p! ⊂ Mp . Note that (M.5) implies that there exists κ > 0 such that p!κ ⊂ Mp , i.e., there p exist c0 , L0 > 0 such that p!κ ≤ c0 L0 Mp , p ∈ N (cf. [17, Lemma 4.1]). For any κ κ > 0, the sequence p! , p ∈ N, satisfies (M.1), (M.2) and (M.5). Additionally, if κ ≥ 1, this sequence satisfies (M.6) as well and, when κ > 1, it satisfies all of the above conditions. Following [17], for p ∈ Z+ , we denote mp = Mp /Mp−1 , and when Mp satisfies (M.1) and mp /C p → ∞, as p → ∞, for any C > 0, we define its associated function by M(ρ) = supp∈N ln+ ρ p /Mp , ρ > 0. It is a nonnegative, continuous, monotonically increasing function, vanishes for sufficiently small ρ > 0, and increases more rapidly than ln ρ n as ρ → ∞, for any n ∈ N. When Mp = p!s , with s > 0, we have M(ρ) ρ 1/s . Denote by R the set of all positive sequences which increase to ∞. CpFor (lp ) ∈ R, we denote as Nlp (·) the associated function for the sequences Mp j =1 lj .1 Let Mp and Ap be two weight sequences such that Mp satisfies (M.1), (M.2) M ,h

and (M.5) and Ap satisfies (M.1), (M.2) and (M.6). We denote by SApp,h , h > 0,

the (B)-space (Banach space) of all ϕ ∈ C ∞ (Rd ) for which the norm σh (ϕ) = sup α

1 here

C0

1 1 h|α| 1eA(h|·|) D α ϕ 1L∞ (Rd ) Mα

and throughout the rest of the article we apply the principle of vacuous products, i.e. = 1.

j =1 lj

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is finite. One easily verifies that for h1 H < h2 the canonical inclusion SApp,h22 → M ,h

SApp,h11 is compact. As locally convex spaces (from now, abbreviated as l.c.s.), we define (M )

M ,h

{M }

M ,h

S p and S{App} (Rd ) = lim S p . S(App) (Rd ) = lim ←− Ap ,h −→ Ap ,h h→∞

h→0

{M }

(M )

Thus S(App) (Rd ) is an (F S)-space and S{App} (Rd ) is a (DF S)-space and consequently they are both Montel spaces. In fact they are both nuclear space (see [29, Proposition 2.11]) and thus, their strong duals (the spaces of tempered ultradistributions of Beurling and Roumieu type respectively) are also nuclear (see [7, 24, 29] for their topological properties). In the sequel we shall employ S†∗ (Rd ) as {M }

(M )

a common notation for S(App) (Rd ) (Beurling case) and S{App} (Rd ) (Roumieu case);

the same goes for their strong duals S†∗ (Rd ). If Mp = Ap , p ∈ N, to ease notation, we write S ∗ (Rd ) = S∗∗ (Rd ) and S ∗ (Rd ) = S∗∗ (Rd ). Finally, we mention that when Mp = Ap , p ∈ N, the Fourier transform is a topological isomorphism on S ∗ (Rd ) and on S ∗ (Rd ).  α d The entire function P (z) = α∈Nd cα z , z ∈ C , is an ultrapolynomial of class (Mp ) (resp. of class {Mp }), whenever the coefficients cα satisfy the estimate 0 and some |cα | ≤ CL|α| /Mα , α ∈ Nd , for some C, L > 0 (resp. for every  L> α C = C(L) > 0). The corresponding operator P (D) = α cα D is called an ultradifferential operator of class (Mp ) (resp. of class {Mp }) and it acts continuously on S†∗ (Rd ) and S†∗ (Rd ). Let K be a regular compact subset of Rd (i.e. int K = K) and assume that Mp satisfies (M.1), (M.2) and (M.3). Following Komatsu [17], for h > 0, we define the (B)-space E Mp ,h (K) as E Mp ,h (K) = {ϕ ∈ C ∞ (K)| sup sup |D α ϕ(x)|/(hα Mα ) < ∞}, α∈Nd x∈K

where C ∞ (K) stands for the space of all smooth functions on int K whose all partial {M },h derivatives extend to continuous functions on K. Furthermore, DK p denotes its subspace of all smooth functions supported by K. We define the l.c.s. E (Mp ) (Rd ), E {Mp } (Rd ), D(Mp ) (Rd ), D{Mp } (Rd ) and their strong duals, the corresponding spaces of ultradistributions of Beurling and Roumieu type; we refer to [17–19] for their properties. Here we just mention that the ultradifferential operators of class ∗ act continuously on all of these space and that D∗ (Rd ) is continuously and densely included into S†∗ (Rd ) (when Mp satisfies (M.1), (M.2) and (M.3)).

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3 Existence of S†∗ -Convolution and D ∗ -Convolution If S ∈ S†∗ (Rd ) and ϕ ∈ S†∗ (Rd ) or, alternatively, S ∈ D∗ (Rd ) and ϕ ∈ D∗ (Rd ), the convolution of S and ϕ is always a well defined element of S†∗ (Rd ) and D∗ (Rd ) respectively given by S ∗ ϕ, ψ = S, ϕˇ ∗ ψ.

(1)

The purpose of this section is to extend this definition in the cases when ϕ is not necessarily a test functions. The problem of extending the definition of convolution of distributions goes back to Schwartz. He introduced the following definition of convolvability of two distributions S and T (tempered distributions, resp.): S and T are said to be D convolvable (S  -convolvable) if for any ϕ ∈ D(Rd ) (for any ϕ ∈ S(Rd )) the distribution (S ⊗ T )ϕ Δ belongs to DL 1 (R2d ), where ϕ Δ (x, y) = ϕ(x + y). In this case, S ∗ T is defined by S ∗ T , ϕ = (S ⊗ T )ϕ Δ , 1

(2)

If either S or T are test functions this is equivalent to defining their convolution by (1). More importantly, it agrees with the definition of convolution of Radon measures as given in Bourbaki [2]. The fact that S ∗ T as defined in (2) is a well defined (tempered) distribution and the sense in which the duality on the right hand side of (2) is given is very subtle; we refer to [32] for the complete account. Although, this definition of convolvability is symmetric, it is usually hard to work with in practice. That is why Schwartz introduced the following alternative definition: (ii) the D -convolution (S  -convolution, resp.) of S and T exists if for every ϕ ∈ ˇ ∈ D 1 (Rd ) D(Rd ) (ϕ ∈ S(Rd ), resp.) (ϕ ∗ S)T L (iii) the D -convolution (S  -convolution, resp.) of S and T exists if for every ϕ ∈ D(Rd ) (ϕ ∈ S(Rd ), resp.) (ϕ ∗ Tˇ )S ∈ DL 1 (Rd ) Furthermore, Chevalley introduce yet another (symmetric) definition of convolvability (iv) the D -convolution (S  -convolution, resp.) of S and T exists if for every ϕ, ψ ∈ ˇ D(Rd ) (ϕ, ψ ∈ S(Rd ), resp.) (ϕ ∗ S)(ψ ∗ T ) ∈ L1 (Rd ) It was Shiraishi [33] who proved that all this definitions are equivalent. The goal of this section is to present results analogous to this for the ultradistributional case. The difficulty in the ultradistributional setting arises in the Roumieu case because of the more complex topology the test spaces carry (when compared to the distributional case) as well as because of the lack of functions with compact support when the sequence Mp is quasianalytic.

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Before we state the results on the existence of convolution we have to introduce the ultradistributional analogue to the space DL 1 (Rd ). Let Mp be a weight sequence which satisfies (M.1), (M.2) and (M.5). For each h > 0, we define the (B)-space M ,h DL∞p (Rd ) as the space of all ϕ ∈ C ∞ (Rd ) for which the following norm is finite sup h|α| D α ϕ L∞ (Rd ) /Mα < ∞.

α∈Nd

As l.c.s. we define (M )

M ,h

DL∞p (Rd ), DL∞p (Rd ) = lim ←− h→∞

{M }

M ,h

DL∞p (Rd ) = lim DL∞p (Rd ). −→ h→0+

{M }

(M )

Then, DL∞p (Rd ) is an (F )-space. Furthermore, DL∞p (Rd ) is complete barrelled and bornological (DF )-space (see [15, Section 4.3]). Let Ap be a weight sequence that satisfies (M.1), (M.2) and (M.6). We denote by B˙ ∗ (Rd ) the closure of S†∗ (Rd ) in DL∗ ∞ (Rd ). Then [15, Theorem 4.16] verifies that B˙ ∗ (Rd ) does not depend on Ap and that B˙ (Mp ) (Rd ) is an (F )-space and B˙ {Mp } (Rd ) is a complete barrelled and bornological (DF )-space (these assertions are not trivial in the Roumieu case). When Mp additionally satisfies (M.3), since D∗ (Rd ) is dense in S†∗ (Rd ), the space B˙ ∗ (Rd ) coincides with the closure of D∗ (Rd ) in DL∗ ∞ (Rd ). We denote by DL∗1 (Rd ) the strong dual of B˙ ∗ (Rd ). We have the following result. Theorem 1 ([15, Theorem 4.17]) The strong bidual of B˙ ∗ (Rd ) is isomorphic to DL∗ ∞ (Rd ) as l.c.s. Moreover, B˙ (Mp ) (Rd ) is a distinguished (F )-space and consequently DL∗1 (Rd ) is barrelled and bornological. Denote by DL∗ ∞ ,c (Rd ) the space DL∗ ∞ (Rd ) equipped with the topology of compact convex circled convergence with respect to the duality DL∗1 (Rd ), DL∗ ∞ (Rd ) (cf. Theorem 1). Then DL∗ ∞ ,c (Rd ) is complete with S†∗ (Rd ) being continuously and densely injected into it (see [29, Proposition 5.3] and the comments after it) and its strong dual is topologically isomorphic to DL∗1 (Rd ) (see [29, Proposition 5.4]). In fact, we can give a system of seminorms which gives the topology of DL∗ ∞ ,c (Rd ). Let C0 (Rd ) be the space of all continuous functions on Rd which vanish at infinity. The space DL∗ ∞ ,c (Rd ) consists of all ϕ ∈ C ∞ (Rd ) such that for every r > 0 and every strictly positive g ∈ C0 (Rd ) (for every (rp ) ∈ R and every strictly positive g ∈ C0 (Rd ), resp.) pg,r (ϕ) = sup α∈Nd

r |α| gD α ϕ L∞ (Rd ) Mα

0, g ∈ C0 (Rd ) strictly positive (pg,(rp ) , (rp ) ∈ R, g ∈ C0 (Rd ) strictly positive, resp.); see [29, Proposition 5.3]. Now we are ready to state the analogous definition to the one given by Schwartz for existence of convolution of two ultradistributions. Definition 1 ([29, Definition 5.7]) Let S, T ∈ S†∗ (Rd ). We say that the S†∗ convolution of S and T exists if for each ϕ ∈ S†∗ (Rd ), (S ⊗ T )ϕ Δ ∈ DL∗1 (R2d ) and we define their convolution by S ∗ T , ϕ = D∗

(R L1

2d )

(S ⊗ T )ϕ Δ , 1D∗ ∞ L

,c

(R2d ) ,

∀ϕ ∈ S†∗ (Rd ).

(3)

The meaning of the right hand side of (3) is well defined by the above consideration. It is a non-trivial fact that S∗T as defined in (3) is indeed a continuous functional on S†∗ (Rd ); see the comments after [29, Definition 5.7] for the details. The main result that we state in this section is the following. Theorem 2 ([29, Theorem 5.8]) Assume the weight sequence Mp satisfies (M.1), (M.2) and (M.5) while the weight sequence Ap satisfies (M.1), (M.2) and (M.6). For S, T ∈ S†∗ (Rd ), the following statements are equivalent (i) (ii) (iii) (iv)

the S†∗ -convolution of S and T exists; ˇ ∈ D∗1 (Rd ); for all ϕ ∈ S†∗ (Rd ), (ϕ ∗ S)T L for all ϕ ∈ S†∗ (Rd ), (ϕ ∗ Tˇ )S ∈ DL∗1 (Rd ); ˇ for all ϕ, ψ ∈ S†∗ (Rd ), (ϕ ∗ S)(ψ ∗ T ) ∈ L1 (Rd ).

Assume that Mp additionally satisfies (M.3). Then one can consider the existence of D∗ -convolution of S, T ∈ D∗ (Rd ). The definition of D∗ -convolvability of two such ultradistributions is analogous to Definition 1 with S†∗ (Rd ) and S†∗ (Rd ) replaced by D∗ (Rd ) and D∗ (Rd ) respectively. Moreover, the following analogous result to Theorem 2 holds. Theorem 3 ([16, Theorem] for the Beurling Case and [25, Theorem 1], [13, Theorem 8.2] for the Roumieu Ccase) Assume the weight sequence Mp satisfies (M.1), (M.2) and (M.3). For S, T ∈ D∗ (Rd ), the following statements are equivalent (i) (ii) (iii) (iv)

the D∗ -convolution of S and T exists; ˇ ∈ D∗1 (Rd ); for all ϕ ∈ D∗ (Rd ), (ϕ ∗ S)T L for all ϕ ∈ D∗ (Rd ), (ϕ ∗ Tˇ )S ∈ DL∗1 (Rd ); ˇ for all ϕ, ψ ∈ D∗ (Rd ), (ϕ ∗ S)(ψ ∗ T ) ∈ L1 (Rd ).

Finally, we mention that if the convolution of the (tempered) ultradistributions S and T exists then (see [29, Remark 5.9]) S ∗ T , ϕ = D∗

(R L1

d)

ˇ , 1D∗ (ϕ ∗ S)T ∞ L

,c

(Rd )

= D∗

(R L1

d)

(ϕ ∗ Tˇ )S, 1D∗ ∞ L

,c

(Rd ) .

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4 Convolution with the Gaussian Kernel The goal of this section is to consider the convolution of ultradistributions with the 2 Gaussian kernel es|x| , s ∈ R\{0}. To state the problem precisely, let Mp be a weight sequence which satisfies (M.1), (M.2) and (M.3). The goal is to identify the largest subspace of D∗ (Rd ) whose elements can be convolved with the Gaussian (in the sense described in the previous section). Incidentally, we will also give a description 2 of the convolution of such elements with es|x| , s ∈ R\{0}, via the Fourier-Laplace transform. Because of this, we will make a slight detour, and present a couple of results concerning the action of the latter transform on ultradistributions.

4.1 The Fourier-Laplace Transform on Ultradistributions For a set B ⊆ Rd denote by ch B the convex hull of B. Theorem 4 ([30, Theorem 2.1]) Assume that Mp satisfies (M.1), (M.2) and (M.3). Let B be a connected open set in Rdξ and T ∈ D∗ (Rdx ) be such that, for all ξ ∈ B, e−xξ T (x) ∈ S ∗ (Rdx ). Then the Fourier transform Fx→η (e−xξ T (x)) is an analytic function of ζ = ξ + iη for ξ ∈ ch B, η ∈ Rd . Furthermore, it satisfies the following estimates: for every K ⊂⊂ ch B there exist k > 0 and C > 0 (for every k > 0 there exists C > 0, resp.) such that |Fx→η (e−xξ T (x))(ξ + iη)| ≤ CeM(k|η|) , ∀ξ ∈ K, ∀η ∈ Rd .

(4)

If, for S ∈ D∗ (Rd ), the conditions of the theorem are fulfilled, we call Fx→η (e−xξ S(x)) the (Fourier-)Laplace transform of S and denote it by L(S). Moreover (see [30, Remark 2.2]), √ 0 / √ 2 2 L(S)(ζ ) = eε 1+|x| e−xζ S(x), e−ε 1+|x| ,

(5)

for ζ ∈ U + iRdη , where U ⊂⊂ ch B and ε depends on U . If for S ∈ D∗ (Rd ) the conditions of the theorem are fulfilled for B = Rd , then the choice of ε can be made uniform for all K ⊂⊂ Rd . Theorem 5 ([30, Theorem 2.5]) Assume that Mp satisfies (M.1), (M.2) and (M.3). Let B be a connected open set in Rdξ and f an analytic function on B + iRdη . Let f satisfy the condition: for every compact subset K of B there exist C > 0 and k > 0 (for every k > 0 there exists C > 0, resp.) such that |f (ξ + iη)| ≤ CeM(k|η|) , ∀ξ ∈ K, ∀η ∈ Rd .

(6)

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Then, there exists S ∈ D∗ (Rdx ) such that e−xξ S(x) ∈ S ∗ (Rdx ), for all ξ ∈ B and   L(S)(ξ + iη) = Fx→η e−xξ S(x) (ξ + iη) = f (ξ + iη), ξ ∈ B, η ∈ Rd . (7) In brief, the above two theorems characterise the analytic functions defined on vertical strips which are Laplace transforms of ultradistributions. Incidentally, the above two theorems imply that if f is analytic on the vertical strip B + iRd with B being open and connected and f satisfies the conditions of Theorem 5 than it is in fact analytic on ch B + iRd . We can give slight generalisations of these two theorems. For this purpose, we (p!) (p!) first remark that if T ∈ S(p!) (Rd ), then exξ T (x) ∈ S(p!) (Rdx ), for all ξ ∈ Rd . Theorem 6 Assume that Mp satisfies (M.1), (M.2), (M.5) and (M.6), and, in the Beurling case, additionally assume that Mp satisfies p! ≺ Mp . Let B be a connected (p!) open set in Rdξ and T ∈ S(p!) (Rdx ) be such that, for all ξ ∈ B, e−xξ T (x) ∈ S ∗ (Rdx ). Then the Fourier transform Fx→η (e−xξ T (x)) is an analytic function of ζ = ξ + iη for ξ ∈ ch B, η ∈ Rd . Furthermore, it satisfies the following estimates: for every K ⊂⊂ ch B there exist k > 0 and C > 0 (for every k > 0 there exists C > 0, resp.) such that |Fx→η (e−xξ T (x))(ξ + iη)| ≤ CeM(k|η|) , ∀ξ ∈ K, ∀η ∈ Rd . The proof is the same as for Theorem 4 and we omit it. Furthermore, from the proof it directly follows that the remark after Theorem 4 is still valid in this case as well. Theorem 6 is indeed a generalisation of Theorem 4, since if Mp satisfies (M.1), (M.2) and (M.3) and T ∈ D∗ (Rd ) satisfies the assumptions of Theorem 4 (M ) (M ) (p!) then e−xξ T (x) ∈ S ∗ (Rdx ) ⊆ S(p!)p (Rd ) and thus T ∈ S(p!)p (Rd ) ⊆ S(p!) (Rd ) (M )

(as x → exξ is a multiplier for S(p!)p (Rdx ), for each fixed ξ ∈ Rd ). The proof of Theorem 5 relies on the existence of ultrapolynomials of class ∗ which do not vanish on a strip along the real axis. The existence of such ultrapolynomials when the sequence Mp satisfies the conditions of Theorem 6 is given in [29, Lemma 2.1] (and especially its proof). Thus, in an analogous way as in the proof of Theorem 5 one can prove the following result. Theorem 7 Assume that Mp satisfies (M.1), (M.2), (M.5) and (M.6), and, in the Beurling case, additionally assume that Mp satisfies p! ≺ Mp . Let B be a connected open set in Rdξ and f an analytic function on B + iRdη . Let f satisfies the condition: for every compact subset K of B there exist C > 0 and k > 0 (for every k > 0 there exists C > 0, resp.) such that |f (ξ + iη)| ≤ CeM(k|η|) , ∀ξ ∈ K, ∀η ∈ Rd . (p!)

Then, there exists S ∈ S(p!) (Rdx ) such that e−xξ S(x) ∈ S ∗ (Rdx ), for all ξ ∈ B and   L(S)(ξ + iη) = Fx→η e−xξ S(x) (ξ + iη) = f (ξ + iη), ξ ∈ B, η ∈ Rd .

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4.2 Convolution with the Gaussian Kernel Now we can characterise the largest subspace of D∗ (Rd ) whose elements can be 2 convolved with es|x| , s ∈ R\{0}. Let B ∗ = {S ∈ D∗ (Rd )| cosh(k|x|)S ∈ S ∗ (Rd ), ∀k ≥ 0} (x → cosh(k|x|) is 2 an entire function) and for s ∈ R\{0}, put Bs∗ = e−s|x| B ∗ . Clearly, B ∗ ⊆ S ∗ (Rd ). Define  A∗ = f ∈ O(Cd )| ∀K ⊂⊂ Rdξ , ∃h, C > 0, resp. ∀h > 0, ∃C > 0, such that  |f (ξ + iη)| ≤ CeM(h|η|) , ∀ξ ∈ K, ∀η ∈ Rd , A∗real = {f|Rd |f ∈ A∗ } and A∗s = es|x| A∗real . In fact, the space B ∗ has the following two equivalent descriptions 2

B ∗ = {S ∈ D∗ (Rd )| cosh(k|x|)S ∈ DL∗1 (Rd ), ∀k ≥ 0},

(8)

B ∗ = {S ∈ D∗ (Rd )| cosh(k|x|)S ∈ OC∗ (Rd ), ∀k ≥ 0},

(9)

where OC∗ (Rd ) stands for the space of convolutors for S ∗ (Rd ), i.e. the space of all T ∈ S ∗ (Rd ) such that ϕ → T ∗ ϕ is well defined and continuous mapping from S ∗ (Rd ) into itself (see [9] for the topological properties of OC∗ (Rd ) and its predual). The validity of (8) and (9) follows from OC∗ (Rd ) ⊆ DL∗1 (Rd ) (cf. [13, Theorem 6.1]), and the fact that x → cosh(k|x|)/ cosh(2k|x|) belongs to S ∗ (Rd ) for all k > 0 (see [26, Lemma 4.1]) together with the fact that ψT ∈ OC∗ (Rd ) for all ψ ∈ S ∗ (Rd ), T ∈ S ∗ (Rd ) (the latter is just a simple verification). The following theorem answers the question posed in the beginning of this section. Theorem 8 ([26, Theorem 4.3]) Let s ∈ R, s = 0. Then a) The convolution of S ∈ D∗ (Rd ) and es|x| exists if and only if S ∈ Bs∗ . b) L : B ∗ → A∗ is well defined and bijective mapping. For S ∈ B ∗ and ξ, η ∈ Rd , e−(ξ +iη)x S(x) ∈ DL∗1 (Rdx ) and the Laplace transform of S is given by 2

L(S)(ξ + iη) = e−(ξ +iη)x S(x), 1x . c) The mapping Bs∗ → A∗s , S → S ∗ es|x| , is bijective and for S ∈ Bs∗ , 2

2

2

2

(S ∗ es|·| )(x) = es|x| L(es|·| S)(2sx).

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413

5 Anti-Wick Quantisation In this section we present one interesting application of Theorem 8, namely we will extend the definition of the Anti-Wick quantisation for symbols that are not necessarily tempered ultradistribution. We start by recalling the definition of the basic ingredient in the Anti-Wick quantisation: the short-time Fourier transform. Let Mp be a weight sequence that satisfies (M.1), (M.2) and (M.6). Denote 1

1

G0 (x) = π −d/4 e− 2 |x| and Gy,η (x) = π −d/4 eixη e− 2 |x−y| , where y and η are parameters in Rd and denote by (·, ·) the inner product in L2 (Rd ). 2

2

Definition 2 For u ∈ S ∗ (Rd ) we define the short-time Fourier transform (from now on abbreviated as STFT) V u of u with window G0 as the tempered ultradistribution in R2d given by V u(y, η) = Ft→η (u(t)G0 (t − y)). Remark 1 In fact, the STFT can be defined when the window is an arbitrary nonzero element of S ∗ (Rd ) (and, in fact, even for a more general class of windows, see [14, Section 4.1]). However, for the Anti-Wick quantisation, we will only need the Gaussian G0 as a window. One easily verifies that V acts continuously S ∗ (Rd ) → S ∗ (R2d ), S ∗ (Rd ) → and L2 (Rd ) → L2 (R2d ). Moreover V u L2 (R2d ) = (2π )d/2 u L2 (Rd ) . Its adjoint map V ∗ : S ∗ (R2d ) → S ∗ (Rd ), S ∗ (R2d )

V ∗ F (t) = (2π )d

 Rd

−1 Fη→t (F (y, η)) G0 (t − y)dy, F ∈ S ∗ (R2d )

extends to a well defined and continuous map S ∗ (R2d ) → S ∗ (Rd ) and L2 (R2d ) → L2 (Rd ) and V ∗ V = (2π )d I . The definition of Anti-Wick operators is the following. Definition 3 Let a ∈ S ∗ (R2d ). We define the Anti-Wick operator with symbol a as the map Aa : S ∗ (Rd ) → S ∗ (Rd ) given by Aa u = (2π )−d V ∗ (aV u), u ∈ S ∗ (Rd ). Notice that, if a is a multiplier for S ∗ (R2d ) than Aa is a continuous operator from into itself. The above formula is equivalent to

S ∗ (Rd )

Aa u, v = (2π )−d a, V uV v,

u, v ∈ S ∗ (Rd ).

(10)

An immediate consequence is that Aa is formally self-adjoint when a ∈ S ∗ (Rd ) is real-valued. Moreover, (10) implies that if a is locally integrable and with ultrapolynomial growth of class ∗, then 1 Aa u(x) = (2π )d

 R2d

  a(y, η) u, Gy,η Gy,η (x)dydη,

  u ∈ S ∗ Rd .

From now on we assume that Mp satisfies (M.1), (M.2) and (M.3). One verifies that (see [20, Proposition 1.7.9], [26, Proposition 3.3]) the Anti-Wick operator Aa

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with symbol a ∈ S ∗ (Rd ) is equal to the operator bw (the Weyl quantisation of b ∈ S ∗ (Rd )) where the symbol b ∈ S ∗ (Rd ) is given by convolving a with the 2 2 Gaussian e−|x| −|ξ | , i.e.   2 2 (11) b(x, ξ ) = π −d a(·, ·) ∗ e−|·| −|·| (x, ξ ). This equality together with Theorem 8, allow us to define Anti-Wick operators Aa : D∗ (Rd ) → D∗ (Rd ), even when a is not necessarily an element of S ∗ (R2d ). If ∗ (and only then) Theorem 8 implies that the ultradistribution b given by (11) a ∈ B−1 is a well defined element of A∗−1 . If this b is such that 1 (2π )d



 Rd



 Rd

Rd

ei(x−y)ξ b

x+y , ξ χ (x, y)dxdydξ 2

(12)

is well defined as oscillatory integral for every χ ∈ D∗ (R2d ), and χ → Kb , χ , D∗ (R2d ) → C, defined by the above integral is a well defined ultradistribution, then the operator associated to that kernel (cf. [18, Theorem 2.3]) ϕ → Kb (x, y), ϕ(y), D∗ (Rd ) → D∗ (Rd ), can be called the Anti-Wick operator with symbol a (because of (11), this is an appropriate generalisation of Anti-Wick operators). The next result gives an example of such symbols. ∗ is such that b, given by (11), satisfies Theorem 9 ([26, Theorem 5.1]) If a ∈ B−1 the following condition: for every K ⊂⊂ Rdx there exists r˜ > 0 such that there exist m, C1 > 0, resp. there exist C1 > 0 and (kp ) ∈ R, (in both cases C1 and m, resp. C1 and (kp ) depend on K) such that

|b(x + iη, ξ )| ≤ C1 eM(m|ξ |) , resp., |b(x + iη, ξ )| ≤ C1 eNkp (|ξ |) ,

(13)

for all x ∈ K, |η| < r˜ , ξ ∈ Rd , then (12) is an oscillatory integral and Kb , given by (12), is a well defined ultradistribution. 2

The conditions of this theorem are met by symbols a of the form el|x| P (ξ ), ∗ and (cf. where l < 1 and P (ξ ) is an ultrapolynomial of class ∗. Clearly, a ∈ B−1 Theorem 8)   2 2 b(x, ξ ) = π −d a(·, ·) ∗ e−|·| −|·| (x, ξ )  1 −|x|2 −|ξ |2  −|·|2 −|·|2 e L e a(·, ·) (−2x, −2ξ ) πd  d/2  π 1 2 2 el|x| /(1−l) e−|η| P (ξ − η)dη. = d d π 1−l R

=

One can easily verify that b satisfies the growth estimate (13) (see [26, Example 5.1]).

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415

6 An Extension of Convolution The aim of this short section is to give a slight extension of our results from Sect. 4.2. More precisely we will determine a class of super exponential type ultradistributions q {p!} from S 1/q  (Rd ) which are convolvable with the function es|x| , 1 < q < q  , {p! } s ∈ R. As we already noted, this part is independent from the previous one and opens a lot of new questions interesting for the further investigations. q

{p!}

Lemma 1 Let q  > 1. For every k > 0, e−k· ∈ S 1/q  (Rd ). Moreover, the {p! } following estimates hold true: ∃δ > 0, ∀k > 0, ∃C ≥ 1, ∃r > 0, q

q

|D α e−kx | ≤ Cr −|α| α!e−δkx , for all x ∈ Rd , α ∈ Nd . Proof For z ∈ Cd , denote z2 = z12 + . . . + zd2 . Take R ≥ 6 large enough such that q  | arg(1 + z2 )| ≤ π/3 when z ∈ W = {z ∈ Cd | |Re z| > R|Im z|}, where arg 2 q  /2 is analytic denotes the principle branch of the argument. Then z → e−k(1+z ) 2 q  /2

k

2 q  /2

on W and |e−k(1+z ) | ≤ e− 2 |1+z | , ∀z ∈ W . For each x ∈ Rd \{0}, we apply 2 q  /2 on the distinguished the Cauchy integral formula to the function z → e−k(1+z ) d | |z − x | ≤ r|x|, j = 1, . . . , d} ⊆ W , with boundary of the polydisc {z ∈ C j j √ r = 1/(2(R + 1) d), to obtain q

q

|D α (e−kx )| ≤ α!r −|α| |x|−|α| e−δkx , x ∈ Rd \{0}, q

where δ ∈ (0, 1) does not depend on x, α and k. Consequently |D α (e−kx )| ≤ q α!4|α| r −|α| e−δkx , for all |x| ≥ 1/4. As Re(1+z2 ) > 0 when |z| < 1, the function  2 q /2 z → e−k(1+z ) is analytic on {z ∈ Cd | |z| < 1} and also bounded on |z| ≤ 3/4. Thus, for |x| ≤ 1/4, we can apply the Cauchy integral√ formula on the distinguished boundary of the polydisc {z ∈ Cd | |zj − xj | ≤ 1/(4 d), j = 1, . . . , d} ⊆ {z ∈ Cd | |z| ≤ 3/4} to obtain √ q q |D α (e−kx )| ≤ Cα!(4 d/r)|α| e−δkx , and the claim in the lemma follows.

 

q {p!} The above lemma implies that for any g ∈ S 1/q  (Rd ) and k > 0, e−k· g is a {p! } q {p!} {p!} convolutor for S 1/q  (Rd ), i.e. the mapping ϕ → (e−k· g) ∗ ϕ, S 1/q  (Rd ) → {p! {p! } } {p!} S 1/q  (Rd ), is continuous (the validity of this can be easily verified for any test {p! } q q {p!} function in place of e−k· ). Consequently, as es|·| ∈ S 1/q  (Rd ) for any 0 < } {p! q q  −k· s|·| q < q , the convolution of f = e g and e is a well defined element of

416

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S. Pilipovi´c and B. Prangoski

{p!} d  (R ) {p!1/q }

and it is given by f ∗ es|·| , ϕ = es|·| , fˇ ∗ ϕ, ϕ ∈ S q

q

{p!} d  (R ). {p!1/q }

We supplement this with the following result. Theorem 10 Let 1 < q < q  , k > 0 and s ∈ R. Then, e−k(1+|·|

2 )q  /2

 s|·|q  {p!} {p!} e ∗ ϕ ∈ S 1/q  (Rd ), ∀ϕ ∈ S 1/q  (Rd ), {p!

{p!

}

}

and the mapping q

ϕ → e−kx (ϕ ∗ es|·| )(x), S q

{p!} d  (R ) {p!1/q }

→S

{p!} d  (R ), {p!1/q }

(14)

q

is continuous. Furthermore, the convolution of es|·| with f = e−k· g where g ∈ {p!} S 1/q  (Rd ), can also be given as q

{p!

}

q

f ∗ es|·| , ϕ = g, e−k· (es|·| ∗ ϕ), ϕ ∈ S q

q

{p!} d  (R ). {p!1/q }

(15)

{p!}

Proof Let B be a bounded subset of S 1/q  . There exist C ≥ 1 and r > 0 such {p! } that q

|D β ϕ(y)| ≤ Cr −|β| β!e−r|x| , for all x ∈ Rd , β ∈ Nd , ϕ ∈ B. As |x − y|q ≤ 2q |x|q + 2q |y|q , one easily verifies that there exist C ≥ 1 and c > 0 such that |D β ϕ ∗ es|·| (x)| ≤ Cr −|β| β!ec|x| , x ∈ Rd , β ∈ Nd , ϕ ∈ B. q

q

Lemma 1 implies q

{e−k· (ϕ ∗ es|·| )| ϕ ∈ B} is a bounded subset of S {p!}

q

{p!} d  (R ). {p!1/q }

(16)

Since S 1/q  (Rd ) is bornological, the mapping (14) is continuous and the proof of {p! } the first part of the theorem is complete. {p!} To prove the equality (15), first notice that for any fixed ϕ ∈ S 1/q  (Rd ) and } {p! k > 0 there exists ε > 0 such that

Convolution and Anti-Wick Quantisation

"

417

q q ε|α| e−k· D α ϕ(x − ·)eε|x|

x ∈ Rd α!

2 {p!} is a bounded subset of S{p! (Rd ). 1/q  }

(17) {p!}

{p!}

Take a sequence gn ∈ S 1/q  (Rd ), n ∈ Z+ , which converges to g in S 1/q  (Rd ) {p! {p! } } and notice that q

q

gn , e−k· (es|·| ∗ ϕ) = es|·| , (e−k· gˇ n ) ∗ ϕ. q

q

q

(18) q

The left hand side tends to g, e−k· (es|·| ∗ ϕ). As (e−k· gˇ n ) ∗ ϕ(x) = q q q gˇ n , e−k· ϕ(x − ·), (17) verifies that (e−k· gˇ n ) ∗ ϕ → (e−k· g) ˇ ∗ ϕ in {p!} S 1/q  (Rd ). Consequently, the right hand side of (18) tends to {p!

q

}

q

es|·| , (e−k· g) ˇ ∗ ϕ, q

and the proof of the theorem is complete.

 

Acknowledgements The work of Stevan Pilipovi´c was supported by the Ministry of Education and Science, Republic of Serbia, project no. 174024. The work of B. Prangoski was partially supported by the bilateral project “Microlocal analysis and applications” funded by the Macedonian and Serbian academies of sciences and arts.

References 1. C. Bargetz, N. Ortner, Convolution of vector-valued distributions: a survey and comparison, Dissertationes Math. 495 (2013) 1–51. 2. N. Bourbaki, Intégration, Ch. 7, 8: Mesure de Haar; convolution et représentation, Hermann, Paris, 1963. 3. M. Cappiello, Fourier integral operators of infinite order and applications to SG-hyperbolic equations, Tsukuba J. Math. 28 (2004), 311–361. 4. M. Cappiello, Fourier integral operators and Gelfand-Shilov spaces, In Recent Advances in Operator Theory and its Applications, 2005 (pp. 81–100), Birkhäuser Basel. 5. M. Cappiello, S. Pilipovi´c, B. Prangoski, Parametrices and hypoellipticity for pseudodifferential operators on spaces of tempered ultradistributions, J. Pseudo-Differ. Oper. Appl. 5 (2014), 491–506. 6. M. Cappiello, S. Pilipovi´c, B. Prangoski, Semilinear pseudodifferential equations in spaces of tempered ultradistributions, J. Math. Anal. Appl. 442 (2016), 317–338. 7. R. Carmichael, A. Kami´nski, S. Pilipovi´c, Boundary Values and Convolution in Ultradistribution Spaces, World Scientific Publishing Co. Pte. Ltd., 2007. 8. A. Debrouwere, J. Vindas, Topological properties of convolutor spaces via the short-time Fourier transform, preprint, arXiv:1801.09246 9. A. Debrouwere, J. Vindas, On weighted inductive limits of spaces of ultradifferentiable functions and their duals, Math. Nachr. 292 (2019), 573–602.

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10. P. Dierolf, S. Dierolf, Topological properties of the dual pair B˙ (Ω) , B˙ (Ω) , Pac. J. Math. 108 (1983), 51–82. 11. P. Dierolf, J. Voigt, Convolution and S  -convolution of distributions, Collect. Math. 29 (1978), 185–196. 12. P. Dimovski, S. Pilipovic, J. Vindas, New distribution spaces associated to translation-invariant Banach spaces, Monatsh. Math. 177 (2015), 495–515. 13. P. Dimovski, S. Pilipovi´c, B. Prangoski, J. Vindas, Convolution of ultradistributions and ultradistribution spaces associated to translation-invariant Banach spaces, Kyoto J. Math. 56 (2016), 401–440. 14. P. Dimovski, S. Pilipovi´c, B. Prangoski, J. Vindas, Translation-Modulation Invariant Banach Spaces of Ultradistributions, J. Fourier Anal. Appl. 25 (2019), 819–841. https://doi.org/10. 1007/s00041-018-9610-x 15. P. Dimovski, B. Prangoski, J. Vindas, On a class of translation-invariant spaces of quasianalytic ultradistributions, Novi Sad J. Math. 45 (2015), 143–175. 16. A. Kami´nski, D. Kovaˇcevi´c, S. Pilipovi´c, The equivalence of various defnitions of the convolution of ultradistributions, Trudy Mat. Inst. Steklov 203 (1994), 307–322 17. H. Komatsu, Ultradistributions, I: Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 20 (1973), 25–105. 18. H. Komatsu, Ultradistributions, II: The kernel theorem and ultradistributions with support in submanifold, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 24 (1977), 607–628. 19. H. Komatsu, Ultradistributions, III: Vector valued ultradistributions and the theory of kernels, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 29 (1982), 653–717. 20. F. Nicola, L. Rodino, Global Psedo-Differential Calculus on Euclidean Spaces, Vol. 4. Birkhäuser Basel, 2010. 21. N. Ortner, On convolvability conditions for distributions, Monatsh. Math. 160 (2010), 313– 335. 22. N. Ortner, P. Wagner, Distribution-Valued Analytic Functions - Theory and Applications, edition swk, Hamburg, 2013. 23. S. Pilipovi´c, On the convolution in the space of Beurling ultradistributions, Comment. Math. Univ. St. Paul 40 (1991), 15–27. 24. S. Pilipovi´c, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1191–1206. 25. S. Pilipovi´c, B. Prangoski, On the convolution of Roumieu ultradistributions through the ! tensor product, Monatsh. Math. 173 (2014), 83–105. 26. S. Pilipovi´c, B. Prangoski, Anti-Wick and Weyl quantization on ultradistribution spaces, J. Math. Pures Appl. 103 (2015), 472–503. 27. S. Pilipovi´c, B. Prangoski, Complex powers for a class of infinite order hypoelliptic operators, Dissertationes Math. 529 (2018), 1–58. 28. S. Pilipovi´c, B. Prangoski, J. Vindas, Spectral asymptotics for infinite order pseudo-differential operators, Bull. Math. Sci. 8 (2018), 81–120. 29. S. Pilipovi´c, B. Prangoski, J. Vindas, On quasianalytic classes of Gelfand Shilov type. Parametrix and convolution, J. Math. Pures Appl. 116 (2018), 174–210. 30. B. Prangoski, Laplace transform in spaces of ultradistributions, Filomat 27 (2013), 747–760. 31. B. Prangoski, Pseudodifferential operators of infinite order in spaces of tempered ultradistributions, J. Pseudo-Differ. Oper. Appl. 4 (2013), 495–549. 32. L. Schwartz, Théorie des distributions á valeurs vectorielles. I, Ann. Inst. Fourier 7 (1957), 1–141. 33. R. Shiraishi, On the definition of convolution for distributions, J. Sci. Hiroshima Univ. Ser. A 23 (1959), 19–32. 34. M. A. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer Verlag, 2001. 35. P. Wagner, Zur Faltung von Distributionen, Math. Ann. 276 (1987), 467–485.

Exact Formulas to the Solutions of Several Generalizations of the Nonlinear Schrödinger Equation Petar Popivanov and Angela Slavova

Abstract This paper deals with the nonlinear Schrödinger equation (NLSE) with logarithmic and power nonlinearities as well as with several generalizations of NLSE—Biswas-Milovi˘c equation and others. Solutions of special form are written explicitly via hyperbolic, Jacobi elliptic, Weierstrass and Legendre elliptic functions of the three kinds. Generalized (distribution) solutions for the logarithmic Schrödinger equation containing one or two delta potentials are constructed too. The Biswas-Milovi˘c equation is ill-posed in the periodic Sobolev space H s with respect to the space variable x for s < 0. Keywords Nonlinear Schrödinger equation with logarithmic and power nonlinearities · Hyperbolis functions · Jacobi elliptic functions · Weierstrass and Legendre ellipric functions · Biswas-Milovi˘c equation

1 Introduction There are many papers and monographs devoted to the nonlinear Schrödinger equation (NLSE) with power, logarithmic and other type nonlinearities (see for example [4, 6, 7, 10]). The construction of exact solutions of NLSE expressed by hyperbolic Jacobi elliptic, ℘- Weierstrass and Legendre elliptic functions of the three kinds and other elementary functions (trigonometric, algebraic, etc.) enables the researchers to understand better the physical nature of the corresponding dynamical processes arising in plasma physics and quantum mechanics.Other possible applications are in the domain of nonlinear optics. Usually travelling wave solutions are found by using the first integral approach, the simplest equation method and other direct methods [8]. At first we shall consider here the Biswas-Milovi˘c equation (BME) (see [3, 12, 13]) with power nonlinearities, i.e.

P. Popivanov · A. Slavova () Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_23

419

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P. Popivanov and A. Slavova

(Φ n )t + α(Φ n )xx + βF (|Φ|2 )Φ n = 0, Φ = Φ(t, x), n ∈ N, α, β ∈ R1 \ 0

(1)

 pk 1 and F (λ) = a0 + m k=1 ak λ , aj ∈ R , a0 = 0, am = 0, 0 < p1 < p2 < . . . < pm . ∞ 0 Thus, F ∈ C (λ > 0) ∩ C (λ ≥ 0). Our first aim is to construct in Theorem 1, Sect. 2 soliton and kink type solutions of (1) under several conditions imposed on α, β, aj , 0 ≤ j ≤ m. To do this we shall use the wave ansatz method. 1 2π Put z = Φ n in (1) ⇒ |Φ| = |z| n and therefore formally Φ = z1/n ei n j , j = 0, . . . , n − 1. Thus, (1) takes the form 2

izt + αzxx + βF (|z| n )z = 0.

(2)

We look for a solution of (2) given by the ansatz z(t, x) = V (x − ct)ei(lx+ωt) , V > 0, ξ = x − ct, c, l, ω ∈ R1 ,

(3)

i.e. 1

i

Φj = V n (x − ct)e n (lx+ωt) ei

2π n

j

, j = 0, 1, . . . , n − 1.

(4)

Substituting (3) into (2) and separating the real and imaginary parts of the corresponding expressions we come to 

2

αV − (ω + αl 2 )V + βF (V n )V = 0, V = V (ξ )

(5)

c = 2lα.

(6)



Multiplying (5) by V , integrating with respect to ξ and taking the integral constant zero we obtain the following first order nonlinear ODE 2pk

βn ak V n ω − a0 β − ) ≡ V 2 (A − H (V )), (V ) = V (l + α α pk + n 

m

2

2 2

(7)

k=1

0 where A = l 2 + ω−βa α , H (V ) = These are our assumptions:

m

k=1 bk V

2pk n

, bk =

βn ak α pk +n .

(i) A > 0 ⇐⇒ l 2 > βa0α−ω (ii) βα am > 0, i.e. bm > 0. Therefore, H (V ) ∈ C ∞ (V > 0) ∩ C 0 (V ≥ 0), H (0) = 0, H (∞) = +∞. Denote by V1 > 0 the first positive root of the equation H1 (V ) = 0, where H1 (V ) = A − H (V ).

Exact Formulas to the Solutions of Several Generalizations of the Nonlinear. . .

421

Theorem 1 Consider BME (1) under the conditions (i), (ii). Then (1) possesses classical solutions of the form (4) which are solitons if V1 > 0 is the first positive simple root of H1 (V ) = 0 and kinks if V1 > 0 is multiple root of H1 (V ) = 0,   H1 (V ) = −H (V ). Theorem 1 will be illustrated by several examples. Example 1 Let b1 > 0, i.e. βα a1 > 0, A > 0 and a2 = . . . = am = 0. Then explicit formula for the soliton solution of (1) will be given below. 2p1

2p2

Example 2 Let H (V ) = b1 V n + b2 V n , p2 = 2p1 , b2 > 0, A > 0, ak = 0 for k ≥ 3. An exact formula for the solution V (ξ ) is given. 2

Example 3 p1 = 1, p2 = 2, p3 = 3, p4 = 4, ak = 0 for k ≥ 5. In this case V n is the inverse function of Legendre elliptic functions of first, second and third type or their linear combinations. We discuss the possibility to express V by the Weierstrass ℘p function and the link between ℘ and the Jacobi elliptic functions (see [1, 11]). Assume that Φ ∈ C 2 ([0, T ] : H s (T1x )), T > 0 and T1x stands for the unit circle, while H s (T1x ) is the Sobolev space of periodic functions with period 2π . We shall prove in Proposition 1(see below) that the Cauchy problem (1) is ill posed for s < 0. At the end of the paper we study logarithmic NLSE (8) with attracting (repelling) delta potential (see [7]) and construct its generalized continuous solution explicitly. One can easily see that when several (more than one) delta potentials participate in (8) then the problem is overdetermined. This is the equation (8): iut + uxx + Buln|u| + γ δ(x − z0 )u = 0, u = u(t, x), γ ∈ R1 \ 0, B > 0(B < 0). (8) The equation (9) is a possible generalization of (8) with double delta potential: iut +uxx +Buln|u|+γ1 δ(x−z0 )+γ2 δ(x−z1 ) = 0, z1 > z0 , γ1 , γ2 ∈ R1 \0.

(9)

Theorem 2 The equation (8) possesses a continuous generalized (distribution) solution that can be written explicitly. The equation (9) is overdetermined, i.e. there is a link among z0 , z1 , γ1 = 0, γ2 = 0. We do not propose in this paper physical interpretation of NLSE, BME and (8), (9) but concentrate only on the mathematical part of the investigation.

2 Proof of Theorem 1 1. We shall prove now Theorem 1 and we shall study the examples and investigate the ill-posedness of the Cauchy problem to (1) in H s (T1x ).

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From (7) and following [2] we get that  ξ − ξ0 = P (V ) =

V V0

γ

dγ , 0 < V0 < V1 , 0 < V < V1 . √ A − H (γ )



Evidently, P (V ) > 0, 0 = P (V0 ) ⇐⇒ ξ = ξ0 as 0 < V < V1 . Moreover, V dγ > 0 for V1 —simple P (+0) = −∞, limV →V1 −0 P (V ) = P (V1 ) = V01 γ √A−H (γ ) 

root of H (γ ) = A, P (V1 ) = +∞, while P (V1 − 0) = +∞ for V1 multiple root of A = H (γ ). In the latter case V (ξ ) = P −1 (ξ − ξ0 ) is a kink. In the first case  (V1 —simple root) we put ξ¯ = ξ0 + P (V1 ), i.e. V (ξ¯ ) = V1 , V (ξ¯ ) = 0 and V is defined in the interval (−∞, ξ¯ ]. We continue V (ξ ) as a smooth solution of (7) on R1 by the formula V (ξ¯ + τ ) = V (ξ¯ − τ ), ∀τ ≥ 0. Evidently, V is a soliton solution 1 of (7) and the same is true for V n (ξ ). In fact, V ∼ V1 − (ξ − ξ¯ )2 B(ξ ), B(ξ ) > 0 1 1 ) ¯ 4 ¯ near ξ¯ ⇒ V n ∼ V1n (1 − n1 (ξ − ξ¯ )2 B(ξ V1 + O((ξ − ξ ) )) for ξ ∼ ξ . To study the Examples we shall try to find appropriate formula for the integral 

m

2pk dV , H1 (V ) = A − bk V n , V > 0, √ V H1 (V ) k=1

ξ= i.e. ξ =



dV m A− 1 bk V

B V

2pk n

=

n 2

 y

√

dy  pk A− m 1 bk y

after the change y = V

(10) 2 n

> 0.

In the case of Example 1 B n b1 V ξ = − √ arcsech A p1 A

p1 n

1

⇒ V n (ξ ) = (

√ n A A 2pn p1 ξ ). ) 1 sech p1 ( b1 n

(see [8] ) In the case of Example 2 according to formula 380.111 from [5] and for p1 = 1, p2 = 2 (2AH1 (y))1/2 + 2A − b1 y n ,y > 0 ξ = − √ ln y 2 A and H1 (y) = A − b1 y − b2 y 2 , b2 > 0, A > 0. Easy computations lead to 2

V n (ξ ) = e 1

√ − 2 nA ξ

4A + (b12

+ 4Ab2 )e

√ 2 A n ξ

. + 2b1

Certainly V n (ξ ) is soliton type solution. Otherwise, we make in (10) the change y p1 = z etc.

Exact Formulas to the Solutions of Several Generalizations of the Nonlinear. . .

423

As it concerns Example 3 the value of the integral 2ξ = n



dy

+

y A − b1 y − b2 y 2 − b3 y 3 − b4 y 4

≡ G(y), A > 0, b4 > 0

(11)

is expressed by the Legendre elliptic functions and it depends on the location of the real zeroes of the polynomial H1 (y) under the square root sign in (11) (see 3.148 2 n and 3.149 from [9]): 2ξ n = G(y), y = V . Remark 1 A larger class of solutions of (1) into explicit form can be proposed for the second order polynomial F (λ) = d0 + d1 λ + d2 λ2 , where λ = |Φ|, dj , 0 ≤ j ≤ 2 being real constants. Taking the ansatz Φ = V (x − ct)ei(lx+ωt) , i.e. Φ n = V n ein(lx+ωt) we deduce easily from (1) that c = 2αnl

(12)





αnV V + n(n − 1)α(V ) + βd1 V + βd2 V − V (nω + αn l − βd0 ) = 0. 2

3

4

2

2 2

Applying the method of the simplest equation with V = a0 + a1 F , where a0 , a1     are real parameters we have: V = a1 F , V = a1 F . The function F (ξ ) is assumed to satisfy the first order nonlinear ODE 

(F )2 = c4 + 4c3 F + 6c2 F 2 + 4c1 F 3 + c0 F 4 ≡ P4 (F )

(13)

and ci , i = 0, 1, 2, 3, 4 are real constants. Certainly, 

F = 2c3 + 6c2 F + 6c1 F 2 + 4c0 F 3 .

(14)

Substituting V = a0 + a1 F in (12) and having in mind (13), (14) we get a fourth order polynomial with respect to F in the left-hand side of (12). Equalizing to 0 the coefficients in front of F j , 0 ≤ j ≤ 4 we obtain a a linear algebraic 5 × 5 system for the coefficients c0 , c1 , . . . , c4 . Solving this system we express cj , 0 ≤ j ≤ 4 by the parameters a0 , a1 , d0 , d1 , d2 , ω and l. The dependence of cj on the parameters is nonlinear. It is well known (see [11]) that the general solution of (13) is given by the Weierstrass ℘ (ξ ) function. More precisely, denote by ℘ (ξ, g2 , g3 ) the Weierstrass function which satisfies the ODE 

(℘ )2 = 4℘ 3 − g2 ℘ − g3 ; g2 = c0 c4 − 4c1 c3 + 3c22 , g3 = c0 c2 c4 + 2c1 c2 c3 − c23 − c0 c32 − c12 c4

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P. Popivanov and A. Slavova

are the so called invariants of P4 . According to [11] the general solution of (13) can be written as √ F (ξ ) = c +





(iv) 1  1 24 P4 (c)) + 24 P4 (c)P4 (c) , (iv) 1  1 2 24 P4 (c)) − 48 P4 (c)P4 (c)

P4 (c)℘ + 12 P4 (c)(℘ − 2(℘ −

(15)



4 where c = const., P4 = dP dF .  Assume that F0 is a simple root of P4 (F ) = 0, i.e. P4 (F0 ) = 0, P4 (F0 ) = 0. Then (15) takes the form 

P4 (F0 ) 1 F (ξ ) = F0 + . 1  4 ℘ (ξ, g2 , g3 ) − 24 P4 (F0 )

(16)

Let Δ = g22 − 27g32 be the discriminant of the cubic polynomial R3 (F ) =  4F 3 − g2 F − g3 , (℘ )2 = R3 (℘). The polynomials P4 and R3 do not have multiple roots if and only if Δ = 0. In each such case the Weierstrass function ℘ (ξ, g2 , g3 ) participating in (16) can be expressed via Jacobi sn and cn elliptic functions (see [1]). Then two cases appear: (a) if Δ > 0 then R3 (F ) = 0 possesses three simple real roots e1 > e2 > e3 . 3 The corresponding Weierstrass function ℘ (ξ, g2 , g3 ) = e3 + sn2 (√ee1 −e , −e ξ,m) 1

3

3 0 < m < ee21 −e −e3 (b) Δ < 0 implies that R3 (F ) = 0 has one simple real root e2 and two complex conjugated roots e1 , e3 , e¯1 = e3 , I me1 = 0. Then

√ 1 + cn(2ξ H2 , m) , ℘ (ξ, g2 , g3 ) = e2 + H2 √ 1 − cn(2ξ H2 , m) : where H2 = 3e22 − g42 , m = 2Re e1 = −e2 ⇒ 0 < m < 1.

1 2

−

3 e2 4 H2 ;

0 < H2 =

:

2e22 + |e1 |2 ≥ 32 |e2 |, as

Acting formally in the case (a) we put m = 1 ⇐⇒ e1 = e2 = e3 . Let e1 = e2 = c > 0 ⇒ e3 = −2c. Thus, g2 = 12c2 , g3 = −8c3 ⇒ ℘ (ξ, 12c2 , −8c3 ) = 3c √ −2c + . In a similar way m = 0 ⇐⇒ e2 = e3 = e1 . Let e1 = 2 (tgh) (ξ 3c)

2c ⇒ e2 = e3 = −c < 0.Then g2 = 12c2 , g3 = 8c3 ⇒ ℘ (ξ, 12c2 , 8c3 ) = √ 1 −c + 2 3c√ = −c + 3c ∞ j =−∞ (ξ 3c−j π )2 . sin (ξ 3c) Suppose that c0 < 0 and e1 > e2 > e3 > e4 are simple roots of P4 (F ) = 0. Then (13) possesses the periodic solution F (ξ ) = e1 −

(e1 − e2 )(e3 − e4 ) (e1 − e4 )(e1 − e3 ) , ,m = (e1 − e3 )(e2 − e4 ) e1 − e3 + (e3 − e4 )sn2 (τ, m)

Exact Formulas to the Solutions of Several Generalizations of the Nonlinear. . .

425

√ m ∈ (0, 1), τ = 12 (e1 − e3 )(e2 − e4 )|c0 |ξ . If e1 > e2 = e3 > e4 , then F (ξ ) = √ −e4 )(e1 −e2 ) e1 − e −e(e1+(e , τ = 12 (e1 − e2 )(e2 − e4 )|c0 |ξ as, m = 1. −e )tgh2 (τ ) 1

2

2

4

2. Consider now Cauchy problem (IVP) for (1) with initial data Φ|t=0 = Φ0 (x) ∈  ). Each periodic distribution u ∈ D (T1x ) can be developed in Fourier H s (T1x ij x and the corresponding Sobolev norm in H s is ||u||2 = series ∞ s j =−∞ aj e ∞ 2 s 2 2 j =−∞ (1 + j ) |aj | . We shall work in the functional class u ∈ C ([0, T ] : H s (T1x )) = XT , T > 0. Definition 1 The Cauchy problem for (1) is called locally well posed in H s (T1x ) if for Φ0 ∈ H s there exists 0 < T = T (||Φ0 ||s ) and a unique solution Φ of the corresponding IVP to (1) and such that (c) Φ ∈ XT (d) the mapping H s - Φ0 (x) → Φ(t, x) ∈ XT is uniformly continuous, i.e. for each ε > 0 there exists δ(ε, M) > 0 and such that ||Φ01 − Φ02 ||H s (T1 ) < δ, ||Φ0i ||H s < M = const. imply ||Φ 1 − Φ 2 ||XT < ε. The condition (c) means that the IVP under consideration possesses a unique solution in XT for some time interval [0, T ), Φ(t, x) is a continuous curve in H s (T1x ) originating in Φ0 and the mapping Φ0 (x) → Φ(t, x) depends continuously upon the initial data Φ0 . Proposition 1 The IVP for the equation (1) is locally ill-posed for Φ0 ∈ H s (T1x ) with s < 0. Proof We will show that the mapping u0 → u(t, .) is not uniformly continuous violating this way condition (d) of Definition 1. There are no difficulties to see that (1) possesses the following two-paramtric family of smooth solutions (travelling waves): ΦA,C = Cei(Ax+Bt) , C > 0, A ∈ N, B ∈ R1 ,

(17)

where βF (C 2 ) = nB + αn2 A2 , i.e. ΦA,C (t, x) = CeiAx+it

βF (C 2 )−n2 A2 α n

, ΦA,C,0 (x) = ΦA,C (0, x) = CeiAx .

(18)

Denote C = aA−s , a > 0 and for a˜ 2 > a˜ 1 fixed put C1 = a˜ 1 A−s , C2 = a˜ 2 A−s . 2 Evidently, for each t ∈ R1 ||ΦA,C ||2H s (T1 ) = C 2 (1 + A2 )s ≤ a 2 as 1 ≤ 1+A ≤2 A2 x for A ≥ 1. On the other hand, ||ΦA,C1 (t, .) − ΦA,C2 (t, .)||2H s (T1 ) = ( x

tβ

2

2

2s |a˜ 1 − a˜ 2 ei n (F (C2 )−F (C1 )) |2

tβ 1 + A2 s 2 2 ) |a˜ 1 − a˜ 2 ei n (F (C2 )−F (C1 )) |2 ≥ A2 (19)

426

P. Popivanov and A. Slavova

and ||ΦA,C1 (0, x) − ΦA,C2 (0, x)||2H s = |C1 − C2 |2 (1 + A2 )s ≤ |a˜ 1 − a˜ 2 |2 .

(20)

Certainly, F (a˜ 22 A−2s ) − F (a˜ 12 A−2s ) ∼ am A−2pm s (a˜ 2

2pm

for A → ∞. Taking tA =

2pm

(a˜ 2

π A2pm s n 2p −a˜ 1 m )|am ||β|

2pm

− a˜ 1

)

we have that tA → +0 , A → ∞ and

tA βn (F (C22 ) − F (C12 )) ∼ π sgn(am β), am = 0, for A → ∞. Therefore, limA→∞ ||ΦA,C1 (tA ) − ΦA,C2 (tA )||2H s (T1 ) ≥ 2s |a˜ 1 − a˜ 2 eiπ sgn(am β) |2 = 2s |a˜ 1 + a˜ 2 |2 x

Combining (19), (20) we prove Proposition 1.

3 Proof of Theorem 2 Put u = eiωt ϕ(x), 0 < ϕ(x), ϕ ∈ C 0 (R1 ), ω ∈ R1 in (8). Then ϕ(x) satisfies the nonlinear ODE 

ϕ + Bϕlnϕ − ωϕ + γ δ(x − z0 )ϕ = 0.

(21)

We shall begin with the classical nonlinear ODE 

y + Byln y + Dy = 0, B = const = 0, D = −ω

(22)

looking for a solution y ∈ C 2 (R1 ), y(x) > 0. Evidently, 

(y )2 = y 2 (

B − D − Bln y) ≥ 0. 2

:  We shall suppose that y = y B2 − D − Bln y, y > 0. Integrating this ODE with separate variables we conclude that B

1

D

y(x) = Ae− 4 (x−x0 ) , A = e 2 − B > 0, x0 ∈ R1 . 2

(23)

Thus, for B > 0 the general solution (23) of (22) is rapidly decreasing at x = ±∞, while y(x) is rapidly increasing at ±∞ for B < 0. To fix the ideas we shall study B 2 the case B < 0 only. Denote by Γ1 the integral curve y = Ae− 4 (x−x0 ) of (22) and

Exact Formulas to the Solutions of Several Generalizations of the Nonlinear. . .

427

− 16 (x1 −x0 ) 1 by Γ2 : y = Ae−B/4(x−x1 ) , x0 = x1 . Then Γ1 ∩ Γ2 = {( x0 +x )}. 2 , Ae x0 +x1 Suppose that z0 = 2 ( z0 participates in (21)), x0 < x1 . Define Γ3 : y = " B 2 e− 4 (x−x0 ) , x ≤ z0 A − B (x−x )2 . Certainly, y ∈ C 0 (R1 )∩C 1 (R1 \{z0 }) and y satisfies (22) 1 , x ≥ z . e 4 0    for x = z0 . The jump of y at z0 , where Γ3 is given by y is: l = y (z0 + 0) − y (z0 − B 2 − 16 (x1 −x0 ) = B2 y(z0 )(x1 − x0 ) = B2 (x1 − x0 )δ(z − z0 )y(x); 0) = AB 2 (x1 − x0 )e l < 0. " B 2  e− 4 (x−x0 ) , x ≥ z0 . Then the jump of y at z0 is (−l). In Put Γ4 : y = A − B (x−x )2 1 4 e , x ≤ z0 .  distribution sense, i.e. in D (R1 ), we have that according to the jump formula for       Γ3 : y = {y }a.e. as y ∈ C 0 (R1 ), while y = {y }a.e. + (y (z0 + 0) − y (z0 −  B 0))δ(x − z0 ) = {y }a.e. + 2 (x1 − x0 )δ(x − z0 )y(x). Consequently, Γ3 satisfies in  D the equation B

2





y + Bln y + Dy = {y + Byln y + Dy}a.e. +

B (x1 − x0 )δ(x − z0 )y. 2

2

(24)

The equation (24) coincides with (21) if γ = − B2 (x1 − x0 ) > 0. In a similar way Γ4 satisfies (21) if γ = B2 (x1 − x0 ) < 0. In the case γ > 0, i.e. Γ3 ,

x1 + x0 = 2z0

x − x = − 2γ 1 0 B



x = z0 − ⇐⇒

1 x0 = z0 +

γ B γ B

> z0 < z0 ,

(25)

i.e. x0 , x1 are determined by γ and z0 and " uniquely γ 2 − B4 (x−z0 − B ) e , x ≤ z0 . Γ3 : y = A − B (x−x + γ )2 0 4 B e , x ≥ z0 . Geometrically: Through each point in the half plane y > A are passing two and only two smooth integral curves of (22). Combining them in Γ3 , Γ4 we obtain two  generalized solutions of (21). One of them has jump of y at z0 equal to l < 0 and  the other - equal to (−l) > 0. There are no other solutions with jumps of y at z0 . Γ3 and Γ4 are peakon type solutions to the logarithmic NLSE with a delta potential. Consider now the equation (9) with two delta potentials for which we suppose that γ1 > 0, γ2 < 0, z0 < z1 < x1 = z0 − γB1 , x0 = z0 + γB1 . We repeat the same procedure as before with the solution ⎧ B − (x−x0 )2 ⎪ , x ≤ z0 ⎨e 4 B 2 Γ5 : y = A e− 4 (x−x1 ) , z0 ≤ x ≤ z1 . ⎪ ⎩ − B (x−z2 )2 e 4 , x ≥ z1

428

P. Popivanov and A. Slavova

Fig. 1 Solution of (9) with A = 1, B = −1, x0 = 0, x1 = 4, z0 = 2, z2 = 3, z1 = 3.5 1 Then y ∈ C 0 (R1 ) as z1 = z2 +x ⇐⇒ z1 − x1 = z2 − z1 , y(z0 ) > y(z1 ) > 0 2   and 0 < y (z1 + 0) − y (z1 − 0) = (z2 − x1 )y(z1 ) B2 . Consequently, Γ5 satisfies (9) iff B2 (z2 − x1 ) = −γ2 > 0, i.e.



z2 = z1 − γ2 < z1 B

x1 = z1 + γ2 > z1 B Geometrically we have Fig. 1. According to (25), (26) x1 = z0 −

γ1 B

= z1 +

z1 = z0 −

γ2 B

γ1 + γ2 . B

(26)

⇐⇒ (27)

Evidently, z1 > z0 ⇐⇒ γ1 + γ2 > 0. Therefore, z0 < z1 , γ1 > 0, γ2 < 0, 2 B < 0 are not independent. There is link among them, namely z1 = z0 − γ1 +γ and B γ1 + γ2 > 0. This way we complete the proof of Theorem 2.

References 1. M.Abramowitz, I.Segun. Handbook of Mathematical Functions. Dover, NY, 1972. 2. V. Arnol’d. Ordinary Differential Equations. Springer, Berlin, 1992. 3. A. Biswas, M. Mirzazadeh, M.Eslami, D.Milovi˘c and M. Beli˘c. Solitons in optical metamaterials by functional variable method and first integral approach. Frequenz 68 (2014), no.11/12, 525–530. 4. J.Bourgain. Nonlinear Schrödinger equations. IAS/Park City Math.Ser., Hyperbolic equations and frequency interactions, L.Cafarelli and E. Weiman editors AMS, 5 (1999), 1–157.

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5. H.Dwight. Tables of Integrals and Other Mathematical Data. McMilan company, NY, 1961. 6. J.Kutz. Mode locked soliton lasers. SIAM review 48 (2006), no.4, 629–678. 7. J.Pava and A.Ardila. Stability of standing waves for the logarithmic Schrödinger equation with attractive delta potentials. Indiana Univ.MathJ. 67 (2018), no.2, 479–493. 8. P.Popivanov and A.Slavova. Nonlinear Waves: A Geometrical Approach. World Scientific, New Jersey, London, Tokyo, 2018. 9. I.Rijik and M.Gradstein. Tables of Integrals, Series and Products. Academic Press, NY, 1980. 10. T.Tao. Nonlinear Dispersive Equations. Local and Global Analysis. CBMS, Regional conference series in Math., 106, AMS, 2006. 11. E.Whittaker and G.Watson. A Course of Modern Analysis. Cambridge Univ.Press, Cambridge, 1927. 12. Q.Zhou. Optical solutions for Biswas-Milovi˘c model with Kerr law and parabolic nonlinearities. Nonlinear Dynamics 84 (2016), no.2, 677–681. 13. Q.Zhou, D.Yao andF.Chen. Analytical study of optical solutions with Kerr and parabolic law nonlinearities. Journal of Modern Optics 60 (2013), no.19, 1652–1657.

Dirichlet-to-Neumann Operator and Zaremba Problem B.-W. Schulze

Abstract Starting with the Zaremba problem for the Laplacian, a boundary value problem with jumping conditions from Dirichlet to Neumann data or also with discontinuous Dirichlet- or Neumann data, a reduction to the boundary in terms of Boutet de Monvel’s calculus gives rise to an interface problem which can be interpreted as a boundary value problem on the Neumann side for the Dirichlet-toNeumann operator. This is a first order elliptic classical pseudo-differential operator on the boundary without the transmission property at the interface. A specific choice of edge quantization admits an interpretation within the edge calculus, and we apply the formalism of the edge algebra together with interface conditions. Keywords Dirichlet-to-Neumann operator · Zaremba problem · Edge quantization · Operator-valued symbols · Weighted edge spaces · Interface problems · Parametrices within the operator algebra

1 Introduction We outline here ideas of the paper [3] jointly with D.-C. Chang and N. Habal on the Laplacian close to an interface Z of codimension 1 on the boundary Y of a smooth domain where we consider Dirichlet- and Neumann boundary conditions. More precisely, Z subdivides Y into a Dirichlet part YD and a Neumann part YN , cf. also [34] and numerous other contributions on this topic. It is well-known, that reducing the corresponding mixed problem {D, N } with jumping boundary conditions from Dirichlet to Neumann across Z gives rise to a classical pseudo-differential operator R of first order on the Neumann side YN . For some chosen sufficiently large Sobolev smoothness s (which can be interpreted later on as a weight) the transformation {D, N } → R is arranged by using techniques from Boutet de Monvel’s algebra [2], B.-W. Schulze () Institut für Mathematik, Universität Potsdam, Potsdam, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_24

431

432

B.-W. Schulze

in such a way that R can be computed in terms of the original mixed boundary value problem and vice versa. The interface Z is in fact the boundary of YN , and the transmission property at Z is violated by the operator R. Since the model cone of local wedges close to Z coincides with the closed half-axis the edge calculus for classical pseudo-differential μ operators in Lcl (Y )|Y \YD works, though such operators do not appear in edgedegenerate form, at least at first glance. However, R admits such an interpretation, according to an observation in joint work with J. Seiler [29, 30]. Thus, for weights = 1/2 modulo Z the operator R ∈ L1cl (Y )|Y \YD may be treated as an upper left corner of an elliptic operator in the edge calculus, consisting of 2×2 block matrices, also containing extra trace and potential operators with respect to Z. Those may be chosen as elliptic boundary conditions at Z, since for R an analogue of the AtiyahBott obstruction for their existence vanishes, cf. also [16]. At the end we indicate some new conclusions on how the original Zaremba problem may be enriched by extra conditions referring to Z and to formulate a larger block matrix operator which is Fredholm between corresponding direct sums of spaces and how we may obtain a parametrix of a similar structure. Since the constructions are close to notions of the edge calculus, we might shift weights to smaller values with a control of extra smoothing contributions. However, this would be a larger work which deserves more effort. At the very end we indicate an alternative approach which is more close to an edge calculus for a manifold with edge and boundary. Precise conclusions also would require more work.

2 Reduction of Neumann Conditions to the Boundary Let G be the closure of a smooth bounded domain in Rn . We reduce boundary problems in G to the boundary in terms of Boutet de Monvel’s calculus. Let Ai = t (A T ), i = 0, 1, denote the column matrix operators representing two elliptic i BVPs for an elliptic operator A with trace (or boundary) operators representing boundary conditions satisfying the Shapiro-Lopatinskii condition. In order to illustrate ideas we assume that A is a second order elliptic differential operator in Rn with smooth coefficients, in the simplest case the Laplacian and T0 u := u|∂G and T1 u := ∂ν u|∂G are Dirichlet and Neumann conditions, respectively, with ∂ν being the derivative normal to the boundary. Let us start with the assumption that A0 is invertible, say, as an operator C ∞ (G) A0 : C (G) → ⊕ C ∞ (∂G) ∞

Dirichlet-to-Neumann Operator and Zaremba Problem

433

which is a classical property for the Laplacian, see also [31], here taken as the main example. By P0 = (P0

K0 )

we denote its inverse which belongs to Boutet de Monvel’s calculus, cf. [23]. Then we have AP0 = 1, AK0 = 0 which entails the relation  A1 P0 =

1 0 . T1 P0 T1 K0

(1)

The composition may be interpreted in Boutet de Monvel’s algebra [2]. Basics on this calculus may be found in [8, 17] and [10, 16]. In particular, lower right corners are classical pseudo-differential operators on the boundary, cf. [11]. Other structures on boundary value problems of more general kind are developed in [4, 10, 28], or [16]. In the present special case we have R := T1 K0 ∈ L1cl (∂G),

σψ (R)(η) = c|η|

(2)

for a constant c ; here σψ (·) denotes the homogeneous principal symbol of the respective operator and η the covariable on ∂G. Remark 1 Let u ∈ C ∞ (G) be harmonic in int G and let uD := u|∂D ∈ C ∞ (∂G) be its Dirichlet data and uN := ∂ν u|∂G ∈ C ∞ (∂G) its Neumann data. Then R induces a map R : uD → uN . Therefore, R is also called the Dirichlet-to-Neumann operator. Let R (−1) be a parametrix of R. Then we obtain a parametrix of A1 P0 , namely,  (A1 P0 )(−1) =

1 0 T1 P0 T1 K0

(−1)

 =

1

0

−R (−1) T1 P0 R (−1)

,

(−1)

and we can construct a parametrix P1 := A1 of the Neumann problem A1 in the form  1 0 (−1) P1 = P0 (A1 P0 ) = (P0 K0 ) −R (−1) T1 P0 R (−1) (3) = (P0 − K0 R (−1) T1 P0

K0 R (−1) ) =: (P1

K1 ).

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B.-W. Schulze

In the considerations below we interpret A0 and A1 as continuous operators  H s−2 (int G) A s : H (int G) → A0 = , ⊕ T0 H s−1/2 (∂G)  H s−2 (int G) A s , A1 = : H (int G) → ⊕ T1 H s−3/2 (∂G) s (U )| first for s > 3/2. Here H s (int G) := Hloc int G for some open neighborhood U s of G. The notation H (int M) for Sobolev spaces on a compact manifold M with boundary will be employed in a similar meaning, where the double 2M may be taken instead of U ; then “loc may be dropped.

3 Reduction of Mixed Problems to the Boundary Let us set Y := ∂G and let Z be an interface on Y of codimension 1 on the boundary which subdivides Y into a Dirichlet part Y− and a Neumann part Y+ which are submanifolds of Y with common boundary Z = Y− ∩ Y+ . Dirichlet and Neumann conditions D∓ and N∓ on int Y∓ are now interpreted as continuous operators D∓ = r∓ T0 : H s (int G) → H s−1/2 (int Y∓ ),

(4)

N∓ = r∓ T1 : H s (int G) → H s−3/2 (int Y∓ ),

(5)

and

respectively. In corresponding restriction operators r∓ of distributions to int Y∓ we assume s > 3/2, since in parametrices of operators containing the Laplacian we employ the condition s − 2 > −1/2. We interpret the Zaremba problem as a continuous operator H s−2 (int G) A ⊕ Am = ⎝D− ⎠ : H s (int G) → H s−1/2 (int Y− ) N+ ⊕ H s−3/2 (int Y+ ). ⎛

⎞

(6)

Dirichlet-to-Neumann Operator and Zaremba Problem

435

Also Dirichlet and Neumann conditions, independently posed over Y∓ , give rise to mixed problems, namely, ⎛

⎛ ⎞ ⎞ A A Ad = ⎝D− ⎠ , An = ⎝N− ⎠ . D+ N+

(7)

Those induce continuous operators H s−2 (int G) ⊕ Ad : H s (int G) → H s−1/2 (int Y− ) , ⊕ H s−1/2 (int Y+ )

(8)

H s−2 (int G) ⊕ An : H s (int G) → H s−3/2 (int Y− ) . ⊕ H s−3/2 (int Y+ )

(9)

We now reduce the operator Am to the boundary by means of Ad . Here we proceed similarly as before in producing the operator R. We first consider an analogue A˜ 0 of the above-mentioned operator A0 , i.e., we set H s−2 (int G) A 0 ⊕ ˜ 0 := ⎝T0 ⎠: A → 0 ⊕ H s−1/2 (Y ) 0 idH s−1/2 (CL(s−1/2) ) ⊕ H s−1/2 (Z, CL(s−1/2) ) H s−1/2 (Z, CL(s−1/2) ) (10) ⎛

⎞

H s (int G)

which is an isomorphism, since A0 itself is an isomorphism. Concerning notation we refer to Lemma 1 and the subsequent constructions below. Another useful isomorphism is an expression on cutting and pasting Sobolev-distributions on Y along the interface Z, namely, for s > 1, s − 1 ∈ /N ⎛ D0 := ⎝

idH s−2 (int G) 0 0

H s−2 (int G) H s−2 (int G) 0 0 ⊕ ⊕ s−1/2 s−1/2 − ⎠ : → (Y ) (int Y− ). H H r d− + ⊕ ⊕ r d+ H s−1/2 (Z, CL(s−1/2) ) H s−1/2 (int Y+ ) ⎞

(11)

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B.-W. Schulze

˜ 0 it follows that For AD := D0 A H s−2 (int G) A 0 H s (int G) ⊕ ⎠ ⎝ AD = D− d− : → H s−1/2 (int Y− ). ⊕ D+ d+ ⊕ H s−1/2 (Z, CL(s−1/2) ) H s−1/2 (int Y+ ) ⎞

⎛

(12)

˜ 0 = A0 ⊕ idH s−1/2 (Z,CL(s−1/2) ) is essentially the Dirichlet problem. The operator A ˜ 0 contains Ad and the extra potential operators d± . Let us write The operator D0 A PD :=

A−1 D

  K0 e+ P0 K0 e− P0 C− C+ d d = . =: 0 c− c+ 0 c− c+

(13)

The involved operators are defined by the following cutting and pasting relation. Lemma 1 Let Y = Y− ∪ Y+ be as before, and let s ∈ R, s > 1/2, s − 1/2 ∈ / N, L(s) := #{l ∈ N : l + 1/2 < s}. Then we have an isomorphism − r r+

d− d+



H s (Y ) H s (int Y− ) : → ⊕ ⊕ s L(s) s H (Z, C ) H (int Y+ )

(14)

where r∓ : H s (Y ) → H s (int Y∓ ) are the respective restriction operators and d∓ are potential operators from Boutet de Monvel’s version of transmission problems. In expression (13) for PD we employ notation 

r− r+

d− d+

−1 =

 − + ed ed . c− c+

(15)

−1

˜ 0 D−1 then follows from The expression for PD = A 0 −1 ˜ −1 A 0 D0

⎞ ⎛ idH s−2 (int G) 0 0 0 P0 K0 +⎠ . ⎝ = 0 e− d ed 0 0 idH s−1/2 (Z,CL(s−1/2) ) 0 c− c+ 

In particular we have continuous operators s−1/2 C∓ = K0 e∓ (int Y∓ ) → H s (int G). d :H

(16)

Dirichlet-to-Neumann Operator and Zaremba Problem

437

Thus ⎛

⎞ ⎞ ⎛ AP0 100 AC− AC+ AD PD = ⎝D− P0 D− C− + d− c− D− C+ + d− c+ ⎠ = ⎝0 1 0⎠ . 001 D+ P0 D+ C− + d+ c− D+ C+ + d+ c+

(17)

We also rephrase Am and form the operator H s−2 (int G) A 0 ⊕ := ⎝D− d− ⎠ : → H s−1/2 (int Y− ). ⊕ N+ 0 ⊕ H s−1/2 (Z, CL(s−1/2) ) H s−3/2 (int Y+ ) ⎛

AM

⎞

H s (int G)

(18)

for N+ := r+ T1 . The extra operator d− has been added for algebraic reasons. It has the form of an edge condition of potential type where Z plays the role of an edge. It follows that ⎛

⎞ ⎛ ⎞ AP0 1 0 0 AC− AC+ ⎜ ⎟ ⎜ ⎟ AM PD = ⎝D− P0 D− C− + d− c− D− C+ + d− c+ ⎠ = ⎝ 0 1 0 ⎠. N+ P 0 N+ C − N+ C + N+ P 0 N+ C − N+ C +

(19) Here N+ C+ : H s−1/2 (int Y+ ) → H s−3/2 (int Y+ ),

(20)

cf. notation in (5) and (13). We give a characterization of the operator N+ C+ in terms of the former R and some smoothing contributions which allows us to interpret N+ C+ as an operator in the edge calculus. Let us we formulate some observation on Sobolev spaces on a smooth compact manifold M with boundary ∂M. Fix an s ∈ R, s > 1/2, s − 1/2 ∈ / N, and set L(s) : #{l ∈ N : l + 1/2 < s.} Let us set Hθs (int M) := {u ∈ H s (2M)|int M :Dxα u|∂M=0 = 0 for all |α| < s − 1/2}. Then there is an isomorphism   Es Ks :

Hθs (int M) → H s (int M) ⊕ H s (∂M, CL(s) )

(21)

438

B.-W. Schulze

for the canonical embedding Es : Hθs (int M) → H s (int M) and a potential operator Ks : H s (∂M, CL(s) ) → H s (int M) of Boutet de Monvel’s calculus. The inverse  −1  Ps = Es Ks Ts

(22)

consists of a projection Ps : H s (int M) → Hθs (int M) along im Ks and a vector Ts of trace operators of Boutet de Monvel’s type.

4 The Relationship to the Edge Calculus The following considerations give a crude impression on several tools on the edge calculus. More selfcontained material may be found, for instance, in [6, 7], and the references there. Several essential contributions also have been developed in joint papers with Schrohe in [19, 19] and also in [21, 22]. We employ (21) and (22) for s −1/2 and s −3/2, respectively, interpret M = Y+ s−1/2 as a manifold with edge Z, and identify Hθ (int Y+ ) with the weighted edge s−3/2 (int Y+ ) with H s−3/2,s−3/2 (int Y+ ). In this space H s−1/2,s−1/2 (Y+ ) and Hθ notation we employ weighted edge Sobolev spaces H s,γ (M) on a compact manifold M with boundary Z of dimension q = n − 2, defined by s H s,γ (M) = {u ∈ Hloc (M \ Z) : ϕu ∈ H s (2M) for every ϕ ∈ C0∞ (M \ Z),

ωu ∈ Ws (Rq , Ks,γ (R+ )) for every cut-off function ω}, (23) cf. also the definition of edge spaces in formula (29) below and Kegel spaces (31). Here M is described close to Z in local coordinates by (z, r) ∈ Rq × R+ and ω(r) is a cut-off function on R+ (i.e., ≡ 1 close to r = 0, and ≡ 0 off some small neighborhood of r = 0). The space of -in general- edge-degenerate pseudodifferential operators of order μ on a manifold M with edge Z is here denoted by Lμ (M, g)

(24)

with g := (γ , γ − μ, Θ) being weight data, for some prescribed weight γ ∈ R and a half-open weight interval Θ := (ϑ, 0] for some −∞ ≤ ϑ < 0 which indicates the width of a strip on the left of a weight line Γβ := {w ∈ C : Re w = β} in the complex plane Cw of the covariable w belonging to the Mellin transform M, transforming functions on R+ in the variable r to functions in w. The corresponding weight strips {w ∈ C : β + ϑ < Re w < β} either for β = γ or β = γ − μ are just the areas in the complex Mellin w-plane, where the edge calculus controls asymptotics of distributions or of Green kernels in terms of poles of meromorphic functions, playing a role in the edge calculus.

Dirichlet-to-Neumann Operator and Zaremba Problem

439

The space (24) is characterized as the set of all operators   + (1 − ωglob )Aint (1 − ωglob )+C A = ωglob Ac ωglob

(25)

for some C ∈ L−∞ (M, g), which implies the continuities C : H s,γ (M) → H ∞,γ −μ+ε (M),

C ∗ : H s,−γ +μ (M) → H ∞,−γ +ε (M) (26)

for all s and some ε = ε(C) > 0 with ∗ indicating the formal H 0,0 (M)-adjoint. When we control asymptotics in the above-mentioned weight strips, indicated by Θ, then the operators (26) are asked to map to subspaces with with discrete/continuous asymptotics, see [27, subsection 2.3]. A complete description of details would exceed the possibilities of the present outline, also since the mentioned features appear in several variants. Therefore, we refer to systematic expositions, e.g., in [15, 25, 27], or [6, 7]. Nevertheless, in order to give some idea, we will sketch some more material below. μ Moreover, Aint ∈ Lcl (M \ Z), while Ac is locally close to the edge Z defined by Opy (a) for an edge amplitude function a(y, η) as is defined for d = 0 below by expressions of the kind (42) and (51), respectively. Finally, writing ϕ ≺ ψ when two real-valued functions ϕ, ψ have the property ψ ≡ 1 on supp ϕ, we assume that   ωglob ≺ ωglob ≺ ωglob

are global cut-off functions on M with respect to Z, which are ≡ 1 close to Z and ≡ 0 off some other small neighborhood of Z. Theorem 1 ([3]) The operator (20) induces an element Ps−3/2 N+ C+ Es−1/2 : H s−1/2,s−1/2 (Y+ ) → H s−3/2,s−3/2 (Y+ )

(27)

in the edge calculus L1 (Y+ , g) for the weight data g := (s − 1/2, s − 3/2, Θ), with the weight interval Θ = (−∞, 0]. The latter theorem is based on the fact that pseudo-differential boundary value problems on a smooth manifold M with boundary may be interpreted as a substructure of the edge calculus, cf. [29], where M is treated as a manifold with edge ∂M, and the operators have not necessarily the transmission property at the boundary, cf. [29, 30]. We will sketch this case of edge-analysis below. The edge calculus in general form, originally introduced in [24], see also special cases in [18] and structures from [5] on Mellin operators on the half-axis in L2 spaces, and then considerably deepened in [6, 7], refers to a manifold M with edge Z of any codimension. The half-axis R+ is replaced by a non-trivial cone X := (R+ × X)/({0} × X). In computations X is often replaced by the open stretched cone X∧ := R+ × X. Moreover, the involved operators are assumed to be

440

B.-W. Schulze

edge-degenerate, and in simplest cases the base X is closed and smooth. Other cases concern base manifolds X which are smooth, compact, with boundary, cf. [19, 20], or [12, 14]. In order to avoid the full complexity of the edge formalism, we first consider the case of a manifold M with smooth boundary of dimension q, locally close to ∂M modeled on R+ × Rq . Let us set μ

μ

Lcl (M)smooth := {A ∈ Lcl (int M) ˜ int M + C, A˜ ∈ Lμ (2M), C ∈ L−∞ (int M)}, : A = A| cl

(28)

with L−∞ (int M) being the space of integral operators the kernels of which belong to C ∞ (int M × int M). Note that operators of this class are more specific than those of the edge calculus in this situation, since edge operators may be edge degenerate at the boundary. Nevertheless, the local symbols involved in (28) may be transformed into the edge-degenerate form which is the case in one-dimensional model cones of local wedges. Recall that this case has been studied already in joint work with Seiler [29], see also [30] which is an update version of [29]. One of the general observations which illustrate truncated operators modulo smoothing contributions (though much more is possible through joint work with Seiler) in terms of edge degenerate symbols is as follows. Remark 2 ([29]) For every μ

A ∈ Lcl (M)smooth ,

μ ∈ R,

and weight data g := (γ , γ − μ, Θ) for any fixed γ ∈ R there exists a Cμ,γ ∈ L−∞ (int M) such that A − Cμ,γ ∈ Lμ (M, g). In other words there are edge quantizations μ

Lcl (M)smooth → Lμ (M, g) on a manifold M with boundary. No transmission property of A is involved here, and resulting continuities refer to weighted edge Sobolev spaces H s,γ (M). As indicated in relation (23) these spaces are locally close to ∂M ∼ = Rq modeled on Ws (Rq , Ks,γ (R+ )).

(29)

For the proof it suffices to focus on the local situation, and consider a symbol 1+q μ a(r, y, ρ, η) ∈ Scl (R+ × Rq × Rρ,η ) where M is locally close to the boundary identified with R+ × Rq - (r, y). For brevity in the following computations the

Dirichlet-to-Neumann Operator and Zaremba Problem

441

edge variable y will be omitted. There is then an asymptotic expansion a(r, ρ, η) ∼

∞

χ (ρ, η)a(μ−j ) (r, ρ, η)

j =0 μ

1+q

in Scl (R+ × Rq × Rρ,η ), for an excision function χ (ρ, η) and homogeneous components a(μ−j ) of order μ − j. This allows us to write a(μ−j ) (r, ρ, η) = r −μ (r j a(μ−j ) (r, rρ, rη)). The functions ˜ η) ˜ := r j a(μ−j ) (r, ρ, ˜ η) ˜ p˜ (μ−j ) (r, ρ, are homogeneous in (ρ, ˜ η) ˜ = 0 and smooth up to r = 0. This allows us to form an asymptotic expansion p(r, ˜ ρ, ˜ η) ˜ ∼

∞

χ (ρ, ˜ η) ˜ p˜ (μ−j ) (r, ρ, ˜ η) ˜

j =0 μ

1+q

and we have p(r, ˜ ρ, ˜ η) ˜ ∈ Scl (R+ × Rρ, ˜ η˜ ). The symbol p(r, rρ, rη) is edgedegenerate, and we have a(r, ρ, η) − p(r, rρ, rη) ∈ S −∞ (R+ × R1+q ρ,η ). Thus the correspondence a(r, ρ, η) → p(r, rρ, rη) represents a first step of edge quantization. From now on other steps can follow, in particular, the holomorphic Mellin-edge quantization, known from the edgecalculus, namely, ˜ w, rη). p(r, rρ, rη) → h(r, In this process we have q μ ˜ w, η) h(r, ˜ ∈ C ∞ (R+ , MO (X; Rη˜ )).

(30)

which is a Mellin symbol, holomorphic in w, indicated by O; clearly X is here of dimension 0, in contrast to the case of non-trivial edges, briefly discussed below. Notation for spaces of holomorphic Mellin symbols including their dependence on parameters and other useful properties may be found in [27] or [6, 7]. If (30) does

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B.-W. Schulze

˜ w, rη) gives rise to an operatornot depend on r for large r then h(r, w, η) := h(r, valued symbol, γ −n/2

r −μ OpM

(h)(η) : Ks,γ (R+ ) → Ks−μ,γ −μ (R+ ).

The latter family of operators belongs to S μ (Rqη ; Ks,γ (R+ ), Ks−μ,γ −μ (R+ )) with spaces Ks,γ (R+ ) := {ωu0 + (1 − ω)u∞ : u0 ∈ Hs,γ (R+ ), u∞ ∈ H s (R+ )}.

(31)

While H s (R+ ) := H s (R)|R+ is based on the Fourier transform on the r axis, the space Hs,γ (R+ ) = r γ Hs (R+ ) with Hs (R+ ) := Hs,0 (R+ ) refers to the Mellin transform  ∞ dr Mf (w) = r w f (r) r 0 for the complex Mellin covariable w ∈ C when supp f is compact in R+ , otherwise w varies on Re w = 0. The space Hs (R+ ), s ∈ R, is defined to be the completion of C0∞ (R+ ) with respect to the norm f H

s

(R+ )

:=



∞

−∞

iρ2s |Mf (iρ)|2 (2π )−1 dρ

1/2

.

Both Hs,γ (R+ ) and Ks,γ (R+ ) are endowed with the group action (κδ u)(r) = δ 1/2 u(δr),

δ > 0.

(32)

Let us now turn to other aspects of the edge calculus, i.e., when the underlying configuration B is a space with edge Z of dimension q, such that B \ Z is a smooth manifold which is locally close to Z modeled on X × Rq for a model cone X := (R+ × X)/({0} × X),

(33)

where X is a closed C ∞ -manifold of dimension n. Globally along Z we assume that there exists a neighborhood of Y in B which has the structure of a locally trivial X bundle over Y where the transition maps between the fibres satisfy a well-known regularity condition at the tips of the respective model cones. An example is the above-mentioned case of 0-dimensional X; then X = R+ may be identified with the inner normal of the respective manifold with smooth boundary Y. For X of

Dirichlet-to-Neumann Operator and Zaremba Problem

443

dimension n we employ Kegel spaces Ks,γ (X∧ )

for X∧ = R+ × X

rather than (31) where the ingredients on the right-hand side of (31) are replaced by s (X ∧ ), respectively, cf. [27, subsection 2.1.4] or [7, page 227], Hs,γ (X∧ ) and Hcone [10, page 123], endowed with the group action (κδ u)(r, x) = δ (n+1)/2 u(δr, x),

δ > 0.

(34)

It also makes sense to assume that X is a smooth manifold with boundary. This is just the situation for the original situation in connection with the Zaremba problem 1 ) for the half-circle itself where the interface is Z but the model cone is just (S+ 1 S+ in the 2-dimensional normal plane of Z in G, generated by the inner normal line of Z pointing transversally from Y to int G and the normal of Z with respect to the embedding Z → Y where Z is of codimension 1. Then the base of the model 1 with boundary having two end-points lying in Z. In cone of the wedge is just S+ any case we see that the Zaremba problem generates several local wedges, not only the trivial one after the reduction to the boundary. Therefore, it is justified to recall some more ideas on the edge calculus, in order to illustrate what is going on for the original problem. Let us first assume that the base X of local model cones on a manifold with edge Z is closed and compact. The original objects may be edgedegenerate families of classical, pseudo-differential operators and their relationship to corresponding families of Mellin operators, namely, p(r, ˜ y, ρ, ˜ η) ˜ ∈ C ∞ (R+ × Rq , Lcl (X; Rρ, ˜ η˜ )),

(35)

p(r, y, ρ, η) := p(r, ˜ y, rρ, rη).

(36)

μ

1+q

and we set

Moreover, Mellin-edge quantization gives us an q μ ˜ y, w, η) h(r, ˜ ∈ C ∞ (R+ × Rq , MOw (X; Rη˜ )),

(37)

˜ y, w, rη) h(r, y, w, η) := h(r,

(38)

such that

satisfies the relation Opr (p)(y, η) − OpM (h)(y, η) ∈ C ∞ (Rq , L−∞ (X∧ ; Rqη )), γ

(39)

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B.-W. Schulze

where the latter remainder is of the form Opr (Q)(y, η) for Q(r, r  , y, η) = (1 − ϕ(r  /r))p(r, r  , y, η), for ϕ ∈ C0∞ (R+ ), ϕ ≡ 1 close to 1, has a particularly desirable regularity structure, such that two different quantizations work. Let us call them “traditional”, according to [24] and “new”, motivated by constructions in [33] and [7], namely, by pointing out classes of symbols μ

2q

2q

μ Rtrad (Ry,η , g; Rdλ ) or Rnew (Ry,η , g; Rdλ ).

(40)

The extra parameter λ makes sense in connection with higher corner variants of the calculus where η is formally replaced by (η, λ). Such a modification is also used on order reducing considerations. The space μ

2q

Rtrad (Ry,η , g; Rdλ )

g = (γ , γ − μ)

for weight data

(41)

of edge symbols from [24] in traditional Mellin-edge quantization is defined to be the set of all operator-functions of the form   atrad (y, η, λ) = σ1 (r) a0 (y, η, λ) + a1 (y, η, λ) σ0 (r) + (1 − σ1 (r))aint (y, η, λ)(1 − σ2 (r)) + (m + g)(y, η, λ)

(42)

for cut-off functions σ2 (r) ≺ σ1 (r) ≺ σ0 (r), and γ − n2

a0 (y, η, λ) := ω1 (r[η, λ])r −μ OpM

(h)(y, η, λ)ω0 (r[η, λ]),

a1 (y, η, λ) := (1 − ω1 (r[η, λ]))r −μ Opr (p)(y, η, λ)(1 − ω2 (r[η, λ])),

(43) (44)

where ω2 (r) ≺ ω1 (r) ≺ ω0 (r), are cut-off functions, and h(r, y, w, η, λ) ∈ C ∞ (R+ × Rq , MO (X; R

q+d )|(η, ˜ λ˜ )=(rη,rλ) , η, ˜ λ˜

(45)

1+q+d ))|(ρ, ˜ η, ˜ λ˜ )=(rρ,rη,rλ) . ρ, ˜ η, ˜ λ˜

(46)

μ

and p(r, y, ρ, η, λ) ∈ C ∞ (R+ × Rq , Lcl (X; R μ

Dirichlet-to-Neumann Operator and Zaremba Problem

445

Moreover, aint (y, η, λ) ∈ C ∞ (Rq , Lcl (X∧ ; Rη,λ )0 ) q+d

μ

for , q+d q+d μ μ Lcl (X∧ ; Rη,λ )0 := aint (η, λ) ∈ Lcl (X∧ ; Rη,λ ) : σ˜ aint (η, λ)σ˜˜ = aint (η, λ) for some cut-off functions σ˜ , σ˜˜ . (47) The operators Ac (λ) = Opy (a)(λ) have symbols a(y, η, λ) in Rtrad (Ry,η , g; Rdλ ) ⊂ S μ (Ry × Rη,λ ; H, H˜ )

(48)

H = Ks,γ (X∧ ), H˜ = Ks−μ,γ −μ (X∧ ), s ∈ R.

(49)

2q

μ

q

q+d

for

Relation (48) can be obtained from another choice of edge amplitude functions, elaborated in [33], and [7], namely, 2q

μ Rnew (Ry,η , g; Rdλ )

for weight data

g = (γ , γ − μ)

(50)

consisting of all operator functions γ −n/2

anew (y, η, λ) := σ1 (r)r −μ OpM

(h)(y, η, λ)σ0 (t)

+ (1 − σ1 (r))aint (y, η, λ)(1 − σ2 (r)) + (m + g)(y, η, λ).

(51)

A result of [7, page 236] is that μ

2q

Rtrad (Ry,η , g; Rdλ ) and

2q

μ Rnew (Ry,η , g; Rdλ )

are equivalent.

(52)

More precisely, setting γ −n/2

aM (y, η, λ) := r −μ OpM

(h)(y, η, λ),

pψ (y, η, λ) := r −μ Opr (p)(y, η, λ) (53)

and assuming that p is related to h via Mellin quantization, for every μ

2q

atrad (y, η, λ) ∈ Rtrad (Ry,η , g; Rdλ )

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B.-W. Schulze

there is an 2q

μ (Ry,η , g; Rdλ ) anew (y, η, λ) ∈ Rnew

(and vice versa) such that atrad (y, η, λ) − anew (y, η, λ) = σ1 ((1 − ω1,η,λ )[pψ (y, η, λ) − aM (y, η, λ)](1 − ω2,η,λ )σ0 , (54) where the remainder on the right-hand side of (54) is a Green symbol of infinite flatness, belonging both to the “traditional” and the “new” edge symbol space. The abbreviation means ωη,λ (r) = ω(r[η, λ]) for any cut-off function ω(r). We do not carry out here all details of the corresponding formalism but refer, for instance, to [7, page 251, Proposition A.4, and page 234, Definition 3.8]. As a consequence we have an analogue of relation (48) also for the symbol space 2q μ Rnew (Ry,η , g; Rdλ ). The parameter-dependent analogue of (24) will be denoted by Lμ (M, g; Rd ).

(55)

It consists of families of continuous operators A(λ) : H s,γ (M) → H s−μ,γ −μ (M).

(56)

There is a slight generalisation of (55), namely, Lμ−m (M, g; Rd ).

(57)

for any m ∈ N with weight data g as before. The definition of (57) can be given in different ways, for instance, by admitting extra powers of r in the symbols involved in Ac and lower order interior contributions over M \ Z. However, the elements of (55) have parameter-dependent homogeneous principal symbols σ (A(λ)) = (σ0 (A(λ)), σ1 (A(λ))) While the interior symbol follows from μ

Lμ (M, g; Rd ) ⊂ Lcl (M \ Z; Rdλ )

(58)

Dirichlet-to-Neumann Operator and Zaremba Problem

447

as the parameter-dependent homogeneous symbols in the standard sense, the component σ1 (A(λ)) has the meaning of the homogeneous principal edge symbol of order μ which is operator-valued and associated with relation (48), though the respective complete symbols are not classical. We then have σ1 (A(λ)) ∈ S (μ) (Rq × (Rη,λ \ {0}); Ks,γ (X∧ ), Ks−μ,γ −μ (X∧ )) q+d

(59)

with S (ν) (Rq × (Rη,λ \ {0}); H, H˜ ) q+d

(60)

being the space of all f (y, η, λ) ∈ C ∞ (Rη,λ \ {0}; L(H, H˜ )) satisfying the homogeneity condition q+d

f (y, δη, δλ) = δ ν κδ f (y, η, λ)κδ−1 for all y and (η, λ) = 0, δ ∈ R+ . Now the space (57) for m = 1 is defined to be the set of all elements A(λ) in (55) such that (58) vanishes, and we can iterate the condition and obtain successively the spaces (57) for all m ∈ N. Let us assume from now on again d = 0. This concept extends edge operator spaces to a calculus of block matrix operators with extra edge conditions of trace and potential type, similarly to ideas from Boutet de Monvel’s calculus, mentioned at the beginning, i.e., we may have, say, for d = 0, operators  H s,γ (M) H s−μ,γ −μ (M) AK A= → , : ⊕ ⊕ T Q s s − + H (Z, J− ) H (Z, J+ )

(61)

for suitable orders s− , s+ and the respective space of operators could be denoted by Lμ (M, g; J− , J+ ).

(62)

Ellipticity of order μ of an upper left corner A ∈ Lμ (M, g) means that A as μ an operator in Lcl (M \ Z) is elliptic in the standard sense on the respective open manifold M \ Z, together with other specific conditions, known in the framework of edge calculus: If A is locally close to Z in coordinates (r, y) ∈ R+ × Rq written 2q μ as Opy (a) for an amplitude function a(y, η) ∈ Rtrad (Ry,η , g), cf. relation (48), we have ˜ η) ˜ = 0 p˜ (μ) (0, y, ρ,

for (ρ, ˜ η) ˜ = 0,

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B.-W. Schulze

where p˜ (μ) (0, y, ρ, ˜ η) ˜ is the homogeneous principal part of order μ of the respective symbol, see formula (35). In addition we ask the Fredholm property of σ1 (A)(y, η) : Ks,γ (X∧ ) → Ks−μ,γ −μ (X∧ )

(63)

for all y and η = 0 and any s. Because of twisted homogeneity of (63) in η = 0 of order μ the operators (63) are completely determined by the values on the unit cosphere bundle S ∗ Z which is a compact space. It is a well-known property of families (63) of Fredholm operators, parametrized by a compact topological space, say, a smooth manifold like S ∗ Z, that there are smooth complex vector bundles G− , G+ on S ∗ Z such that there is a family of 2 × 2-block matrix operators  Ks,γ (X∧ ) Ks−μ,γ −μ (X∧ ) σ1 (A) σ1 (K) σ1 (A)(y, η) := (y, η) : → . ⊕ ⊕ σ1 (T ) σ1 (Q) G− G+

(64)

consisting of isomorphisms, cf. [10, Subsction 3.3.4]. Recall that the K-theoretic index indS ∗ Z of the Fredholm family (63) is defined as [G+ ] − [G− ] ∈ K(S ∗ Z), and under the condition indS ∗ Z σ1 (A) ∈ πZ∗ K(Z)

(65)

for the pull back πZ∗ : K(Z) → K(S ∗ Z) with respect to the canonical projection πZ : S ∗ Z → Z we can replace G± by bundle pull backs πZ∗ J± for some vector bundles J± over Z. Remark 3 The condition (65) holds, in particular, for the case M = Y+ and A := Ps−3/2 N+ C+ Es−1/2 , cf. formula (20). In fact, this is a consequence of relation (2). In the general set-up the extra entries in (64), after extensions from S ∗ Z by suitable twisted homogeneities to T ∗ Z \ 0, give rise to operator-valued symbols, in local coordinates aK (y, η) ∈ Sclord K (Ry × Rqη ; CL− , Ks−μ,γ −μ (X∧ )),

(66)

aT (y, η) ∈ Sclord T (Ry × Rqη ; Ks,γ (X∧ ), CL+ ),

(67)

q

q

q

aQ (y, η) ∈ Sclord Q (Ry × Rqη ; CL− , CL− ),

(68)

and associated continuous operators K : H s− (Z, J− ) → H s−μ,γ −μ (M),

(69)

Dirichlet-to-Neumann Operator and Zaremba Problem

449

see [25, 32], T : H s,γ (M) → H s+ (Z, J+ ),

(70)

Q : H s− (Z, J− ) → H s+ (Z, J+ ).

(71)

The orders of symbols in (66), (67), (68) are uniquely determined by s, s−μ, s− , s+ in (61). Recall that the group actions in the spaces involved in (66), (67), (68) are adapted to the continuities in (61), except for those in K-spaces. The arizing operator (61) then belongs to Lμ (M, g; J− , J+ ). As a consequence of the general edge calculus we know that the operator (61) is Fredholm when it is elliptic, which is defined as the above-mentioned ellipticity of the upper left corner A together with the condition that Ks,γ (X∧ ) Ks−μ,γ −μ (X∧ ) σ1 (A) : → ⊕ ⊕ πZ∗ J− πZ∗ J+

(72)

is a family of isomorphisms which is an analogue of the Shapiro-Loptinskii condition. Ellipticity of A in this sense is also necessary for the Fredholm property of (61) which is shown in [1]. Then, applying known properties of the edge calculus, we have a two-sided parametrix P ∈ L−μ (M, g −1 ; J+ , J− ) such that PA − I = CL ,

AP − I = CR

(73)

for smoothing remainders CL , CR in the block matrix version of the edge calculus which are compact when M itself is compact. Such a situation is natural when the operator A in the upper left corner is elliptic with respect to the involved symbols. Let us now return to the operator (20). Using (21) and (22) we can pass to A=

   Ps−3/2 N+ C+ Es−1/2 Ks−1/2 Ts−3/2

(74)

which is a continuous operator H s−1/2,s−1/2 (Y+ ) H s−3/2,s−3/2 (Y+ ) A: → ⊕ ⊕ s−1/2 L(s−1/2) s−3/2 (Z, C ) (Z, CL(s−3/2) ) H H

(75)

for s indicated in connection with (20). The upper left corner of (75) shows that s plays a twofold role close to Z, namely, of smoothness and weight (up to corresponding translations by 1/2, and 3/2, respectively). Information of that kind is known from the edge calculus. The extra entries referring to Sobolev spaces on Z play the role of edge conditions of trace and potential type with respect to

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the edge. Those also contribute to σ1 (A), the corresponding block matrix-valued edge symbol, and for sufficiently large fibre dimensions of J− , J+ in (72) the trivial bundles CL(s−1/2) and CL(s−3/2) over Z may be interpreted as subbundles of J− and J− , respectively. Alternatively, in (72) we can also replace J− by CL(s−1/2) ⊕ J− and J+ by CL(s−3/2) ⊕ J+ . The original Zaremba problem (6) is related to AM by an extra conditions along the interface Z, interpreted as an edge (though embedded in ∂G) on the manifold with boundary G. In other words, G is a specific manifold with edge on the boundary which fits to the ideas of the general edge calculus, see terminology in [9, 10], or [13]. The replacement of the operator (6) by (18) containing the additional potential operator d− is a manipulation in such a sense, i.e., an admitted operator along the edge Z. Clearly AM can be recovered by its reduction to the interface AM PD since PD is an isomorphism. Note that there are also other suitable edge condition of similar kind. In other words, it suffices to deal with the operator (18). Multiplication by the isomorphism PD = (AD )−1 just gave us the operator (19) which is of triangular form. The situation resembles the relationship between the operators A1 and R in Sect. 2. Let us now add some hints on further using this process, though at the very end we sketch an alternative way to obtaining Fredholm operators and parametrices. First recall that B := N+ C+ satisfies an ellipticity condition and as indicated before we may pass to a Fredholm 2 × 2 block-matrix B containing B in the upper left corner, cf. the formal analogy to Boutet de Monvel’s calculus, namely, B :=

 B k , t q

(76)

with extra elliptic edge conditions along the edge Z where t and k which are of trace and potential type, respectively, and q is a pseudo-differential contribution on Z. We now interpret the right-hand side of (19) as an operator within our calculus. As we saw, the reduction of (18) to the boundary by means of the modified Dirichlet problem (12) gave us this matrix, for brevity denoted by ⎛

⎞ 1 0 0 A M PD = ⎝ 0 1 0 ⎠ r0 r B

(77)

for r0 := N+ , r := N+ C− . Then we pass to the larger block matrix ⎛

EMD

1 ⎜0 := ⎜ ⎝r0 0

0 0 1 0 rB 0 t

⎞ 0 0⎟ ⎟ k⎠ q

(78)

which is certainly Fredholm between the respective direct sums of spaces, and it (−1) obvously has a parametrix EMD also compatible with the chosen framework. When

Dirichlet-to-Neumann Operator and Zaremba Problem

451

we now form the operator AM AM := ⊕ id

(79)

with id being the identity operator in the fourth Sobolev space of distributional sections over Z arizing in the image under (78) we can consider the composition (−1) AM EMD containing one more row and column of operators compared with AM (−1) itself. Clearly, AM EMD is not Fredholm in general. Nevertheless we see the direction for a Fredholm answer. We also may ignore the fourth row and colum and replace B by B1 := B + G1 + M1 omitting the extra conditions t, k, q, and arrange the Green operator G1 and the smoothing Mellin operator M1 in such a way that B1 is Fredholm between the spaces in formula (20). Here we employ the fact, that M1 admits manipulating the index of the associated edge symbol, cf. analogous constructions in [26, Proposition 2.1.89]. Such a trick has been a crucial element in [1] and it also played a role in joint work with Seiler, cf. [29]. Then the inverse reduction to the boundary gives us back a new “Zaremba operator” AM,1 containing different additional terms which are explicitly generated and have a similar status as Green, etc., operators in usual edge problems, but here referring to the original mixed problem itself. This needs some computation, but it is evident that the resulting operator is then Fredholm and has a parametrix within our structure. Recall that we are basically in a framework of edge calculus which admits weight shifts on the expense of several kinds of smoothing Mellin contributions, depending on position and multiplicities of involved poles or when we look at non-bijectivity points of holomorphic non-smoothing Mellin symbols. Let us finally note that there are other approaches to mixed boundary value problems which are more directly modeled on the edge theory using Boutet de Monvel’s calculus on a half-circle with different conditions at the end points, together with associated cone operators on the half-plane \{0}. This requires other constructions which are not the topic of the present discussion, but it is an interesting program to make the present calculations more transparent by using such ideas. More results on the formal structure of conclusions may also be found in a cycle of papers jointly with Schrohe on the edge symbol calculus when the base of the model cone has a smooth boundary, see, e.g., [21, 22], moreover, a joint book with Harutyunyan [10] or papers jointly with Khalil [13, 14].

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References 1. S. Behm, Pseudo-Differential Operators with Parameters on Manifolds with Edges, Ph-D thesis, Potsdam, 1995. 2. L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51. 3. D.-C. Chang, N. Habal, and B.-W. Schulze, The edge algebra structure of the Zaremba problem, NCTS Preprints in Mathematics 2013-6-002, Taiwan, 2013. J. Pseudo-Differ. Oper. Appl. 5 (2014), 69–155, DOI: https://doi.org/10.1007/s11868-013-0088-7 4. N. Dines, X. Liu, and B.-W. Schulze, Edge quantisation of elliptic operators, Monatshefte für Math. 156 (2009), 233–274. 5. G.I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Transl. of Nauka, Moskva, 1973, Math. Monographs, Amer. Math. Soc. 52, Providence, Rhode Island 1980. 6. J.B. Gil, B.-W. Schulze, and J. Seiler, Holomorphic operator-valued symbols for edgedegenerate pseudo-differential operators, in “Differential Equations, Asymptotic Analysis and Mathematical Physics”, eds. M. Demuth et al. Mathematical Research, Vol. 100, Akademie Verlag (1997), pp. 113–137. 7. J.B. Gil, B.-W. Schulze, and J. Seiler, Cone pseudodifferential operators in the edge symbolic calculus, Osaka J. Math. 37 (2000), 221–260. 8. G. Grubb, Functional calculus of pseudo-differential boundary problems, Second Edition, Birkhäuser Verlag, Boston, 1996. 9. N. Habal, Operators on singular manifolds, Ph.D. thesis, University of Potsdam, 2013. 10. G. Harutyunyan and B.-W. Schulze, Elliptic mixed, transmission and singular crack problems, European Mathematical Soc., Zürich, 2008. 11. L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. 83, 1 (1966), 129–200. 12. S. Khalil and B.-W. Schulze, Boundary Problems on a Manifold with Edge, Asian-European Journal of Mathematics, AEJM, Vol. 10, No.2 (2017) 1750087 World Scientific Publ. Company DOI: https://doi.org/10.1142/S1793557117500875 13. S. Khalil and B.-W. Schulze, Calculus on a manifold with edge and boundary, CAOT 13 (2019), 2627–2670, DOI: https://doi.org/10.1007/s11785-018-0800-y. 14. S. Khalil and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus on manifolds with edge, Proc. Conf. “ Mathematics, Informatics in Natural Sciences and Engineering”, AMINSE, December 6–9, 2017, Tbilisi, Georgia, (eds. G. Jaiani, D. Natroshvili), Springer. 15. T. Krainer, The calculus of Volterra Mellin pseudo-differential operators with operator-valued symbols, Oper. Theory Adv. Appl. 138, Adv. in Partial Differential Equations “Parabolicity, Volterra Calculus, and Conical Singularities” (S. Albeverio, M. Demuth, E. Schrohe, and B.W. Schulze, eds.), Birkhäuser Verlag, Basel, 2002, pp. 47–91. 16. X. Liu and B.-W. Schulze, Boundary Value Problems with Global Projection Conditions, Operator Theory, Advances and Applications, Vol. 265, Springer/Birkhäuser, Basel. 2018. 17. S. Rempel and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982; North Oxford Acad. Publ. Comp., Oxford, 1985. (Transl. to Russian: Mir, Moscow, 1986). 18. S. Rempel and B.-W. Schulze, Asymptotics for elliptic mixed boundary problems (pseudodifferential and Mellin operators in spaces with conormal singularity), Math. Res. 50, Akademie-Verlag, Berlin, 1989. 19. E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities I, Adv. in Partial Differential Equations “PseudoDifferential Calculus and Mathematical Physics”, Akademie Verlag, Berlin, 1994, pp. 97–209.

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20. E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II, Adv. in Partial Differential Equations “Boundary Value Problems, Schrödinger Operators, Deformation Quantization”, Akademie Verlag, Berlin, 1995, pp. 70–205. 21. E. Schrohe and B.-W. Schulze, A symbol algebra for pseudo-differential boundary value problems on manifolds with edges, in “Differential Equations, Asymptotic Analysis and Mathematical Physics”, eds. M. Demuth et al. Mathematical Research, Vol. 100, Akademie Verlag (1997), pp. 292–324. 22. E. Schrohe and B.-W. Schulze, Edge-degenerate boundary value problems on cones, Proc. “Evolution Equations and their Applications in Physical and Life Sciences”, Bad Herrenalb, Karlsruhe, 2000. 23. B.-W. Schulze, Topologies and invertibility in operator spaces with symbolic structures, Teubner-Texte zur Mathematik 111, “Problems and Methods in Mathematical Physics”, BSB Teubner, Leipzig, 1989, pp. 257–270. 24. B.-W. Schulze, Pseudo-differential operators on manifolds with edges, Teubner-Texte zur Mathematik 112, Symp. “Partial Differential Equations, Holzhau 1988”, BSB Teubner, Leipzig, 1989, pp. 259–287. 25. B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, North-Holland, Amsterdam, 1991. 26. B.-W. Schulze, Pseudo-differential boundary value problems, conical singularities, and asymptotics, Akademie Verlag, Berlin, 1994. 27. B.-W. Schulze, Boundary value problems and singular pseudo-differential operators, J. Wiley, Chichester, 1998. 28. B.-W. Schulze and J. Seiler, Pseudodifferential boundary value problems with global projection conditions, J. Funct. Anal. 206, 2 (2004), 449–498. 29. B.-W. Schulze and J. Seiler, The edge algebra structure of boundary value problems, Ann. Glob. Anal. Geom. 22 (2002), 197–265. 30. B.-W. Schulze and J. Seiler, Truncation quantization in the edge calculus, (in preparation). 31. B.-W. Schulze and G. Wildenhain, Methoden der Potentialtheorie für Elliptische Differentialgleichungen beliebiger Ordnung, Akademie-Verlag, Berlin; Birkhäuser Verlag, Basel, 1977. 32. J. Seiler, Continuity of edge and corner pseudo-differential operators, Math. Nachr. 205 (1999), 163–182. 33. J. Seiler, Pseudodifferential calculus on manifolds with non-compact edges, Ph.D. thesis, University of Potsdam, 1997. 34. S. Zaremba, Sur in probleme mixte relatif a l’équation de Laplace, Bull. de l’Académie des Sciences de Cracovie, Classe des Sciences Mathématiques et Naturelles, Series A, 1910, pp. 313–344.

Extended Gevrey Regularity via the Short-Time Fourier Transform Nenad Teofanov and Filip Tomi´c

Abstract We study the regularity of smooth functions whose derivatives are σ dominated by sequences of the form Mpτ,σ = pτp , τ > 0, σ ≥ 1. We show that such functions can be characterized through the decay properties of their shorttime Fourier transforms (STFT), and recover (Cordero et al., Trans. Am. Math. Soc., 367 (2015), 7639–7663; Theorem 3.1) as the special case when τ > 1 and σ = 1, i.e. when the Gevrey type regularity is considered. These estimates lead to a PaleyWiener type theorem for extended Gevrey classes. In contrast to the related result from Pilipovi´c et al. (Sarajevo Journal of Mathematics, 14 (2) (2018), 251–264; J. Pseudo-Differ. Oper. Appl. (2019)), here we relax the assumption on compact support of the observed functions. Moreover, we introduce the corresponding wave front set, recover it in terms of the STFT, and discuss local regularity in such context. Keywords Gevrey classes · Paley-Wiener theorem · Modulation spaces · Wave front sets · Ultradistributions

1 Introduction Classes of extended Gevrey functions and the corresponding wave front sets are introduced and investigated in [23–25, 34]. Such classes consist of smooth functions, and they are larger than any Gevrey class. This turned out to be important e.g. in the study of strictly hyperbolic equations, see [3]. Paley-Wiener type theorems

N. Teofanov () Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia e-mail: [email protected] F. Tomi´c Department of Fundamental Sciences, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_25

455

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N. Teofanov and F. Tomi´c

for compactly supported extended Gevrey regular functions are given in [26, 27], and it is shown that the Fourier-Laplace transform of such functions have certain logarithmic decay at infinity which can be expressed in terms of the Lambert W function. This fact is used to resolve wave front sets in the context of extended Gevrey regularity. We refer to [24, 25] for related theorems on propagation of singularities. The aim of this paper is twofold. Firstly, we give another version of the PaleyWiener theorem for extended Gevrey regularity and formulate the result in terms of the short time Fourier transform (STFT) (cf. [14]). More precisely, we prove a generalization of [7, Theorem 3.1], where the STFT estimates are related to Gevrey type regularity, and obtain a Paley-Wiener type result as its corollary. Secondly, we give a description of (micro)local regularity related to the extended Gevrey regularity by the means of the STFT. This result is inspired by recent characterization of C ∞ wave front sets via the STFT given in [21]. The paper is organized as follows: We fix some notation in Sect. 1.1. In Sect. 2 we collect the main notions and tools for our analysis: Sect. 2.1 contains basic facts concerning the extended Gevrey classes, and in Sect. 2.2 we introduce extended associated functions which appear in the formulation of our main results. The correct asymptotic behavior of the extended associated function is given by the means of the Lambert W function, see Theorem 1. In Sect. 2.3 we introduce the short-time Fourier transform, and modulation spaces defined by means of the decay and integrability conditions of the STFT of ultradistributions. We also recall some basic properties of modulation spaces. In Sect. 3 we prove Theorem 4. It is a generalization of [7, Theorem 3.1] which is used in the study of pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity, cf. [1, 2, 7]. As a corollary of Theorem 4 we obtain a Paley-Wiener type theorem for element of modulation spaces related to the extended Gevrey classes. This result extends [27, Theorem 3.1] in the sense that the condition on compact support is replaced by appropriate decay property given by a modulation space norm, when the Fourier-Laplace transform is replaced by the STFT. In Sect. 4 we recall the notion of wave front sets related to extended Gevrey regularity. We prove that such wave front sets can be characterized by the decay properties of the STFT of a distribution with respect to a suitably chosen window function, Theorem 6. As a consequence we derive a result on local extended Gevrey regularity, Theorem 7. Our results are proved for the so-called Roumieu case, and we note that proofs for the Beurling case are similar and therefore omitted.

1.1 Basic Notation We denote by N, Z+ , R, C the sets of nonnegative integers, positive integers, real numbers and complex numbers, respectively. For x ∈ Rd we put x = (1+|x|2 )1/2 . The integer parts (the floor and the ceiling functions) of x ∈ R+ are denoted by

Extended Gevrey Regularity via the STFT

457

x := max{m ∈ N : m ≤ x} and CxD := min{m ∈ N : m ≥ x}. For a multiindex α = (α1 , . . . , αd ) ∈ Nd we write ∂ α = ∂ α1 . . . ∂ αd , D α = (−i)|α| ∂ α , and |α| = |α1 | + · · · + |αd |. Open ball of radius r > 0 centered at x0 is denoted by Br (x0 ). As usual, C ∞ (Rd ) is the space of smooth functions, the Schwartz space of rapidly decreasing functions is denoted by S(Rd ), and S  (Rd ) denotes its dual space of tempered distributions. Lebesgue spaces over an open set Ω ⊆ Rd are denoted by Lp (Ω), 1 ≤ p < ∞, and the norm of f ∈ Lp (Ω) is denoted by f Lp . The Fourier transform is normalized to be  fˆ(ω) = Ff (ω) = f (t)e−2π itω dt. We f, g to denote the extension of the inner product f, g =  use the brackets f (t)g(t)dt on L2 (Rd ) to the dual pairing between a test function space A and its dual A : ·, · = A ·, ·A . Translation and modulation operators, T and M respectively, when acting on f ∈ L2 (Rd ) are defined by Tx f (·) = f (· − x)

and

Mx f (·) = e2π ix· f (·), x ∈ Rd .

Then for f, g ∈ L2 (Rd ) the following relations hold: My Tx = e2π ix·y Tx My , (Tx f )ˆ = M−x fˆ, (Mx f )ˆ = Tx fˆ, x, y ∈ Rd . These operators are extended to other spaces of functions and distributions in a natural way. Throughout the paper, A  B denotes A ≤ cB for a suitable hidden constant c > 0, whereas A B means that c−1 A ≤ B ≤ cA for some c ≥ 1. The symbol B1 → B2 denotes the continuous and dense embedding of the topological vector space B1 into B2 .

2 Preliminaries In this section we collect the main notions and auxiliary results which will be used in the sequel. More precisely, we introduce the test function spaces related to the σ sequences of the form Mpτ,σ = pτp , p ∈ Z+ , for a given τ > 0 and σ > 1. Notice that, when τ > 1 and σ = 1, then Mpτ,1 = pτp , p ∈ Z+ , is (equivalent to) the Gevrey sequence p!τ . Then we discuss associated functions to such sequences, which are the main tool of our analysis. To describe precise asymptotic behavior of those associated functions at infinity is a nontrivial problem, which can be resolved by the use of Lambert’s W functions. Finally, we recall the definition and some elementary properties of the STFT and modulation spaces defined by mixed weighted Lebesgue norm conditions on the STFT.

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2.1 Extended Gevrey Regularity In this subsection we introduce extended Gevrey classes and discuss their basic properties. We employ Komatsu’s approach [19] to spaces of ultradifferentiable σ functions, and consider defining sequences of the form Mpτ,σ = pτp , p ∈ N, depending on parameters τ > 0 and σ > 1, [24]. Essential properties of the defining sequences are listed in the following lemma. We refer to [23] for the proof. In the general theory of ultradistributions (see [19]) different properties of defining sequences give rise to particular structural properties of the corresponding spaces of ultradifferentiable functions, see [28] for a detalied survey. σ

Lemma 1 Let τ > 0, σ > 1 and Mpτ,σ = pτp , p ∈ Z+ , M0τ,σ = 1. Then there exists an increasing sequence of positive numbers Cq , q ∈ N, and a constant C > 0 such that: τ,σ τ,σ (M.1) (Mpτ,σ )2 ≤ Mp−1 Mp+1 , p ∈ Z+ τ,σ (M.2) Mp+q ≤ Cp

(M.2) (M.3)

τ,σ Mp+q

≤

Mpτ 2

pσ Cq Mpτ,σ ,

∞ M τ,σ

p−1 p=1

σ +q σ

Mpτ,σ

σ −1 ,σ

Mqτ 2

σ −1 ,σ

, p, q ∈ N,

p, q ∈ N, τ,σ Mp−1

< ∞. Moreover,

Mpτ,σ

≤

1 σ −1 (2p)τ (p−1)

, p ∈ N.

Let τ, h > 0, σ > 1 and let K ⊂⊂ Rd be a regular compact set. By Eτ,σ,h (K) we denote the Banach space of functions φ ∈ C ∞ (K) such that φ Eτ,σ,h (K) = sup sup

α∈Nd x∈K

|∂ α φ(x)| σ τ,σ < ∞. h|α| M|α|

(1)

The set of functions φ ∈ Eτ,σ,h (K) whose support is contained in K is denoted K by Dτ,σ,h . Let U be an open set Rd and K ⊂⊂ U . We define families of spaces by introducing the following projective and inductive limit topologies: E{τ,σ } (U ) = lim ← −

lim Eτ,σ,h (K), − →

K⊂⊂U h→∞

E(τ,σ ) (U ) = lim lim Eτ,σ,h (K), ← − ← − K⊂⊂U h→0

K K D{τ,σ } (U ) = lim D{τ,σ ( lim Dτ,σ,h ), } = lim − → − → − → K⊂⊂U

K⊂⊂U h→∞

K K D(τ,σ ) (U ) = lim D(τ,σ ( lim Dτ,σ,h ). ) = lim − → − → ← − K⊂⊂U

K⊂⊂U h→0

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We will use abbreviated notation τ, σ for {τ, σ } (the Roumieu case) or (τ, σ ) (the K and D Beurling case) . The spaces Eτ,σ (U ), Dτ,σ τ,σ (U ) are nuclear, cf. [23]. We refer to [23–25, 33, 34, 37] for other properties of those spaces. Remark 1 If τ > 1 and σ = 1, then E{τ,1} (U ) = E{τ } (U ) is the Gevrey class, and D{τ,1} (U ) = D{τ } (U ) is its subspace of compactly supported functions in E{τ } (U ). In particular, lim E{t} (U ) → Eτ,σ (U ) → C ∞ (U ), τ > 0, σ > 1, − →

t→∞

so that the regularity in Eτ,σ (U ) can be thought of as an extended Gevrey regularity. If 0 < τ ≤ 1, then Eτ,1 (U ) consists of quasianalytic functions. In particular, Dτ,1 (U ) = {0} when 0 < τ ≤ 1, and E{1,1} (U ) = E{1} (U ) is the space of analytic functions on U . The non-quasianalyticity condition (M.3) provides the existence of partitions of unity in E{τ,σ } (U ), i.e. for any given τ > 0 and σ > 1, there  exists a compactly supported function φ ∈ E{τ,σ } (U ) such that 0 ≤ φ ≤ 1 and Rd φ dx = 1, see [23] for a construction of a compactly supported φ ∈ D{τ,σ } (U ) \ D{t} (U ), t > 1. Note that the additional exponent σ , which appears in the power of term h in (1), makes the definition of Eτ,σ (U ) different from the definition of Carleman classes, cf. [18]. This difference appears to be essential in many calculations, and in particular when dealing with the operators of “infinite order“, cf. [24].

τ,σ

2.2 The Associated Function to the Sequence Mp

= p τp

σ

In this subsection we recall the definition and asymptotic properties of extended σ associated function to the sequence Mpτ,σ = pτp , p ∈ N, τ > 0, σ > 1, cf. [27]. We also recall the Paley-Wiener theorem related to the extended Gevrey regularity. σ

Definition 1 Let τ > 0, σ > 1 and Mpτ,σ = pτp , p ∈ Z+ , M0τ,σ = 1. The extended associated function related to the sequence Mpτ,σ , is given by σ

Tτ,σ,h (k) = sup ln+ p∈N

where ln+ A = max{0, ln A}, for A > 0.

hp k p , h, k > 0, Mpτ,σ

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Obviously Tτ,σ,h (k), τ, h > 0, σ > 1, is positive for sufficiently large k > 0. 1/p In fact, for any sequence of positive numbers Mp , p ∈ N, such that Mp is bounded from below and M0 = 1, its associated function is defined to be T (k) = sup ln p∈N

kp , Mp

k > 0.

Therefore, for τ > 0 and σ = 1, Tτ,1,h (k) := Tτ (hk) is the associated function to the Gevrey sequence pτp , p ∈ N (we may assume h = 1 without loosing generality). It is well known (cf. [13, 29]) that Ak 1/τ − B ≤ Tτ (k) ≤ Ak 1/τ ,

k > 0,

(2)

for suitable A, B > 0. In particular, the growth of eTτ (k) for τ > 1 is subexponential. Moreover, for any t, τ > 0 and σ > 1, by [27, Lemma 2.3] it follows that there is a constant C > 0 such that Tτ,σ,1 (k) < Ck 1/t ,

k > 0.

Therefore the function eTτ,σ,h (k) has a less rapid growth at infinity than any subexponential function. The precise asymptotic behavior of Tτ,σ,h (k) at infinity is a challenging problem. We use an auxiliary special function to resolve that problem. The Lambert W function is defined as the inverse function of zez , z ∈ C, wherefrom the following property holds: x ≥ 0.

x = W (x)eW (x) ,

We denote its principal (real) branch by W (x), x ≥ 0 (see [8]). It is a continuous, increasing and concave function on [0, ∞), W (0) = 0, W (e) = 1, and W (x) > 0, x > 0. It can be shown that W can be represented in the form of the absolutely convergent series W (x) = ln x − ln(ln x) +

∞ ∞

k=0 m=1

ckm

(ln(ln x))m , (ln x)k+m

x ≥ x0 > e,

with suitable constants ckm and x0 , wherefrom the following estimates hold: ln x − ln(ln x) ≤ W (x) ≤ ln x −

1 ln(ln x), 2

x ≥ e.

(3)

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The equality in (3) holds if and only if x = e. We refer to [8, 17] for more details about the Lambert W function. Theorem 1 ([27]) Let there be given τ, h > 0, σ > 1 and let Cτ,σ,h = σ −1 σ −1 h− τ e σ στ−1 σ . Then σ − 1 σ   1 1 σ σ −1 W − σ −1 (Cτ,σ,h ln k) ln σ −1 k  eTτ,σ,h (k) exp (2σ −1 τ )− σ −1 σ  σ − 1  1  1 σ σ −1  exp W − σ −1 (Cτ,σ,h ln k) ln σ −1 k , k > e. τσ

(4)

If, moreover 1 < σ < 2, then we have the precise asymptotic formula eTτ,σ,h (k) exp

 σ − 1  1  1 σ σ −1 W − σ −1 (Cτ,σ,h ln k) ln σ −1 k , τσ

k > e.

The hidden constants in (4) depend on τ, σ and h. Remark 2 Note that, in the view of (3) we have σ

W

1 − σ −1

(C ln k) ln

σ σ −1

k

ln σ −1 k 1

ln σ −1 (C ln k)

σ



ln σ −1 k 1

,

ln σ −1 (ln k)

k → ∞,

(5)

for any given σ > 1, and the last behavior follows from ln(C ln k) ln(ln k), k → ∞, for any given C > 0. Since limk→∞ (ln k)1/(σ −1) (ln(C ln k))−1/(σ −1) = ∞, for every C > 0, (5) implies that for every M > 0 there exists B > 0 (depending on h and M) such that 1

σ

W − σ −1 (C ln k)) ln σ −1 k > M ln k,

k > B.

K when 1 < σ < 2. For the Next we recall the Paley-Wiener theorem for Dτ,σ proof we refer to [26], and a more general case when σ ≥ 2 is considered in [27]. K Theorem 2 Let τ > 0, 1 < σ < 2, U be open set in Rd and K ⊂⊂ U . If ϕ ∈ D{τ,σ } K (resp. ϕ ∈ D(τ,σ ) ) then its Fourier-Laplace transform is an entire function and there exists constants A, B > 0 (resp. for every B > 0 there exists A > 0) such that

  σ  σ − 1 1  1 σ −1 | ϕ (η)| ≤ A exp − W − σ −1 B ln(e + |η|) ln σ −1 (e + |η|) + HK (η) τσ h > 0, η ∈ Cd , where HK (η) = sup Im(y · η). y∈K

(6)

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Conversely, if there exists A, B > 0 (resp. for every B > 0 there exists A > 0) such that an entire function  ϕ (η) satisfies (6) then  ϕ (η) is the Fourier-Laplace K K transform of ϕ ∈ D{2 σ −1 τ,σ } (resp. D(2σ −1 τ,σ ) ). The following corollary is an immediate consequence of Theorem 2 and (5). Corollary 1 Let 1 < σ < 2, U be open set in Rd and K ⊂⊂ U . Then the entire function  ϕ (η), η ∈ Cd , is the Fourier-Laplace transform of K K (resp. ϕ ∈ lim Dτ,σ ) ϕ ∈ lim Dτ,σ ← − − → τ →∞

τ →0

if and only if there exist constant A, B > 0 (resp. for every B > 0 there exists A > 0) such that " | ϕ (η)| ≤ A exp −B

2

σ

ln σ −1 (e + |η|) ln

1 σ −1

(ln(e + |η|))

+ HK (η) , η ∈ Cd ,

where HK (η) = sup Im(x · η). x∈K

2.3 Modulation Spaces The modulation spaces were initially (and systematically) introduced in [9]. See also [14, Ch. 11–13] and the original literature quoted there for various properties and applications of the so called standard modulation spaces. It is usually sufficient to observe weighted modulation spaces with weights which may grow at most polynomially at infinity. However, for the study of ultra-distributions a more general approach which includes weights of exponential or even superexponential growth is needed, cf. [6, 36]. We refer to [10, 11] for related but even more general constructions, based on the general theory of coorbit spaces. For our purposes it is sufficient to consider weights of exponential growth. Therefore we begin with the Gelfand-Shilov space of analytic functions S (1) (Rd ) given by f ∈ S (1) (Rd ) ⇐⇒ sup |f (x)eh·|x| | < ∞ and sup |fˆ(ω)eh·|ω| | < ∞, x∈Rd

ω∈Rd

for every h > 0. Any f ∈ S (1) (Rd ) can be extended to a holomorphic function f (x + iy) in the strip {x + iy ∈ Cd : |y| < T } some T > 0, [13, 20]. The dual  space of S (1) (Rd ) will be denoted by S (1) (Rd ). In fact, S (1) (Rd ) is isomorphic  to the Sato test function space for the space of Fourier hyperfunctions S (1) (Rd ), see [4].

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463

Let there be given f, g ∈ L2 (Rd ). The short-time Fourier transform (STFT) of f with respect to the window g is given by  Vg f (x, ω) =

e−2π itω f (t)g(t − x)dt, x, ω ∈ Rd .

(7)

It restricts to a mapping from S (1) (Rd ) × S (1) (Rd ) to S (1) (R2d ), which is proved in the next Lemma. Lemma 2 Let f, g ∈ S (1) (Rd ), and let the short-time Fourier transform (STFT) of f with respect to g be given by (7). Then Vg f (x, ω) ∈ S (1) (R2d ), that is |Vg f (x, ω)| < Ce−s (x,ω) , x, ω ∈ Rd , for every s > 0. Proof The proof is standard, see e.g. [14] for the proof in the context of S(Rd ). We use the arguments based on the structure of S (1) (Rd ) as follows. Let f ⊗ g be the tensor product f ⊗ g(x, t) = f (x) · g(t), let T denote the asymmetric coordinate transform T F (x, t) = F (t, t − x), and let F2 be the partial Fourier transform  F2 F (x, ω) =

Rd

F (x, t)e−2π itω dt,

x, ω ∈ Rd ,

of a function F on R2d . Then Vg f (x, ω) = F2 T (f ⊗ g)(x, ω),

(x, ω) ∈ R2d .

ˆ (1) (Rd ) (see e.g. [32] for the kernel theorem in Since S (1) (R2d ) ∼ = S (1) (Rd )⊗S Gelfand-Shilov spaces) and since S (1) (R2d ) is invariant under the action of T and F2 , we conclude that |Vg f (x, ω)| < Ce−s (x,ω) , x, ω ∈ Rd , for every s > 0.   Weight Functions. In the sequel v will always be a continuous, positive, even, submultiplicative function (submultiplicative weight), i.e., v(0) = 1, v(z) = v(−z), and v(z1 + z2 ) ≤ v(z1 )v(z2 ), for all z, z1 , z2 ∈ R2d . Moreover, v is assumed to be even in each group of coordinates, that is, v(x, ω) = v(−ω, x) = v(−x, ω), for any (x, ω) ∈ R2d . Submultipliciativity implies that v(z) is dominated by an exponential function, i.e. ∃ C, k > 0

such that

v(z) ≤ Cek z ,

z ∈ R2d ,

(8)

and z is the Euclidean norm of z ∈ R2d . For example, every weight of the form b

v(z) = es z (1 + z )a logr (e + z ) for parameters a, r, s ≥ 0, 0 ≤ b ≤ 1 satisfies the above conditions.

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Associated to every submultiplicative weight we consider the class of so-called v-moderate weights Mv . A positive, even weight function m on R2d belongs to Mv if it satisfies the condition m(z1 + z2 ) ≤ Cv(z1 )m(z2 ) ∀z1 , z2 ∈ R2d . We note that this definition implies that v1  m  v, m = 0 everywhere, and that 1/m ∈ Mv . The widest class of weights allowing to define modulation spaces is the weight class N . A weight function m on R2d belongs to N if it is a continuous, positive function such that 2

for |z| → ∞,

m(z) = o(ecz ),

∀c > 0, b

with z ∈ R2d . For instance, every function m(z) = es|z| , with s > 0 and 0 ≤ b < 2, is in N . Thus, the weight m may grow faster than exponentially at infinity. For example, the choice m ∈ N \ Mv is related to the spaces of quasianalytic functions, [5]. We notice that there is a limit in enlarging the weight class for modulation 2 spaces, imposed by Hardy’s theorem: if m(z) ≥ Cecz , for some c > π/2, then the corresponding modulation spaces are trivial [16]. We refer to [15] for a survey on the most important types of weights commonly used in time-frequency analysis. Definition 2 Let v be a submultiplicative weight, m ∈ Mv , and let g be a non-zero p,q window function in S (1) (Rd ). For 1 ≤ p, q ≤ ∞ the modulation space Mm (Rd )  p,q (1) d d consists of all f ∈ S (R ) such that Vg f ∈ Lm (R ) (weighted mixed-norm p,q spaces). The norm on Mm (Rd ) (making it a Banach space) is given by  f Mmp,q = Vg f Lp,q = m

 Rd

Rd

1/q

q/p |Vg f (x, ω)|p m(x, ω)p dx

dω

(with obvious changes if either p = ∞ or q = ∞). If p, q < ∞, the modulation p,q p,q space Mm (Rd ) is the norm completion of S (1) (Rd ) in the Mm -norm. If p = ∞ p,q or q = ∞, then Mm (Rd ) is the completion of S (1) (Rd ) in the weak∗ topology. Note that for f, g ∈ S (1) (Rd ) the above integral is convergent so that S (1) (Rd ) ⊂ Namely, in view of (8), for a given m ∈ Mv there exist l > 0 such that m(x, ω) ≤ Cel (x,ω) and therefore

  q/p





p p |Vg f (x, ω)| m(x, ω) dx dω



Rd

Rd

  q/p





p lp (x,ω) ≤C

|Vg f (x, ω)| e dx dω < ∞,

Rd

d R

p,q Mm (Rd ).

since by Lemma 2 it follows that |Vg f (x, ω)| < Ce−s (x,ω) for every s > 0.

Extended Gevrey Regularity via the STFT

465

p

p,p

If p = q, we write Mm instead of Mm , and if m(z) ≡ 1 on R2d , then we write p,q p,p and M p for Mm and Mm respectively, and so on. p,q In the next proposition we show that Mm (Rd ) are Banach spaces whose definition is independent of the choice of the window g ∈ Mv1 (Rd ) \ {0}. In order to do so, we need the adjoint of the short-time Fourier transform. p,q For a given window g ∈ S (1) (Rd ) and a function F (x, ξ ) ∈ Lm (R2d ) we define ∗ Vg F by M p,q

Vg∗ F, f  := F, Vg f , whenever the duality is well defined. Then [14, Proposition 11.3.2] (see also [6]) can be rewritten as follows. Proposition 1 Fix m ∈ Mv and g, ψ ∈ S (1) , with g, ψ = 0. Then 1. Vg∗ : Lm (R2d ) → Mm (Rd ), and p,q

p,q

. Vg∗ F Mmp,q ≤ C Vψ g L1v F Lp,q m 2. The inversion formula holds: IMmp,q = g, ψ−1 Vg∗ Vψ , where IMmp,q stands for the identity operator. p,q 3. Mm (Rd ) are Banach spaces whose definition is independent on the choice of g ∈ S (1) \ {0}. 4. The space of admissible windows can be extended from S (1) (Rd ) to Mv1 (Rd ). When m is a polynomial weight of the form m(x, ω) = xt ωs we will use the p,q notation Ms,t (Rd ) for the modulation spaces which consists of all f ∈ (S (1) ) (Rd ) such that  p,q ≡ f Ms,t

 Rd

Rd

1/q

q/p |Vφ f (x, ω)x ω | dx t

s p

dω

0, σ ≥ 1, by the rate of decay of the STFT. In particular, we extend [7, Theorem 3.1], which is formulated in terms of Gevrey sequences and the corresponding spaces of test functions. In the proof of Theorem 4 in several occasions we will use the following simple inequalities: |α|σ + |β|σ ≤ |α + β|σ ≤ 2σ −1 (|α|σ + |β|σ ),

α, β ∈ Nd , σ > 1,

(9)

and √ |(1/ d)ξ ||α| ≤ |ξ α | ≤ |ξ ||α| ,

α ∈ Nd , ξ ∈ Rd .

(10)

Theorem 4 Let τ > 0, σ ≥ 1, let v be a submultiplicative weight, m ∈ Mv , and 1,1 let g ∈ Mv⊗1 (Rd )\{0} such that for some Cg > 0, ∂ α g L1v (Rd )  Cg|α| |α|τ |α| , σ

σ

α ∈ Nd .

(11)

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467

For a smooth function f the following conditions are equivalent: i) There exists a constant Cf > 0 such that |α|σ

∂ α f L∞ (Rd )  m(x)Cf

|α|τ |α| , α ∈ Nd ; σ

(12)

ii) There exists a constant Cf,g > 0 such that |α|σ

|ξ ||α| |Vg f (x, ξ )|  m(x)Cf,g |α|τ |α| , x, ξ ∈ Rd , α ∈ Nd ; σ

iii) There exists a constant C > 0 such that |Vg f (x, ξ )|  m(x)e−Tτ,σ,C (|ξ |) , x, ξ ∈ Rd . Proof When σ = 1 we obtain [7, Theorem 3.1], where the function C|x|1/τ appears instead of Tτ,σ,C (|ξ |). However, this makes no difference, since from Tτ,1,C (|ξ |) := Tτ (C|ξ |) (see also (2)) it follows that iii) is equivalent to |Vg f (x, ξ )|  m(x)e−C|ξ |

1/τ

,

x, ξ ∈ Rd ,

for some C > 0. Note also that due to (10) the condition (ii) on |ξ α Vg f (x, ξ )| given by (38) in [7] is equivalent to ii) here above. Let σ > 1. We follow the proof of [7, Theorem 3.1], with necessary modifications, since we consider a more general situation. i) ⇒ ii) Since Vg f (x, ξ ) = F(f Tx g)(ξ ), we can formally write ξ α Vg f (x, ξ ) =

1 Ff (∂ α (f Tx g))(ξ ) (2π i)|α|

α 1 F(∂ α−β f ∂ β (Tx g))(ξ ), = β (2π i)|α|

x, ξ ∈ Rd

β≤α

(where we used the Leibnitz formula), and the formalism can be justified as follows.

α 1 F(∂ α−β f Tx (∂ β g)) L∞ |ξ Vg f (x, ξ )|  β (2π )|α| α

β≤α

α 1 ∂ α−β f Tx (∂ β g) L1 .  β (2π )|α| β≤α

Since m is a positive v−moderate weight, from (11), inequality we obtain

(12), and Hölder’s

m(x)(∂ β Tx g) L1 ∂ α−β f Tx (∂ β g) L1 ≤ ∂ α−β f L∞ 1/m |α−β|σ

 Cf

|α − β|τ |α−β| m(x) v(· − x)∂ β g(· − x) L1 σ

468

N. Teofanov and F. Tomi´c |α−β|σ

 m(x)Cf

|α − β|τ |α−β| · Cg|β| |β|τ |β| σ

σ

σ

|α|σ

 m(x)C˜ f,g |α|τ |α| , σ

σ

where we used the fact that the sequence Mp = pτp satisfies τ,σ Mqτ,σ ≤ Mpτ,σ , (M.1) : Mp−q

q ≤ p,

p, q ∈ N,

which follows from (M.1), see Lemma 1, and also [19]. Thus √ |ξ ||α| |Vg f (x, ξ )|  ( d)|α| |ξ α Vg f (x, ξ )|  √ |α|

α |α|σ d σ m(x) |α|τ |α|  C˜ β f,g 2π β≤α

 √ |α| d σ σ |α|σ |α|σ  m(x)|α|τ |α| C˜ f,g = Cf,g |α|τ |α| , π

x, ξ ∈ Rd , α ∈ Nd ,

where we used (10), and ii) follows. ∞,1 d ii) ⇒ i) Note that (12) means that f ∈ Mm −1 ⊗1 (R ). Hence we may use the inversion formula for STFT (cf. Proposition 1), and since f is a smooth function, we may assume that it holds everywhere. So we formally write  1 α ∂ f (t) = Vg f (x, ξ )∂ α (Mξ Tx g)(t) dxdξ g 2L2 R2d   1 α = Vg f (x, ξ )(2π iξ )β Mξ Tx (∂ α−β g)(t) dxdξ, β R2d g 2 2 L β≤α

α ∈ Nd , t ∈ Rd , where we used the Leibnitz formula. The estimates below also justify the exchange of the order of derivation and integration. Therefore,   1 α |β| (2π ) |∂ α f (t)|  |Vg f (x, ξ )ξ β ||Tx (∂ α−β g)(t)| dxdξ, 2d β g 2 2 R L β≤α

 1 α (2π )|β| Iα,β (t),  g 2L2 β≤α β

where we put  Iα,β (t) =

R2d

|Vg f (x, ξ )ξ β ||Tx (∂ α−β g)(t)| dxdξ,

t ∈ Rd .

t ∈ Rd ,

Extended Gevrey Regularity via the STFT

469

We note that ii) is equivalent with |α|σ

ξ |α| |Vg f (x, ξ )|  m(x)Cf,g |α|τ |α| , x, ξ ∈ Rd , α ∈ Nd , σ

and estimate Iα,β (t) as follows:  Iα,β (t) ≤

R2d

ξ β |Vg f (x, ξ )|



ξ d+1 (α−β) |g (t − x)| dxdξ ξ d+1

ξ β+d+1 m(x) (α−β) |g (t − x)| dxdξ m(x) ξ d+1 R2d  1 σ |β+d+1|σ  Cf,g |β + d + 1|τ |β+d+1| m(x)|g (α−β) (t − x)| dxdξ d+1 R2d ξ   |β|σ τ |β|σ m(t) v(t − x)|g (α−β) (t − x)| dx  C · Cf,g |β| 

|Vg f (x, ξ )|

Rd

|β|σ

= C · Cf,g |β|τ |β| m(t) g (α−β) L1v , σ

t ∈ Rd , σ

where C depends on τ, σ and d, and we used (M.2) property of the sequence pτp , p ∈ N, τ > 0, σ > 1, cf. Lemma 1. Therefore, by (11) we obtain  m(t) α σ |β|σ (2π )|β| C˜ · Cf,g |β|τ |β| g (α−β) L1v , |∂ α f (t)|  2 β g 2 L β≤α

 C˜

 m(t) α σ σ σ |β|σ (2π )|β| Cf,g |β|τ |β| Cg|α−β| |α − β|τ |α−β| , 2 g L2 β≤α β C

m(t) ˜ |α|σ τ |α|σ , C |α| g 2L2 f,g

t ∈ Rd ,

which gives (12). Here above we used (9) in several occasions. The equivalence between ii) and iii) follows immediately from Definition 1. Details are left for the reader.   Note that the condition (11) is weaker than the corresponding condition in [7, Theorem 3.1], so by [7, Proposition 3.2] one can choose elements from GelfandShilov spaces as window functions. K , τ > 1, σ ≥ 1, then it obviously satisfies the condition We note that if f ∈ Dτ,σ i) in Theorem 4, i.e. |∂ α f (x)|  m(x)C |α| |α|τ |α| , σ

σ

α ∈ Nd ,

K . so that Theorem 4 gives the decay properties of the STFT of elements from Dτ,σ

470

N. Teofanov and F. Tomi´c

We use this remark to extend Theorem 2. Recall the Paley-Wiener type result K describes the decay properties of the Fourier-Laplace transform for f ∈ Dτ,σ in the context of the extended Gevrey regularity. The role of compact support in Paley-Wiener type theorems is essential. In the following Corollary we weaken the assumptions from [27] (see also [26, Corollary 3.2]) and allow the global growth condition given by (12). Then, instead of cut-off functions, which are usually used 1,1 (Rd )\{0}, and give a Paleyin localization procedures, we take a window in Mv⊗1 Wiener type result by using the STFT. Corollary 2 Let τ > 0, 1 < σ < 2, let v be a submultiplicative weight, m ∈ Mv , 1,1 and let g ∈ Mv⊗1 (Rd )\{0} such that (11) holds for some Cg > 0. Then a smooth function f satisfies (12) if and only if  σ − 1  1 ln σ σ−1 (e + |ξ |)  σ −1 , |Vg f (x, ξ )|  m(x) exp − 1 τσ ln σ −1 (ln(e + |ξ |)) 

x, ξ ∈ Rd .

The proof is an immediate consequence of Theorems 1 and 4, and Remark 2. We finish this section with a version of Theorem 4 and Corollary 2 in the context of Beurling type ultradifferentiable functions. The proofs are left for the reader as an exercise. Theorem 5 Let τ > 0, σ ≥ 1, let v be a submultiplicative weight, m ∈ Mv , and 1,1 let g ∈ Mv⊗1 (Rd )\{0} such that for some C > 0, ∂ α g L1v (Rd ) ≤ C |α|

σ +1

|α|τ |α| , σ

α ∈ Nd .

For a smooth function f the following conditions are equivalent: i) For every h > 0 there exists A > 0 such that ∂ α f L∞ (Rd ) ≤ m(x)Ah|α| |α|τ |α| , α ∈ Nd ; σ

σ

(13)

ii) For every h > 0 there exists A > 0 such that |ξ ||α| |Vg f (x, ξ )| ≤ m(x)Ah|α| |α|τ |α| , x, ξ ∈ Rd , α ∈ Nd ; σ

σ

iii) For every h > 0 there exists A > 0 such that |Vg f (x, ξ )| ≤ m(x)Ae−Tτ,σ,h (|ξ |) , x, ξ ∈ Rd . Corollary 3 Let τ > 0, 1 < σ < 2, and let g ∈ M 1,1 (Rd )\{0} satisfies condition (11). Then a smooth function f satisfies (13) if and only if for every H > 0 there exists A > 0 such that σ  σ − 1 1 ln σ −1 (e + |ξ |) σ −1 , x, ξ ∈ Rd . |Vg f (x, ξ )| ≤ m(x)A exp − 1 τσ W σ −1 (H ln(e + |ξ |))



Extended Gevrey Regularity via the STFT

471

Remark 3 A more general versions of Corollaries 2 and 3 when σ ≥ 2 can be proved by using the asymptotic formulas (4) from Theorem 1. This will give different necessary and sufficient conditions for f in terms of the decay properties of the STFT. We leave details for the reader.

4 Wave Front Sets WFτ,σ and STFT In this section we characterize wave front sets related to the classes introduced in Sect. 2.1, by the means of the STFT and extended associated function from Sect. 2.2. We start with the following definition of the wave front set WFτ,σ (u) of a distribution u with respect to the extended Gevrey regularity, see also [24–27, 34] for details. Definition 3 Let U ⊆ Rd be open, τ > 0, σ > 1 or τ > 1 and σ = 1, u ∈ D (U ), and let (x0 , ξ0 ) ∈ Rd × Rd \{0}. Then (x0 , ξ0 ) ∈ WF{τ,σ } (u) (resp. (x0 , ξ0 ) ∈ WF(τ,σ ) (u)) if and only if there exists a conic neighborhood Γ of ξ0 , a compact K K neighborhood K of x0 , and φ ∈ D{τ,σ } (resp. φ ∈ D(τ,σ ) ) such that φ = 1 on some neighborhood of x0 , and there exists A, h > 0 (for every h > 0 there exists A > 0 such that) σ

. |φ u(ξ )| ≤ A

σ

hN N τ N , |ξ |N

N ∈ N,ξ ∈ Γ .

K it can be proved that Definition 3 By using the Paley-Wiener theorem for Dτ,σ K , see [27]. does not depend on the choice of the cut-off function φ ∈ Dτ,σ Note that when τ > 1 and σ = 1 we have WF{τ,1} (u) = WFτ (u), where WFτ (u) denotes Gevrey wave front set, cf. [29]. We refer to [24] for a relation between WFτ,σ (u) from Definition 3 and classical, analytic and Gevrey wave front sets. By using the ideas presented in [21] we resolve WFτ,σ (u) of a distribution u via decay estimates of its STFT as follows.

Theorem 6 Let u ∈ D (Rd ), τ > 0, σ > 1. The following assertions are equivalent: i) (x0 , ξ0 ) ∈ WF{τ,σ } (u) (resp. (x0 , ξ0 ) ∈ WF(τ,σ ) (u)) . ii) There exists a conic neighborhood Γ of ξ0 , a compact neighborhood K of x0 K K such that for every φ ∈ D{τ,σ } (resp. φ ∈ D(τ,σ ) ) there exists A, h > 0 (resp. for every h > 0 there exists A > 0) such that σ

. |φ u(ξ )| ≤ A

σ

hN N τ N , |ξ |N

N ∈ N, ξ ∈ Γ ;

(14)

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iii) There exists a conic neighborhood Γ of ξ0 , a compact neighborhood K of x0 K−{x } K−{x } such that for every φ ∈ D{τ,σ } 0 (resp. φ ∈ D(τ,σ ) 0 ) there exists A, h > 0 (resp. for every h > 0 there exists A > 0) such that |Vφ u(x, ξ )| ≤ Ae−Tτ,σ,h (|ξ |) ,

x ∈ K, ξ ∈ Γ,

(15)

where K − {x0 } = {y ∈ Rd | y + x0 ∈ K}. Proof We give the proof for the Roumieu case and leave the Beurling case to the reader. The equivalence i) ⇔ ii) is proved in Theorem 4.2. in [27]. ii) ⇒ iii) Without loss of generality we may assume that there exists a conic neighborhood Γ of ξ0 , compact neighborhood K1 = Br (x0 ), r > 0, such that for K1 every φ ∈ Dτ,σ (14) holds. K−{x } Set K = Br/2 (x0 ) and note that if φ(t) is an arbitrary function Dτ,σ 0 and K1 . x ∈ K, then Tx φ(t) is a function in Dτ,σ Using the definition of STFT and (14) we have that σ

σ

hN N τ N = Ae−Tτ,σ,1/ h (|ξ |) , |Vφ u(x, ξ )| = |F(uTx φ)| ≤ A inf N ∈N |ξ |N x ∈ K, ξ ∈ Γ, for some constant A > 0, and iii) follows. iii) ⇒ i) Since u ∈ D (Rd ), we may choose the window function φ ∈ Dτ,σ (Rd ) in iii) to be centered near any point in Rd . Let φ be centered near 0, then clearly ψ = Tx0 φ is centered near x0 and (15) implies σ

σ

(1/ h)N N τ N , N ∈N |ξ |N

. )| = |Vφ u(x0 , ξ )|  e−Tτ,σ,h (|ξ |)  inf |ψu(ξ for some h > 0 and the proof is finished.

ξ ∈ Γ,  

Finally, we discuss local extended Gevrey regularity via the STFT. To that end we introduce the singular support as follows (cf. [33]). Definition 4 Let there be given x0 ∈ Rd , u ∈ D (U ), τ > 0 and σ > 1. Then x0 ∈ singsuppτ,σ (u) if and only if there exists open neighborhood Ω ⊂ U of x0 such that u ∈ Eτ,σ (Ω). The local regularity is related to the wave front set as follows. Proposition 2 Let τ > 0 and σ > 1, u ∈ D (U ). Let π1 : U × Rd \{0} → U be the standard projection given by π1 (x, ξ ) = x. Then singsuppτ,σ (u) = π1 (WFτ,σ (u)) . We refer to [33] for the proof, see also [24, Theorem 3.1] and [12, Proposition 11.1.1].

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Theorem 7 Let there be given x0 ∈ Rd , u ∈ D (U ), τ > 0 and σ > 1. Then x0 ∈ singsuppτ,σ (u) if and only if there exists a compact neighborhood K of x0 K−{x } K−{x } such that for every φ ∈ D{τ,σ } 0 (resp. φ ∈ D(τ,σ ) 0 ) there exists A, h > 0 (resp. for every h > 0 there exists A > 0) such that |Vφ u(x, ξ )| ≤ Ae−Tτ,σ,h (|ξ |) ,

x ∈ K, ξ ∈ Rd \{0},

(16)

where K − {x0 } = {y ∈ Rd | y + x0 ∈ K}. Proof We give the proof for the Roumieu case only. If x0 ∈ singsupp{τ,σ } (u) then by Proposition 2 it follows that (x0 , ξ ) ∈ WF{τ,σ } (u) for any ξ ∈ Rd \{0}, so that (16) follows from Theorem 6 iii). Now assume that (16) holds, then it holds for any cone Γ . By Theorem 6 we conclude that (x0 , ξ ) ∈ WF{τ,σ } (u) for any ξ ∈ Rd \{0}. Now Proposition 2 implies that x0 ∈ singsupp{τ,σ } (u), and the proof is completed.   Acknowledgements This work is supported by MPNTR through Project 174024.

References 1. A. Abdeljawad, M. Cappiello, J. Toft, Pseudo-differential calculus in anisotropic GelfandShilov setting, Integr. Equ. Oper. Theory, 91 (26) (2019), https://doi.org/10.1007/s00020-0192518-2 2. M. Cappiello, J. Toft, Pseudo-differential operators in a Gelfand-Shilov setting, Math. Nachr., 290 (2017), 738–755 3. M. Cicognani, D. Lorenz, Strictly hyperbolic equations with coefficients low-regular win time and smooth in space, J. Pseudo-Differ. Oper. Appl., 9 (2018), 643–675. 4. J. Chung, S.-Y. Chung, D. Kim, A characterization for Fourier hyperfunctions, Publ. Res. Inst. Math. Sci., 30 (1994), 203–208 5. E. Cordero, S. Pilipovi´c, L. Rodino, N. Teofanov, Localization operators and exponential weights for modulation spaces, Mediterranean Journal of Mathematics, 2 (2005), 381–394 6. E. Cordero, S. Pilipovi´c, L. Rodino, N. Teofanov, Quasianalytic Gelfand-Shilov spaces with application to localization operators, Rocky Mountain Journal of Mathematics, 40 (2010), 1123–1147 7. E. Cordero, F. Nicola, L. Rodino, Gabor representations of evolution operators, Trans. Am. Math. Soc., 367 (2015), 7639–7663. 8. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329–359. 9. H. G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983. and also in M. Krishna, R. Radha, S. Thangavelu (eds.), Wavelets and Their Applications, Allied Publishers, 99–140 (2003) 10. H. G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal., 86 (1989), 307–340 11. H. G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions II, Monatsh. f. Math. 108 (1989), 129–148 12. G. Friedlander, M. Joshi, The Theory of Distributions, Cambridge University Press, Cambridge, 1998.

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13. I.M. Gelfand, G.E. Shilov, Generalized Functions II, Academic Press, New York, 1968. 14. K. Gröchenig, Foundations of Time-frequency analysis, Birkhäuser, Boston, 2001. 15. K. Gröchenig, Weight functions in time-frequency analysis, in: L. Rodino, B.-W., Schulze, M. W. Wong (eds.) Pseudodifferential Operators: Partial Differential Equations and TimeFrequency Analysis, Fields Institute Comm., 52 (2007), 343–366 16. K. Gröchenig, G. Zimmermann, Hardy’s theorem and the short-time Fourier transform of Schwartz functions, J. London Math. Soc., 63 (2001), 205–214 17. A. Hoorfar, M. Hassani, Inequalities on the Lambert W function and hyperpower function, J. Inequalities in Pure and Applied Math., 9 (2008), 5pp. 18. L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. I: Distribution Theory and Fourier Analysis, Springer-Verlag, 1983. 19. H. Komatsu, Ultradistributions, I: Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 20 (1973), 25–105 20. F. Nicola, L. Rodino, Global Pseudo-differential calculus on Euclidean spaces, PseudoDifferential Operators. Theory and Applications 4, Birkhäuser Verlag, 2010. 21. S. Pilipovi´c, B. Prangoski, On the characterizations of wave front sets via short-time Fourier transform, Math. Notes, 105 (1–2) (2019), 153–157 22. S. Pilipovi´c, N. Teofanov, Wilson bases and ultra-modulation spaces, Math. Nachr., 242 (2002), 179–196 23. S. Pilipovi´c, N. Teofanov, and F. Tomi´c, On a class of ultradifferentiable functions, Novi Sad Journal of Mathematics, 45 (2015), 125–142. 24. S. Pilipovi´c, N. Teofanov, F. Tomi´c, Beyond Gevrey regularity, J. Pseudo-Differ. Oper. Appl., 7 (2016), 113–140. 25. S. Pilipovi´c, N. Teofanov, and F. Tomi´c, Superposition and propagation of singularities for extended Gevrey regularity, Filomat, 32 (2018), 2763–2782. 26. S. Pilipovi´c, N. Teofanov, and F. Tomi´c, Regularities for a new class of spaces between distributions and ultradistributions, Sarajevo Journal of Mathematics, 14 (2) (2018), 251–264. 27. S. Pilipovi´c, N. Teofanov, and F. Tomi´c, A Paley-Wiener theorem in extended Gevrey regularity, J. Pseudo-Differ. Oper. Appl. (2019), https://doi.org/10.1007/s11868-01900298-y 28. A. Rainer, G. Schindl, Composition in ultradifferentiable classes, Studia Math, 224 (2) (2014), 97–131. 29. L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, 1993. 30. M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, second edition (2001) 31. N. Teofanov, Modulation spaces, Gelfand-Shilov spaces and pseudodifferential operators, Sampl. Theory Signal Image Process, 5 (2006), 225–242 32. N. Teofanov, Gelfand-Shilov spaces and localization operators, Funct. Anal. Approx. Comput. 7 (2015), 135–158 33. N. Teofanov, F. Tomi´c, Inverse closedness and singular support in extended Gevrey regularity, J. Pseudo-Differ. Oper. Appl., 8 (3) (2017), 411–421. 34. N. Teofanov, F. Tomi´c, Ultradifferentiable functions of class Mpτ,σ and microlocal regularity, in: Oberguggenberger M., Toft J., Vindas J., Wahlberg P. (eds) Generalized Functions and Fourier Analysis. Operator Theory: Advances and Applications, vol 260. Birkhuser, Cham (2017), 193–213. 35. J. Toft, The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. Pseudo-Differ. Oper. Appl., 3 (2012), 145–227 36. J. Toft, Images of function and distribution spaces under the Bargmann transform, J. PseudoDiffer. Oper. Appl. 8 (2017), 83–139 37. F. Tomi´c, A microlocal property of PDOs in E(τ,σ ) (U ), in The Second Conference on Mathematics in Engineering: Theory and Applications, Novi Sad, (2017), 7–12.

Wiener Estimates on Modulation Spaces Joachim Toft

Abstract We characterise modulation spaces by Wiener estimates on the shorttime Fourier transforms. We use the results to refine some formulae for periodic distributions with Lebesgue estimates on their coefficients. Keywords Wiener spaces · Modulation spaces · Gelfand-Shilov · Quasi-Banach spaces · Coorbit spaces Mathematics Subject Classification (2010) Primary 42C20, 43A32, 42B35, 46E10; Secondary 46A16, 35A22, 37A05, 46E35

1 Introduction An essential step when linking modulation spaces to Gabor theory concerns the fact p,q p,q that Wiener norms of W(L(ω) ) = W(L∞ , L(ω) ) on short-time Fourier transforms p,q are equivalent to the modulation space norm of M(ω) on corresponding (ultra-) distributions, when p, q ∈ (0, ∞]. (See [13, 17] and Section 2 for notations.) We refer to Chapter 12 in [13] and the references therein for approaches in the Banach space case, p, q ≥ 1, and to [10] for the general case when p, q > 0. In the paper we show that the local component L∞ in the Wiener norm above can be replaced by any Lr space with r ∈ (0, ∞]. That is we show that the p,q (quasi-)norm of W(Lr , L(ω) ) on the short-time Fourier transforms are equivalent p,q to the modulation space norm of M(ω) on corresponding (ultra-)distributions, when p, q, r ∈ (0, ∞]. We apply the results to deduce some refined formulae on periodic functions and distributions, given in [30].

J. Toft () Department of Mathematics, Linnæus University, Växjö, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_26

475

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J. Toft p,q

The (classical) modulation space M(ω) , p, q ∈ [1, ∞] and ω(x, ξ ) being a moderate weight function on the phase space, as introduced by Feichtinger in [4] for some choices of ω, consists of all (ultra-)distributions whose short-time Fourier p,q transforms (STFT) have finite mixed L(ω) norm. It follows that the parameter p and the asymptotic behaviour of ω(x, ξ ) with respect to the configuration variable x to some extent quantify the degree of asymptotic decay, and that the parameter q and the asymptotic behaviour of ω(x, ξ ) with respect to the momentum variable p,q ξ quantify the degree of singularity of the distributions in M(ω) . The theory of modulation spaces was developed further and generalized in [6–8, 13], where Feichtinger and Gröchenig established the theory of coorbit spaces. After the pioneering paper [4], the usefulness of the modulation spaces in timefrequency analysis was confirmed, where they occur naturally. For example, in p,q [8, 13, 15], it is shown that all modulation spaces M(ω) admit reconstructible sequence space representations using Gabor frames, thereby they fit well in Gabor p,q theory. These properties were extended in [10] to any modulation space M(ω) with p and q belonging to the full interval (0, ∞] instead of [1, ∞], when ω is a polynomial type weight. (See [27] for an approach with general moderate weights.) Today, modulation spaces are important in several fields, e. g. the theory of partial differential equations, pseudo-differential calculus and statistics (cf. e. g. [25, 31]). An essential issue when extending the Gabor theory from L2 spaces to modulation spaces concerns estimates of short-time Fourier transforms by suitable Wiener norms. Let f and g be (Borel-)measurable (complex-valued) functions on Rd , and B1 and B2 be quasi-Banach spaces of measurable functions on Rd , which are solid, i. e. f Bj  g Bj when |f | ≤ |g|, j = 1, 2. Also let E = {e1 , . . . , ed } be an ordered basis of Rd , ΛE be the lattice spanned by E and κ(E) be the closed parallelepiped spanned by E. Then the Wiener quasi-norm f W(B1 ,B2 ) = f WE (B1 ,B2 ) of f , with B1 as local component and B2 as global component, is given by f WE (B1 ,B2 ) ≡ fB1 ,E B2 , where fB1 ,E (x) ≡

(1.1)

f · χj +κ(E) B1 · χj +κ(E) (x).

j ∈ΛE

Here χΩ is the characteristic function of the set Ω. The set WE (B1 , B2 ) is the quasi-Banach space which consists of all measurable f such that f WE (B1 ,B2 ) is finite. Usually, B1 = Lr (Rd ), for some r ∈ (0, ∞]. For this reason we set WrE (B2 ) = WE (Lr , B2 ) = WE (Lr (Rd ), B2 ), and note that WrE (B2 ) decreases with r because of the solidity of B2 and Hölder’s inequality. We also set WE (B2 ) = W∞ E (B2 ), since the case r = ∞ is especially

Wiener Estimates on Modulation Spaces

477

important. We observe that for measurable f , WE (B2 ) ⊆ B2

and

f B2  f WE (B2 )

(1.2)

d since WE (B2 ) is continuously embedded in L∞ loc (R ) ∩ B2 , again because of the solidity of B2 . In our situations, B2 is a mixed (quasi-)normed space of Lebesgue type, which also seems to be the most common situation in the applications (see e. g. [13]). p,q For example, when linking the modulation space M(ω) (Rd ) above to Gabor theory, p,q especially the space WE (L(ω) )1 is important, with E being the standard basis of p,q R2d . That is B2 = L(ω) (R2d ). For such choices of B2 , it follows by suitable applications of Hölder’s and Minkowski’s inequalities, that (1.2) is refined into p,q

p,q

p,q

WrE2 (L(ω) (R2d )) ⊆ L(ω) (R2d ) ⊆ WrE1 (L(ω) (R2d )) F Wr1 (Lp,q )  F Lp,q  F Wr2 (Lp,q ) ,

and

E

when

(ω)

(ω)

E

(1.3)

(ω)

r1 ≤ min(p, q), r2 ≥ max(p, q),

when F is measurable on R2d . An essential step for the link between Gabor theory and modulation spaces is p,q p,q to show that the sets of short-time Fourier transforms in L(ω) and WE (L(ω) ) are the same. That is, if φ1 and φ2 are fixed and suitable (window) functions, then additionally to the relation (1.3) we have Vφ1 f Lp,q Vφ2 f WE (Lp,q ) . (ω)

(1.4)

(ω)

In the case p, q ≥ 1, (1.4) is essentially reached by the convolution estimate F ∗ G WE (Lp,q )  F Lp,q G WE (L1 (ω)

(v) )

(ω)

in combination with Vφ φ0 WE (L1

(v) )

φ0 M 1 φ M 1 (v)

(v)

and |Vφ f |  |Vφ0 f | ∗ |Vφ φ0 |.

(1.5)

(Cf. Chapter 11 in [13].) Here v is chosen such that ω is v-moderate. p,q

p,q

p,q

p,q

often write WE (E,(ω) ) and WrE (E,(ω) ) instead of WE (L(ω) ) and WrE (L(ω) ) in order for p,q emphasizing that we apply the discrete E,(ω) (ΛE ) norm instead of corresponding continuous norm in (1.1). (Cf. Definition 8.)

1 We

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J. Toft

In the general case, p, q > 0, the preceding approach does not work because of lack of convexity of the topology in involved spaces when p < 1 or q < 1. In this situation the relationship (1.4) is first deduced for Gaussian windows, using suitable arguments based on subharmonic mean-value estimates of analytic functions, and then combining F ∗ G WE (Lp,q )  F WE (Lp,q ) G WE (L1 ,Lr ) , (ω)

r = min(1, p, q),

(v)

(ω)

with (1.5) to deduce Vφ f WE (Lp,q )  Vφ0 f WE (Lp,q ) Vφ φ0 WE (L1 ,Lr ) , (ω)

(v)

(ω)

r = min(1, p, q). (1.6)

This leads to that (1.4) is carried over from Gaussian windows to other suitable windows φ1 and φ2 . As a consequence of (1.2) we have f M p,q Vφ1 f Lp,q Vφ2 f WE (L∞ ,Lp,q ) , (ω)

(ω)

(ω)

(1.7)

which links Vφ2 f WE (L∞ ,Lp,q ) to the modulation space norm of f . On the other (ω)

hand, in (1.6), the factor Vφ φ0 WE (L1 ,Lr ) involves the L1 norm instead of the L∞ (v) norm as the local component in the Wiener norm. This leads to questions wether it is possible to link the Wiener norm f → Vφ2 f WE (Lr ,Lp,q ) (ω)

to modulation space norms for other values than r = ∞. In Sect. 3 we give an afirmative answer and extend (1.7) into f M p,q Vφ1 f Lp,q Vφ2 f WE (Lr ,Lp,q ) , (ω)

(ω)

(ω)

(1.7)

for any p, q, r ∈ (0, ∞]. In the end of Sect. 3 we apply (1.7) to characterize periodic functions and distributions in modulation spaces. It follows from [30] that if q ∈ (0, ∞] and f is a 2π -periodic Gelfand-Shilov distribution on Rd with Fourier coefficients c(f, α), α ∈ Zd , then {c(f, α)}α∈Zd ∈ q

⇔ f ∈ M ∞,q ,

or equivalently, {c(f, α)}α∈Zd ∈ q

⇔

ξ → Vφ f ( · , ξ ) L∞ (Rd ) ∈ Lq .

(1.8)

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We note that a proof of (1.8) in the case q ∈ [1, ∞] can be found in e. g. [26], and with some extensions in [23]. By observing that periodicity of f induce the same periodicity for x → |Vφ f (x, ξ )|, it follows that (1.8) is the same as {c(f, α)}α∈Zd ∈ q

⇔

ξ → Vφ f ( · , ξ ) L∞ ([0,2π ]d ) ∈ Lq .

(1.8)

In Sect. 3 we show that the latter equivalence hold true with Lr ([0, 2π ]d ) norm in place of L∞ ([0, 2π ]d ) norm for every r ∈ (0, ∞]. That is, we improve (1.8) into {c(f, α)}α∈Zd ∈ q

⇔

ξ → Vφ f ( · , ξ ) Lr ([0,2π ]d ) ∈ Lq .

(1.8)

In particular, if q = r < ∞, then we obtain

 |c(f, α)| < ∞ q

⇔

α∈Zd

[0,2π ]d ×Rd

|Vφ f (x, ξ )|q dxdξ < ∞. (1.9)

More generally, we deduce weighted versions of these identities. Since our weights include general moderate weights which are allowed to possess exponential type of growth and decay, we formulate our results in the framework of Gelfand-Shilov spaces of functions and distributions.

2 Preliminaries In this section we recall some basic facts. We start by discussing Gelfand-Shilov spaces and their properties. Thereafter we recall some properties of modulation spaces and discuss different aspects of periodic distributions.

2.1 Gelfand-Shilov Spaces and Gevrey Classes Let 0 < s, σ ∈ R be fixed. Then the Gelfand-Shilov space Ssσ (Rd ) (Σsσ (Rd )) of Roumieu type (Beurling type) with parameters s and σ consists of all f ∈ C ∞ (Rd ) such that σ ≡ sup f Ss,h

|x α ∂ β f (x)| h|α+β| α!s β!σ

(2.1)

is finite for some h > 0 (for every h > 0). Here the supremum should be taken over all α, β ∈ Nd and x ∈ Rd . We equip Ssσ (Rd ) (Σsσ (Rd )) with the canonical

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inductive limit topology (projective limit topology) with respect to h > 0, induced by the semi-norms in (2.1). The Gelfand-Shilov distribution spaces (Ssσ ) (Rd ) and (Σsσ ) (Rd ) are the dual spaces of Ssσ (Rd ) and Σsσ (Rd ), respectively. As for the Gelfand-Shilov spaces there is a canonical projective limit topology (inductive limit topology) for (Ssσ ) (Rd ) ((Σsσ ) (Rd )).(Cf. [11, 18, 20].) For conveniency we set Ss = Sss ,

Ss = (Sss ) ,

Σs = Σss

and

Σs = (Σss ) .

From now on we let F be the Fourier transform which takes the form  d f (x)e−ix,ξ  dx (F f )(ξ ) = f(ξ ) ≡ (2π )− 2 Rd

when f ∈ L1 (Rd ). Here  · , ·  denotes the usual scalar product on Rd . The map F extends uniquely to homeomorphisms on S  (Rd ), from (Ssσ ) (Rd ) to (Sσs ) (Rd ) and from (Σsσ ) (Rd ) to (Σσs ) (Rd ). Furthermore, F restricts to homeomorphisms on S (Rd ), from Ssσ (Rd ) to Sσs (Rd ) and from Σsσ (Rd ) to Σσs (Rd ), and to a unitary operator on L2 (Rd ). Gelfand-Shilov spaces may in convenient ways be characterized by suitable estimates on the involved functions and their Fourier transforms (see [2]). In the same way, Gelfand-Shilov spaces and their distribution spaces can be characterized by estimates of short-time Fourier transforms, (see e. g. [16, 28]). More precisely, let s ≥ 12 and φ ∈ Ss (Rd ) be fixed. Then the short-time Fourier transform Vφ f of f ∈ Ss (Rd ) with respect to the window function φ is the Gelfand-Shilov distribution on R2d , defined by Vφ f (x, ξ ) = F (f φ( · − x))(ξ ).

(2.2)

If f, φ ∈ Ss (Rd ), then it follows that − d2

Vφ f (x, ξ ) = (2π )



f (y)φ(y − x)e−iy,ξ  dy.

The characterizations (1) and (2) in the following proposition are special cases of [16, Theorem 2.7] and [28, Proposition 2.2] respectively. The proofs are therefore omitted. Here and in what follows, g  h means that g(θ ) ≤ ch(θ ) holds uniformly for all θ in the intersection of the domains of g and h for some constant c > 0, and we write g h when g  h  g. Proposition 1 Let f be a Gelfand-Shilov distribution on Rd , s ≥ φ ∈ Ss (Rd ) \ 0 (φ ∈ Σs (Rd ) \ 0).

1 2

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Then the following is true: (1) f ∈ Ss (Rd ) (f ∈ Σs (Rd )), if and only if |Vφ f (x, ξ )|  e−r(|x|

1 1 s +|ξ | s )

,

(2.3)

,

(2.4)

for some r > 0 (for every r > 0); (2) f ∈ Ss (Rd ) (f ∈ Σs (Rd )), if and only if |Vφ f (x, ξ )|  er(|x|

1 1 s +|ξ | s )

for every r > 0 (for some r > 0). Next we consider Gevrey classes on Rd . Let σ ≥ 0. For any compact set K ⊆ h > 0 and f ∈ C ∞ (K) let

Rd ,

 f K,h,σ ≡ sup α∈Nd

∂ α f L∞ (K) h|α| α!σ

(2.5)

.

The Gevrey class Eσ (K) (E0,σ (K)) of order σ and of Roumieu type (of Beurling type) is the set of all f ∈ C ∞ (K) such that (2.5) is finite for some (for every) h > 0. We equip Eσ (K) (E0,σ (K)) with the inductive (projective) limit topology with respect to h > 0, supplied by the seminorms in (2.5). Finally if {Kj }j ≥1 is an exhaustive set of compact subsets of Rd , then let Eσ (Rd ) = proj lim Eσ (Kj )

and

E0,σ (Rd ) = proj lim E0,σ (Kj ).

j

j

In particular, Eσ (Rd ) =



Eσ (Kj )

and

E0,σ (Rd ) =

j ≥1



E0,σ (Kj ).

j ≥1

It is clear that E0,0 (Rd ) contains all constant functions on Rd , and that E0 (Rd ) contains all non-constant trigonometric polynomials.

2.2 Ordered, Dual and Phase Split Bases Our discussions involving periodicity, modulation spaces and Wiener spaces are done in terms of suitable bases.

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Definition 1 Let E = {e1 , . . . , ed } be an ordered basis of Rd . Then E  denotes the basis of e1 , . . . , ed in Rd which satisfies ej , ek  = 2π δj k

for every

j, k = 1, . . . , d.

The corresponding lattices are given by ΛE = { n1 e1 + · · · + nd ed ; (n1 , . . . , nd ) ∈ Zd }, and ΛE = ΛE  = { ν1 e1 + · · · + νd ed ; (ν1 , . . . , νd ) ∈ Zd }. The sets E  and ΛE are called the dual basis and dual lattice of E and ΛE , respectively. If E1 , E2 are ordered bases of Rd such that a permutation of E2 is the dual basis for E1 , then the pair (E1 , E2 ) are called permuted dual bases (to each others on Rd ). Remark 1 Evidently, if E is the same as in Definition 1, then there is an invertible d × d matrix TE with E as the image of the standard basis in Rd . Then E  is the image of the standard basis under the map TE  = 2π(TE−1 )t . Two ordered bases on Rd can be used to construct a uniquely defined ordered basis for R2d as in the following definition. Definition 2 Let E1 , E2 be ordered bases of Rd , V1 = { (x, 0) ∈ R2d ; x ∈ Rd },

V2 = { (0, ξ ) ∈ R2d ; ξ ∈ Rd }

and let πj from R2d to Rd , j = 1, 2, be the projections π1 (x, ξ ) = x

and

π2 (x, ξ ) = ξ.

Then E1 × E2 is the ordered basis {e1 , . . . , e2d } of R2d such that {e1 , . . . , ed } ⊆ V1 , {ed+1 , . . . , e2d } ⊆ V2

E1 = {π1 (e1 ), . . . , π1 (ed )}, and

E2 = {π2 (ed+1 ), . . . , π2 (e2d )}.

In the phase space R2d ) Rd × Rd it is convenient to consider phase split bases, which are defined as follows. Definition 3 Let V1 , V2 , π1 and π2 be as in Definition 2, E be an ordered basis of the phase space R2d and let E0 ⊆ E. Then E is called phase split (with respect to E0 ), if the following is true: (1) the span of E0 and E \ E0 equal V1 and V2 , respectively; (2) let E1 = π1 (E0 ) and E2 = π2 (E \ E0 ) be the bases in Rd which preserves the orders from E0 and E \ E0 . Then (E1 , E2 ) are permuted dual bases.

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If E is a phase split basis with respect to E0 and E0 consists of the first d vectors in E, then E is called strongly phase split (with respect to E0 ). In Definition 3 it is understood that the orderings of E0 and E \ E0 are inherited from the ordering in E. Remark 2 Let E and Ej , j = 0, 1, 2 be the same as in Definition 3. It is evident that E0 and E \ E0 consist of d elements, and that E1 and E2 are uniquely defined. The pair (E1 , E2 ) is called the pair of permuted dual bases, induced by E and E0 . On the other hand, suppose that (E1 , E2 ) is a pair of permuted dual bases to each others on Rd . Then it is clear that for E1 × E2 = {e1 , . . . , e2d } in Definition 2 and E0 = {e1 , . . . , ed }, we have that E0 and E fulfils all properties in Definition 3. In this case, E1 × E2 is called the phase split basis (of R2d ) induced by (E1 , E2 ). It follows that if E  , E1 and E2 are the dual bases of E, E1 and E2 , respectively, then E  = E1 × E2 .

2.3 Invariant Quasi-Banach Spaces and Spaces of Mixed Quasi-Normed Spaces of Lebesgue Types We recall that a quasi-norm · B of order r ∈ (0, 1] on the vector-space B over C is a nonnegative functional on B which satisfies 1

f + g B ≤ 2 r −1 ( f B + g B ), f, g ∈ B, α · f B = |α| · f B ,

α ∈ C,

(2.6) f ∈B

and f B = 0

⇔

f = 0.

The space B is then called a quasi-norm space. A complete quasi-norm space is called a quasi-Banach space. If B is a quasi-Banach space with quasi-norm satisfying (2.6), then by [1, 24] there is an equivalent quasi-norm to · B which additionally satisfies f + g rB ≤ f rB + g rB ,

f, g ∈ B.

(2.7)

From now on we always assume that the quasi-norm of the quasi-Banach space B is chosen in such way that both (2.6) and (2.7) hold. Before giving the definition of v-invariant spaces, we recall some facts on weight functions. d A weight or weight function on Rd is a positive function ω ∈ L∞ loc (R ) such that d 1/ω ∈ L∞ loc (R ). The weight ω is called moderate, if there is a positive weight v on

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Rd such that ω(x + y)  ω(x)v(y),

x, y ∈ Rd .

(2.8)

If ω and v are weights on Rd such that (2.8) holds, then ω is also called v-moderate. It follows that (2.8) implies that ω fulfills the estimates v(−x)−1  ω(x)  v(x),

x ∈ Rd .

(2.9)

We let PE (Rd ) be the set of all moderate weights on Rd . It can be proved that if ω ∈ PE (Rd ), then ω is v-moderate for some v(x) = er|x| , provided the positive constant r is large enough (cf. [14]). In particular, (2.9) shows that for any ω ∈ PE (Rd ), there is a constant r > 0 such that e−r|x|  ω(x)  er|x| ,

x ∈ Rd .

We say that v is submultiplicative if v is even and (2.8) holds with ω = v. In the sequel, v and vj for j ≥ 0, always stand for submultiplicative weights if nothing else is stated. Next we shall discuss suitable classes of function spaces, and begin to consider those spaces that ramify sequence spaces. Suppose that E is an ordered basis of Rd and let Λ be a lattice, spanned by the vectors {e0,1 , . . . , e0,d } which are parallel with the vectors in E. Then we let 0 (Λ) be the set of all formal sequences {a(j )}j ∈Λ ⊆ C, and we let 0 (Λ) be the set of all such sequences such that at most finite numbers of a(j ) are non-zero. For any integer N ≥ 1, we let · 0,N (Λ) be the semi-norm on 0 (Λ), given by a → a 0,N (Λ) ≡

sup

j ∈Λ∩BN

|a(j )|,

(2.10)

where Br = Br (0) is the open ball with radius r > 0 and centered at origin. It follows that 0 (Λ) is a Fréchet space under the topology, induced from these seminorms. For the integer N ≥ 1, we also let 0,N (Λ) be the set of all sequences in 0 (Λ) with support contained in BN . It follows that 0,N (Λ) is a finite dimensional Banach space with norm given by (2.10), and that 0 (Λ) =



0,N (Λ).

N ≥1

We let the topology of 0 (Λ) be the inductive limit topology of 0,N (Λ) with respect to N ≥ 1. The next definition is similar to [6, Section 3] in the Banach space case.

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Definition 4 Let r ∈ (0, 1], v ∈ PE (Rd ) and let B = B(Rd ) ⊆ Lrloc (Rd ) be a quasi-Banach space such that Σ1 (Rd ) ⊆ B(Rd ). Then B is called v-invariant on Rd , translation invariant Quasi-Banach Function Space on Rd (with respect to v) or invariant QBF-space on Rd of order r, if the following is true: (1) x → f (x + y) belongs to B for every f ∈ B; (2) there is a constant C > 0 such that f1 B ≤ C f2 B when f1 , f2 ∈ B are such that |f1 | ≤ |f2 |. Moreover, f ( · + y) B  f B v(y),

f ∈ B, y ∈ Rd .

Let B be as in Definition 4, E be a basis for Rd and let κ(E) be the closed parallelepiped spanned by E. The discrete version, B ,E = B ,E (ΛE ), of B with respect to E is the set of all a ∈ 0 (ΛE ) such that a B,E

1 1 1 1 1 1 1 ≡1 a(j )χ j +κ(E) 1 1 1j ∈ΛE 1

B

is finite. Here recall that χΩ denotes the characteristic function of the set Ω. An important example on v-invariant spaces concerns mixed quasi-norm spaces of Lebesgue type, given in the following definition. Definition 5 Let E = {e1 , . . . , ed } be an ordered basis of Rd , κ(E) be the parallelepiped spanned by E, ω ∈ PE (Rd ) q = (q1 , . . . , qd ) ∈ (0, ∞]d and r = min(1, q). If f ∈ Lrloc (Rd ), then f Lq

E,(ω)

≡ gd−1 Lqd (R)

where gk : Rd−k → R, k = 0, . . . , d − 1, are inductively defined as g0 (x1 , . . . , xd ) ≡ |f (x1 e1 + · · · + xd ed )ω(x1 e1 + · · · + xd ed )|, and gk (zk ) ≡ gk−1 ( · , zk ) Lqk (R) ,

zk ∈ Rd−k , k = 1, . . . , d − 1.

q

If Ω ⊆ Rd is measurable, then LE,(ω) (Ω) consists of all f ∈ Lrloc (Ω) with finite quasi-norm f Lq

E,(ω) (Ω)

q

≡ fΩ Lq

d , E,(ω) (R )

fΩ (x) ≡

⎧ ⎨f (x),

when x ∈ Ω

⎩0,

when x ∈ / Ω.

The space LE,(ω) (Ω) is called E-split Lebesgue space (with respect to ω, q and Ω).

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J. Toft p

p

We let E,(ω) (ΛE ) be the discrete version of B = LE,(ω) (Rd ) when p ∈ (0, ∞]d . q

q

Remark 3 Evidently, LE,(ω) (Ω) and E,(ω) (Λ) in Definition 5 are quasi-Banach spaces of order min(1, p). We set q

q

LE = LE,(ω)

and

q

q

E = E,(ω)

when ω = 1. For conveniency we identify q = (q, . . . , q) ∈ (0, ∞]d with q ∈ (0, ∞] when considering spaces involving Lebesgue exponents. In particular, q

q

q

q

q

q

LE,(ω) = LE,(ω) , LE = LE , E,(ω) = E,(ω)

q

q

and

E = E

and

q ,

for such q, and notice that these spaces agree with q

q

Lq ,

L(ω) ,

(ω)

respectively, with equivalent quasi-norms.

2.4 Modulation and Wiener Spaces We consider a general class of modulation spaces given in the following definition (cf. [5]). Here recall Sect. 2.1 for the definition of Fourier transforms and short-time Fourier transforms. Definition 6 Let ω, v ∈ PE (R2d ) be such that ω is v-moderate, B be a v-invariant quasi-Banach space on R2d , and let φ ∈ Σ1 (Rd ) \ 0. Then the modulation space M(ω, B) consists of all f ∈ Σ1 (Rd ) such that f M(ω,B ) ≡ Vφ f · ω B

(2.11)

is finite. An important family of modulation spaces which contains the classical modulation spaces, introduced by Feichtinger in [4], is given next. Definition 7 Let p, q ∈ (0, ∞]d , E1 and E2 be ordered bases of Rd , E = E1 ×E2 , φ ∈ Σ1 (Rd ) \ 0 and let ω ∈ PE (R2d ). For any f ∈ Σ1 (Rd ) set f M p,q ≡ H1,f,E1 ,p,ω Lq , E,(ω)

E2

where

H1,f,E1 ,p,ω (ξ ) ≡ Vφ f ( · , ξ )ω( · , ξ ) Lp

E1

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487

and f W p,q ≡ H2,f,E2 ,q,ω Lp , E1

E,(ω)

where p,q

H2,f,E2 ,q,ω (x) ≡ Vφ f (x, · )ω(x, · ) Lq . E2

p,q

The modulation space ME,(ω) (Rd ) (WE,(ω) (Rd )) consist of all f ∈ Σ1 (Rd ) such that f M p,q ( f W p,q ) is finite. E,(ω)

E,(ω)

The theory of modulation spaces has developed in different ways since they were introduced in [4] by Feichtinger. (Cf. e. g. [5, 10, 13, 27].) For example, let p, q, E, p,q ω and v be the same as in Definition 6 and 7, and let B = LE (R2d ) and r = p,q d min(1, p, q). Then M(ω, B) = ME,(ω) (R ) is a quasi-Banach space. Moreover, p,q p,q f ∈ ME,(ω) (Rd ) if and only if Vφ f · ω ∈ LE (R2d ), and different choices of φ give rise to equivalent quasi-norms in Definition 7. We also note that for any such B, then p,q

Σ1 (Rd ) ⊆ ME,(ω) (Rd ) ⊆ Σ1 (Rd ). p,q

Similar facts hold for the space WE,(ω) (Rd ). (Cf. [10, 27].) We shall consider various kinds of Wiener spaces involved later on when finding different characterizations of modulation spaces. The following type of Wiener spaces can essentially be found in e. g. [6, 10, 13], and is related to coorbit spaces of Lebesgue spaces. (See also the introduction for an overview.) Definition 8 Let r ∈ (0, ∞]d , ω0 ∈ PE (Rd ), ω ∈ PE (R2d ), φ ∈ Σ1 (Rd ) \ 0, E ⊆ Rd be an ordered basis, and let κ(E) be the closed parallelepiped spanned by E. Also let B = B(Rd ) and B0 = B0 (Rd ) be invariant QBF-spaces on Rd , f and F be measurable on Rd respective R2d , Fω = F · ω, and let B ,E (ΛE ) be the discrete version of B with respect to E. (1) the quasi norm f → f WrE (ω0 ,B,E ) is given by f WrE (ω0 ,B,E ) ≡ hE,ω0 ,q,f B,E (ΛE ) , hE,ω0 ,q,f (j ) = f LrE (j +κ(E)) ω0 (j ),

j ∈ ΛE .

The set WrE (ω, B ,E ) consists of all measurable f on Rd such that f WrE (ω0 ,B,E ) < ∞; (2) the quasi norms F → F Wrk,E (ω,B,E ,B0 ) , k = 1, 2, are given by F Wr1,E (ω,B,E ,B0 ) ≡ ϕF,ω,r,B ,E B0 , where

ϕF,ω,r,B ,E (ξ ) = Fω ( · , ξ ) WrE (1,B,E ) ,

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J. Toft

and F Wr2,E (ω,B,E ,B0 ) ≡ ψF,ω,B0 WrE (1,B,E ) , where

ψF,ω,B0 (x) = Fω (x, · ) B0 .

The set Wrk,E (ω, B ,E , B0 ) consists of all measurable F on R2d such that F Wrk,E (ω,B,E ,B0 ) < ∞, k = 1, 2. The space WrE (ω0 , B ,E ) in Definition 8 is the Wiener amalgam space with LrE as local (quasi-)norm and B or B ,E (ΛE ) as global component. They are also related to coorbit spaces. (See [3, 6, 7, 9, 21, 22].) Remark 4 Let p, ω0 , ω, E, B, B0 , f and F be the same as in Definition 8. Evidently, by using the fact that ω0 is v0 -moderate for some v0 , it follows that f · ω0 Wq (1,B,E ) f Wq (ω0 ,B,E ) E

E

and F · ω Wq

k,E (1,B,E

,B0 )

= F Wq

k,E (ω,B,E

,B0 )

for k = 1, 2. Furthermore, q

q

W1,E (ω, B ,E , B0 ) = ω−1 · WE (1, B ,E ; B0 ) and q

q

W2,E (ω, B ,E , B0 ) = ω−1 · B0 (Rd ; WE (1, B ,E )). Here and in what follows, B(Rd ; B0 ) = B(Rd ; B0 (Rd0 )) is the set of all functions g in B with values in B0 , which are equipped with the quasi-norm g B (Rd ;B0 ) ≡ g0 B ,

g0 (x) ≡ g(x) B0 ,

when B(Rd ) and B0 (Rd0 ) are invariant QBF-spaces. Later on we discuss periodicity in the framework of certain modulation spaces which are related to spaces which are defined by imposing L∞ -conditions on the configuration variable x of corresponding short-time Fourier transforms in (2.2). Definition 9 Let E, r, B0 and ω ∈ PE (R2d ) be the same as in Definition 8, and let φ ∈ Σ1 (Rd ) \ 0. Then MrE (ω, B0 ) and WrE (ω, B0 )) are the sets of all f ∈ Σ1 (Rd ) such that f MrE (ω,B0 ) ≡ Vφ f Wr1,E (ω,∞ E ,B0 )

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respectively f WrE (ω,B0 ) ≡ Vφ f Wr2,E (ω,∞ E ,B0 ) are finite. Remark 5 For the spaces in Definition 8 we set Wq0 ,r 0 = Wr , when r 0 = (r1 , . . . , rd ) ∈ (0, ∞]d ,

and

r = (q0 , . . . , q0 , r1 , . . . , rd ) ∈ (0, ∞]2d ,

and similarly for other types of exponents and for the spaces in Definitions 7 and 9. (See also Remark 3.) We also set ∞,q

∞,q

ME,(ω) = ME2 ,(ω)

and

∞,q

∞,q

WE,(ω) = WE2 ,(ω)

when E1 , E2 are ordered bases of Rd and E = E1 × E2 , for spaces in Definition 7, since these spaces are independent of E1 . In Sect. 3 we prove that if B0 is an E-split Lebesgue space on Rd and ω(x, ξ ) ∈ PE (R2d ) which is constant with respect to the x variable, then MrE (ω, B0 ) and WrE (ω, B0 ) are independent of r and agree with modulation spaces of the form in Definition 6 (cf. Theorem 1). The next result is a reformulation of [27, Proposition 3.4], and indicates how Wiener spaces are connected to modulation spaces. The proof is therefore omitted. 1 Here, x ≡ (1 + |x|2 ) 2 when x ∈ Rd and  (Θρ v)(x, ξ ) = v(x, ξ )(x, ξ ) , ρ

where

ρ ≥ 2d

1 −1 , r

(2.12)

for any submultiplicative v ∈ PE (R2d ) and r ∈ (0, 1]. It follows that L1(Θρ v) (R2d ) 1 r (Rd ). (Rd ) ⊆ M(v) is continuously embedded in Lr(v) (R2d ), giving that M(Θ ρ v)

Proposition 2 Let E be a phase split basis for R2d , p ∈ (0, ∞]2d , r ∈ (0, min(1, p)], ω, v ∈ PE (R2d ) be such that ω is v-moderate, ρ and Θρ v be as 1 in (2.12) with strict inequality when r < 1, and let φ1 , φ2 ∈ M(Θ (Rd ) \ 0. Then ρ v) f M p

E,(ω)

Vφ1 f Lp

E,(ω)

Vφ2 f W∞ (ω,p ) , E

E

f ∈ Σ1 (Rd ).

In particular, if f ∈ Σ1 (Rd ), then p

f ∈ ME,(ω) (R2d )

⇔

p

Vφ1 f ∈ LE,(ω) (R2d )

⇔

p

Vφ2 f ∈ W∞ E (ω, E (ΛE )). p

In Sect. 3 we extend this result in such way that we may replace W∞ E (ω, E ) by p r WE (ω, E ) for any r > 0.

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2.5 Classes of Periodic Elements We consider spaces of periodic Gevrey functions and their duals. Let s, σ ∈ R+ be such that s + σ ≥ 1, f ∈ (Ssσ ) (Rd ), E be a basis of Rd and let E0 ⊆ E. Then f is called E0 -periodic if f (x + y) = f (x) for every x ∈ Rd and y ∈ E0 . We note that for any E-periodic function f ∈ C ∞ (Rd ), we have the absolutely convergent Fourier series expansion f =

c(f, α)ei · ,α ,

(2.13)

α∈ΛE

where c(f, α) are the Fourier coefficients given by c(f, α) ≡ |κ(E)|−1 (f, ei · ,α )L2 (E) . E (Rd ) and E E (Rd ) be the sets of all For any s ≥ 0 and basis E ⊆ Rd we let E0,σ σ E-periodic elements in E0,σ (Rd ) and in Eσ (Rd ), respectively.

Remark 6 Let E = {e1 , . . . , ed } be an ordered basis on Rd and V be a topological space of functions or (ultra-)distributions on Rd . Then we use the convention that V E (E as upper case index) denotes the E periodic elements in V , while VE (E as lower case index) is the set of all functions or ultra-distributions f on Rd such that Rd - (x1 , . . . , xd ) → f (x1 e1 + · · · + xd ed ) belongs to V . Let s, s0 , σ, σ0 > 0 be such that s + σ ≥ 1, s0 + σ0 ≥ 1 and (s0 , σ0 ) = ( 12 , 12 ). E ) (Rd ) of E E (Rd ) and E E (Rd ), Then we recall that the duals (EσE ) (Rd ) and (E0,σ σ 0,σ0 0 respectively, can be identified with the E-periodic elements in (Ssσ ) (Rd ) and (Σsσ00 ) (Rd ) respectively via unique extension of the form (f, φ)E =

c(f, α)c(φ, α)

α∈ΛE E (Rd ) × E E (Rd ). We also let (E E ) (Rd ) be the set of all formal expansions on E0,σ 0,σ0 0 0 in (2.13) and E0E (Rd ) be the set of all formal expansions in (2.13) such that at most finite numbers of c(f, α) are non-zero (cf. [30]). We refer to [19, 30] for more E and their duals. characterizations of EσE , E0,σ The following definition takes care of spaces of formal expansions (2.13) with coefficients obeying specific quasi-norm estimates.

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Definition 10 Let E be a basis of Rd , B be a quasi-Banach space continuously embedded in 0 (ΛE ) and let ω0 be a weight on Rd . Then LE (ω0 , B) consists of all f ∈ (E0E ) (Rd ) such that f LE (ω0 ,B ) ≡ {c(f, α)ω0 (α)}α∈ΛE B is finite. If ω0 ∈ PE (Rd ) and ω(x, ξ ) = ω0 (ξ ), then f MrE (ω,B ) = gω0 B , when

f ∈ (E0E ) (Rd ),

g(ξ ) = Vφ f ( · , ξ ) LrE (κ(E)) ,

(2.14)

and f WrE (ω,B ) = h LrE (κ(E)) , when

f ∈ (E0E ) (Rd ),

h(x) = Vφ f (x, · )ω0 B ,

(2.15)

because that the E-periodicity of x → |Vφ f (x, ξ )| when f is E periodic gives g(ξ ) = Vφ f ( · , ξ ) LrE (κ(E)) = Vφ f ( · , ξ ) LrE (x+κ(E)) , (2.16) h LrE (κ(E)) = h LrE (x+κ(E)) ,

x ∈ Rd .

The following result characterizes periodic elements in modulation spaces in terms of estimates of the Fourier coefficients of the involved distributions. We omit the proof since it is given in [30]. (Cf. [30, Theorem 3.1].) Proposition 3 Let E be a basis of Rd , r ∈ (0, 1], B ⊆ Lrloc (Rd ) be an E  -split Lebesgue space, B ,E (ΛE ) be its discrete version, ω0 ∈ PE (Rd ) and let ω(x, ξ ) = ω0 (ξ ) when x, ξ ∈ Rd . Then LE (ω0 , B ,E ) = M∞ E (ω, B)



(E0E ) (Rd ) = W∞ E (ω, B)



(E0E ) (Rd ).

p

Earlier it was as announced that the WrE (ω, E ) norm of short-time Fourier transforms are independent of r ∈ (0, ∞]d in Sect. 3. At the same time it will follow that if B is a suitable quasi-norm space of Lebesgue type, then MrE1 (ω, B) = WrE2 (ω, B)

when

for every r 1 , r 2 ∈ (0, ∞]d (see Proposition 6).

ω ∈ PE (R2d )

(2.17)

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Remark 7 The link between periodic Gelfand-Shilov distributions and formal Fourier series expansions is given by the formula

d

f, φ = (2π ) 2

(−α). c(f, α)φ

(2.18)

α∈ΛE

3 Estimates on Wiener Spaces and Periodic Elements in Modulation Spaces In this section we deduce equivalences between various Wiener (quasi-)norm estimates on short-time Fourier transforms. Especially we prove that (2.17) holds for every r 1 , r 2 ∈ (0, ∞]d .

3.1 Estimates of Wiener Spaces We begin with the following inclusions between the different Wiener spaces in the previous section. Proposition 4 Let (E1 , E2 ) be permuted dual bases of Rd , E = E1 ×E2 , p, q, r ∈ (0, ∞]d , r1 ∈ (0, min(p, q, r)], r2 ∈ (0, min(q)], and let ω ∈ PE (R2d ). Then p,q

p

q

r d Wr,∞ E (ω, E (ΛE )) → W1,E1 (ω, E1 (ΛE1 ), LE2 (R )) p,q

→ WrE1 (ω, E (ΛE ))

(3.1)

and q,p

p

q

r2   d W∞ E  (ω, E  (ΛE )) → W2,E  (ω, E  (ΛE2 ), LE  (R )) 2

2

1

q,p

→ WrE2 (ω, E  (ΛE )).

(3.2)

Remark 8 For the involved spaces in Proposition 4 it follows from Hölder’s inequality that p

q

p

Wr1,E1 (ω, E1 (ΛEk ), LE2 (Rd )),

q

Wr2,E  (ω, E  (ΛE2 ), LE  (Rd )) 2

and p

WrE (ω, E (ΛE ))

2

1

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493

increase with respect to p and decrease with respect to r. We need the following lemma for the proof of Proposition 4. Lemma 1 Let ω ∈ PE (Rd ), E be an ordered basis of Rd , κ(E) the parallelepiped spanned by E, p ∈ (0, ∞]d , r ∈ (0, min(p)] and let f be measurable on Rd . Then a p (ΛE )  f Lp E

E,(ω)

a(j ) = f LrE (j +κ(E)) ω(j ).

,

(3.3)

In the proof and in what follows we let Qk = [0, 1]k be the k dimensional closed unit cube and Q = Q1 . Proof Let gk be the same as in Definition 5, TE be the linear map which maps the standard basis into E and p k = (pk+1 , . . . , pd ), when k ≥ 1. Then f LrE (TE (j )+κ(E)) ω(j ) f · ω LrE (TE (j )+κ(E)) g0 LrE (j +Qd ) ,

j ∈ Zd .

This reduces the situation to the case that E is the standard basis, κ(E) = Qd and ω = 1. Moreover, by replacing |f |r with f and pj r by pj , j = 1, . . . , d, we may assume that r = 1 (and that each pj ≥ 1). By induction it suffices to prove that if ak (l) = gk L1 (l+Qd−k ) ,

l ∈ Zd−k ,

then ak pk (Zd−k ) ≤ ak+1 pk+1 (Zd−k−1 ) ,

k = 0, . . . , d − 1,

(3.4)

since a0 p0 (Zd ) is equal to the left-hand side of (3.3), and ad = ad ∞ (Z0 ) is equal to the right-hand side of (3.3). Let m ∈ Zd−k−1 be fixed. We only prove (3.4) in the case pk+1 < ∞. The case pk+1 = ∞ will follow by similar arguments and is left for the reader. By first using Minkowski’s inequality and then Hölder’s inequality we get ⎛ ak ( · , m) pk+1 (Z) = ⎝

l1 ∈Z

⎛ =⎝

⎞

 l1 ∈Z

p

gk Lk+1 1 ((l

≤ m+Qd−k−1

⎠



⎛ ⎝

l1 +Q1

m+Qd−k−1



1 ,m)+Qd−k )

1 pk+1



l1 ∈Z

l1 +Q1

gk (t, y) dt

pk+1 dy

pk+1 gk (t, y) dt

⎞ ⎠

⎞

1 pk+1

⎠

1 pk+1

dy

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J. Toft

⎛



⎝

≤

⎞

 l1 ∈Z l1 +Q1

m+Qd−k−1

gk (t, y)pk+1 dt ⎠



 =

gk (t, y) m+Qd−k−1

pk+1

1 pk+1

dt

1 pk+1

R

dy

 dy =

gk+1 (y) dy. m+Qd−k−1

Hence, ak ( · , m) pk+1 (Z) ≤ ak+1 (m),

m ∈ Zd−k−1 .

By applying the pk+1 (Zd−k−1 ) norm on (3.5) we get (3.4), and thereby (3.3).

(3.5)  

Proof of Proposition 4 Since the map F → F · ω is homeomorphic between the involved spaces and their corresponding non-weighted versions, we may assume that ω1 = ω2 = 1. Furthermore, by a linear change of variables, we may assume that E1 is the standard basis and E2 = 2π E1 . Then κ(E1 ) = Qd , E1 = E2 and E2 = E1 . Let F be measurable on R2d , f1,r (ξ, j ) = F ( · , ξ ) Lr (j +κ(E1 )) ,

g1 (ξ ) = f1,r (ξ, · ) p

and G1 (j, ι) = F Lr,∞

(j,ι)+κ(E1 ×E2 )

.

Then g1 ≤ g ≡

  g L∞ (ι+2π Q) · χι+2π Q , ι+ΛE2

and F Wr1,E

1

(1,p ,Lq )

= g1 Lq ≤ g Lq = G1 p,q F Wr,∞ (1,p,q ) . E

p,q ) → Wr p q This implies that Wr,∞ E (1,  1,E1 (1,  , L ), and the first inclusion in (3.1) follows.

Wiener Estimates on Modulation Spaces

495

In order to prove the second inclusion in (3.1), we may assume that r1 < ∞, since otherwise the result is trivial. Let ψ(ξ ) = f1,r1 (ξ, · ) p ,

a(ι) = ψ Lq (ι+κ(E2 ))

and H1 (j, ι) = f1,r1 ( · , j ) Lr1 (ι+κ(E2 )) . Then ψ Lq (Rd ) = F Wr1

1,E1 (1,

p ,Lq )

and

H1 p,q = F Wr1 (1,p,q ) . E

By Minkowski’s inequality and the fact that min(p) ≥ r1 we get

H1 ( · , ι) p

1 11 1 r1 1 1 1 r1 =1 f1,r1 (ξ, · ) dξ 1 1 1 ι+κ(E2 )

p

1 1 = 1 1

1 1 f1,r1 (ξ, · ) dξ 1 1

1

r1

r1

ι+κ(E2 )

p/r1

 ≤

1 f1,r1 (ξ, · ) p/r1 dξ r1

ι+κ(E2 )

r1

= a(ι).

Hence H1 p,q ≤ a q . By Lemma 1 it follows that a q ≤ ψ Lq , and the second inclusion of (3.1) follows by combining these relations. It remains to prove (3.2). Again we may assume that r2 < ∞, since otherwise the result is trivial. Let f2,q (x, ι) = F (x, · ) Lq (ι+κ(E2 )) ,

f3 (x) = F (x, · ) Lq (Rd ) ,

H2,q1 ,q2 (ι, j ) = f2,q1 ( · , ι) Lq2 (j +κ(E1 )) ,

and

H2,q = H2,q,q

when q, q1 , q2 ∈ (0, ∞]. Then the fact that r2 ≤ min(q), Minkowski’s inequality and Lemma 1 give  H2,r2 ( · , j ) q ≤

j +κ(E1 )

1

r2

f2,r2 (x, · ) r2q dx  ≤

j +κ(E1 )

1 F (x,

· ) rL2q

dx

r2

.

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J. Toft

By applying the p norm on the latter inequality we get F Wr2 (1,q,p ) ≤ F Wr2

2,E2 (1,

E

p ,Lq )

,

and the second relation in (3.2) follows. On the other hand, we have  j +κ(E1 )

1 F (x,

· ) rL2q (Rd ) dx

r2

 

1 j +κ(E1 )

f2,∞ (x,

· ) r2q

dx

r2

≤ H2,∞ ( · , j ) q Again, by applying the p norm with respect to the j variable, we get F Wr2

2,E2 (1,

p ,Lq )

≤ F W∞ (1,q,p ) , E

 

and the first relation in (3.2) follows.

3.2 Wiener Estimates on Short-Time Fourier Transforms, and Modulation Spaces Essential parts of our analysis are based on Lebesgue estimates of the semi-discrete convolution with respect to (the ordered) basis E in Rd , given by

a(j )f (x − j ), (3.6) (a ∗[E] f )(x) = j ∈ΛE

when f ∈ Σ1 (Rd ) and a ∈ 0 (ΛE ). We notice that such estimates are in background in the whole analysis in [10] and [27] when extending the Gabor theory to modulation spaces which might fail to be Banach spaces (see [27, Proposition 2.1] and [10, Lemma 2.6]). Since we need to take care of functions f which are periodic in some directions, the results in [10] and [27] are not sufficient. The next result is an extension of [27, Proposition 2.1] and [10, Lemma 2.6], but a special case of [29, Theorem 2.1]. The proof is therefore omitted. Here the domain of integration is of the form I = { x1 e1 + · · · + xd ed ; xk ∈ Jk },

Jk =

⎧ ⎨[0, 1],

ek ∈ E0

⎩R,

ek ∈ / E0 .

(3.7)

Wiener Estimates on Modulation Spaces

497

Proposition 5 Let E be an ordered basis of Rd , E0 ⊆ E, I be given by (3.7), ω, v ∈ PE (Rd ) be such that ω is v-moderate, and let p, r ∈ (0, ∞]d be such that rk ≤ min(1, pm ). m≤k

p

Also let f be measurable on Rd such that |f | is E0 -periodic and f ∈ LE,(ω) (I ). Then the map a → a ∗[E] f p

from 0 (ΛE ) to LE,(ω) (I ) extends uniquely to a linear and continuous map from p rE,(v) (ΛE ) to LE,(ω) (I ), and a ∗[E] f Lp

E,(ω) (I )

≤ C a rE,(v) (ΛE ) f Lp

E,(ω) (I )

(3.8)

,

for some constant C > 0 which is independent of a ∈ rE,(v) (ΛE ) and measurable f on Rd such that |f | is E0 -periodic. We have now the following result, which agrees with Proposition 2 when r = (∞, . . . , ∞). Proposition 2 Let E be a phase split basis for R2d , p, r ∈ (0, ∞]2d , r ∈ (0, min(1, p)], ω, v ∈ PE (R2d ) be such that ω is v-moderate, ρ and Θρ v ρ be as in (2.12) in Sect. 2 with strict inequality when r < 1, and let φ1 , φ2 ∈ 1 M(Θ (Rd ) \ 0. Then ρ v) f M p

E,(ω)

Vφ1 f Lp

E,(ω)

Vφ2 f Wr (ω,p ) , E

E

f ∈ Σ1 (Rd ).

In particular, if f ∈ Σ1 (Rd ), then p

f ∈ ME,(ω) (R2d )

⇔

p

Vφ1 f ∈ LE,(ω) (R2d )

⇔

p

Vφ2 f ∈ WrE (ω, E (ΛE )).

We need the following lemma for the proof. Lemma 2 Let p ∈ (0, ∞], r > 0, (x0 , ξ0 ) ∈ R2d be fixed, and let φ ∈ Σ1 (Rd ) be a Gaussian. Then |Vφ f (x0 , ξ0 )| ≤ C Vφ f Lp (Br (x0 ,ξ0 )) ,

f ∈ Σ1 (Rd ),

where the constant C is independent of (x0 , ξ0 ) and f . When proving Lemma 2 we first reduce to the case that the Gaussian φ should be centered at origin, by straight-forward arguments involving pullbacks with translations. The result then follows by the same arguments as in [10, Lemma 2.3]

498

J. Toft

and its proof, based on the fact that z → Fw (z) ≡ ec1 |z|

2 +c

2 (z,w)+c3 |w|

3

Vφ f (x, ξ ),

z = x + iξ

is an entire function for some choice of the constant c1 (depending on φ). Here (z, w) =

d

xj w j ,

z = (z1 , . . . , zd ) ∈ Cd , w = (w1 , . . . , wd ) ∈ Cd .

j =1

Proof of Proposition 2 Let F = Vφ f , F0 = Vφ0 f , κ(E) be the (closed) parallelepiped which is spanned by E = {e1 , . . . , e2d }, and let κM (E) = { x1 e1 + · · · + x2d e2d ; |xk | ≤ 2, k = 1, . . . , 2d }. Also choose r0 > 0 small enough such that κ(E) + Br0 (0, 0) ⊆ κM (E) The result holds when r = (∞, . . . , ∞), in view of Proposition 2. By Hölder’s inequality we also have Vφ f Wr (ω,p )  Vφ f W∞ (ω,p ) . E

E

(3.9)

E

E

We need to prove the reversed relation Vφ f W∞ (ω,p )  Vφ f Wr (ω,p ) , E

E

(3.10)

E

E

and it suffices to prove this for r = (r, . . . , r) for some r ∈ (0, 1] in view of Hölder’s inquality. d

|x|2

First we consider the case when φ = φ0 , where φ0 (x) = π − 4 e− 2 . If r > 0 is small enough and j ∈ ΛE , then Lemma 2 gives for some (xj , ξj ) ∈ j + κ(E) that Vφ0 f L∞ (j +κ(E)) = |Vφ0 f (xj , ξj )|  Vφ0 f Lr (Br (xj ,ξj )) ≤ Vφ0 f Lr (j +κM (E)) Hence, Vφ0 f W∞ (ω,p ) = { Vφ0 f L∞ (j +κ(E) ω(j )}j ∈ΛE p (ΛE ) E

E

E

 { Vφ0 f Lr (j +κM (E) ω(j )}j ∈ΛE p (ΛE ) E

{ Vφ0 f Lr (j +κ(E) ω(j )}j ∈ΛE p (ΛE ) = Vφ0 f Wr (ω,p ) , E

E

E

Wiener Estimates on Modulation Spaces

499

and (3.10) holds for φ = φ0 . 1 (Rd ) is arbitrary and choose the integer n ≥ 1 large Next suppose that φ ∈ MΘ ρv enough such that if En =

e 1 e2d  1 ·E ≡ ,..., n n n

and

Λ=

1 ΛE = ΛEn , n

then {φ( · − k)ei · ,κ }(k,κ)∈Λ is a frame with canonical dual frame {ψ( · − k)ei · ,κ }(k,κ)∈Λ , 1 (Rd ). This is possible in view of [13, Theorem S]. In fact, since where ψ ∈ MΘ ρv 1 1 φ ∈ M(Θ , it follows that its canonical dual ψ also belongs to M(Θ as well (cf. ρ v) ρ v) p

[12, Theorem S]). Consequently, any f ∈ ME,(ω) (Rd ) possess the expansions

f =

Vφ f (k, κ)ψ( · − k)ei · ,κ

(k,κ)∈Λ

=

Vψ f (k, κ)φ( · − k)ei · ,κ .

(3.11)

(k,κ)∈Λ

Here the series converge unconditionally when max(p) < ∞ and with respect to ∞ (Rd ) otherwise. the weak∗ topology of M(1/v) Let F0 = |Vφ0 f | · ω,

F = |Vφ f | · ω,

and

a(k) = |Vψ φ0 (−k)|.

As in the proofs of [10, Theorem 3.1] and [27, Proposition 3.1] we use the fact that d

|Vφ0 f | ≤ (2π )− 2 a ∗[En ] |Vφ f |, which follows from d

|Vφ0 f (x, ξ )| = (2π )− 2 |(f, ei · ,ξ  φ0 ( · − x))| d

≤ (2π )− 2

(k,κ)∈Λ

|(Vψ φ0 )(k, κ)||(f, ei · ,ξ +κ φ( · − x − k))|

(3.12)

500

J. Toft

d

= (2π )− 2

|(Vψ φ0 )(k, κ)||Vφ f (x + k, ξ + κ)|

(k,κ)∈Λ d

= (2π )− 2 (a ∗[En ] |Vφ f |)(x, ξ ). Here we have used (3.11) with φ0 in place of f , in the inequality. By using that ω(X)  v(k)ω(X + k),

X ∈ R2d , k ∈ Λ,

(3.12) gives F0  (a · v) ∗[En ] F.

(3.13)

If we set  b0 (j ) =

 |F0 (X)| dX r

j +κ(E)

and

b(j ) =

j +κ(E)

|F (X)|r dX,

j ∈ Λ,

integrate (3.13) and use the fact that r ≤ 1, we get for j ∈ Λ that 

 b0 (j ) 

j +κ(E)



r a(k)v(k)|F (X − k)|

dX

k∈Λ

 (a(k)v(k))

r j +κ(E)

k∈Λ

|F (X − k)|r dX = ((a · v)r ∗ b))(j ),

where ∗ is the discrete convolution with respect to the lattice Λ. Let q = p/r. Then min(q) ≥ 1, and Young’s inequality applied on the last inequality gives 1

1

F0 Wr (1,p ) = b0r p (ΛE )  (a · v)r ∗ b rq (Λ E

E

E

E

E)

1  1 r ≤ (a · v)r ∗ b rq (Λ) ≤ (a · v)r 1 (Λ) b q (Λ) E

E

1

a r(v) (Λ) b rq (Λ) ≤ a 1 E

(Θρ v) (Λ)

1

b r p (Λ) E

 φ0 M 1

(Θρ v)

1

b r p (Λ) . E

(3.14)

Wiener Estimates on Modulation Spaces

501

In the last steps we have used Hölder’s inequality and Vψ φ0 L1

2d (Θρ v) (R )

{ Vψ φ0 L∞ (j +κ(E) (Θρ v)(j )}j ∈ΛE 1 (ΛE ) φ0 M 1

(Θρ v)

.

We have 1

b r p (Λ) = { F Lr (j +κ(E)) }j ∈Λ p (Λ) , E



E

+ κ(E)) = R2d , and Λ is n times as dense as ΛE . From these facts it follows by straight-forward computations that j ∈ΛE (j

{ F Lr (j +κ(E)) }j ∈Λ p (Λ) { F Lr (j +κ(E)) }j ∈ΛE p (ΛE ) E

E

{ Vφ f Lr (j +κ(E)) ω(j )}j ∈ΛE p (ΛE ) = F Wr (ω,p ) . E

E

E

Here the second relation follows from the fact that ω(x) ω(j ) when j ∈ ΛE and x ∈ j + κ(E), which follows from (2.8). A combination of these relations with (3.14) gives F0 Wr (1,p )  F Wr (1,p ) . E

E

E

E

By Proposition 2, and that (3.10) is already proved for φ = φ0 , we get Vφ f W∞ (ω,pσ ) Vφ0 f W∞ (ω,pσ ) F0 W∞ (1,pσ )  F0 Wr (1,pσ ) E

E

E

E

 F Wr (1,pσ ) Vφ f Wr (ω,pσ ) . E

E

  By combining Propositions 2 , 4 and Remark 8 we get the following. Theorem 1 Let E0 be a basis for Rd , E0 be its dual basis, E = E0 × E0 , q, r ∈ (0, ∞]d , ω0 , v0 ∈ PE (Rd ) be such that ω0 is v0 -moderate, ω(x, ξ ) = ω0 (ξ ), v(x, ξ ) = v0 (ξ ), Θρ v be as in (2.12) in Sect. 2 with strict inequality when r < 1, 1 and let φ ∈ M(Θ (Rd ) \ 0. Then ρ v) ∞,q

q

∞,q

ME,(ω) (Rd ) = MrE0 (ω0 , LE  (Rd )),

q

WE,(ω) (Rd ) = WrE0 (ω0 , LE  (Rd )),

0

0

and f M ∞,q Vφ f Wr E,(ω)

∞ q 1,E0 (ω,E ,LE  ) 0

,

f W ∞,q Vφ f Wr E,(ω)

∞ q 2,E0 (ω,E ,LE  ) 0

.

502

J. Toft

3.3 Periodic Elements in Modulation Spaces By a straight-forward combination of Proposition 3 and Theorem 1 we get the following. The details are left for the reader. Proposition 6 Let E0 be a basis for Rd , E0 be its dual basis, E = E0 × E0 , q, r ∈ (0, ∞]d , ω0 , v0 ∈ PE (Rd ) be such that ω0 is v0 -moderate, ω(x, ξ ) = ω0 (ξ ), v(x, ξ ) = v0 (ξ ), Θρ v be as in (2.12) in Sect. 2 with strict inequality when r < 1, 1 and let φ ∈ M(Θ (Rd ) \ 0. Then ρ v) f M ∞,q f W ∞,q f Mr E,(ω)

q E0 (ω,LE  ) 0

E,(ω)

f Wr

q E0 (ω,LE  ) 0

f LE (ω0 ,q  (Λ E0

E0 ))

,

f ∈ (E0E ) (Rd ).

(3.15)

As an immediate consequence of the previous result we get the following extension of Proposition 3. The details are left for the reader. Proposition 3 Let E be a basis of Rd , r ∈ (0, ∞]d , r ∈ (0, 1], B ⊆ Lrloc (Rd ) be an E  -split Lebesgue space, B ,E (ΛE ) its discrete version, and let ω ∈ PE (Rd ). Then   LE (ω, B ,E ) = MrE (ω, B) (E0E ) (Rd ) = WrE (ω, B) (E0E ) (Rd ). Remark 9 Let E0 , q, r, ω, v and φ be the same as in Proposition 6, r0 ≤ min(r), and f ∈ f ∈ (E0E ) (Rd ) with Fourier series expansion (2.13) in Sect. 2. Also let E1 = E0 × E0 = {e1 , . . . , e2d } and

E2 = {ed+1 , . . . , e2d , e1 , . . . , ed }.

Then (2.14)–(2.16) in Sect. 2 and (3.15) imply that Vφ f · ω Lr,q (κ(E0 )×Rd ) Vφ f · ω Lq,r (Rd ×κ(E0 )) c(f, · ) q  E1

E2

.

(3.16)

E0 ,(ω0 )

Let · be the quasi-norm on the left-hand side of (3.16), after the orders of the q involved Lek (R) and Lrekk (κ(ek )) quasi-norms have been permuted in such way that k

q

the internal order of the hitting Lek (R) quasi-norms remains the same. Then k

F Lr0 ,q (κ(E E

d 0 )×R )

 F  F L∞,q (κ(E0 )×Rd ) , E

(3.17)

Wiener Estimates on Modulation Spaces

503

by repeated application of Hölder’s inequality. A combination of (3.16) and (3.17) give Vφ f · ω c(f, · ) q 

(3.18)

.

E0 ,(ω0 )

In particular, if ej is the same as above, E3 = {e1 , ed+1 , . . . , ed , e2d }, Ω = { y1 e1 + · · · + y2d e2d ; yj ∈ [0, 1] and yd+j ∈ R, j = 1, . . . , d } and q 0 = (q1 , q1 , q2 , q2 , . . . , qd , qd ) ∈ (0, ∞]2d , then Vφ f · ω Lq 0 (Ω) c(f, · ) q  E3

.

(3.19)

E0 ,(ω0 )

Remark 10 With the same notation as in the previous remark, we note that if E0 is the standard basis of Rd , Xj = (xj , ξj ), j = 1, . . . , d, I = R × [0, 2π ] and max(q) < ∞, then (3.19) is the same as ⎛ ⎞1 pd ppd q2    d−1 q1 ⎜ ⎟ q1 |Vφ f (x, ξ )ω0 (ξ )| dX1 ··· dXd ⎠ ··· ⎝ I

I

⎛

⎛

⎛

⎞ q2

⎜ ⎜ ⎜ |c(f, α)ω0 (α)|q1 ⎠ ⎝· · · ⎝ ⎝ αd ∈Z

α1 ∈Z

q1

1 ⎞ ppd ⎞ pd d−1 ⎟ ⎟ ⎟ . ···⎠ ⎠

(3.19)

Acknowledgements I am very grateful to Professor Hans Feichtinger for reading parts of earlier versions of the paper and giving valuable comments, leading to improvements of the content and the style.

References 1. T. Aoki Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588– 594. 2. J. Chung, S.-Y. Chung, D. Kim Characterizations of the Gelfand-Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc. 124 (1996), 2101–2108. 3. H. G. Feichtinger Banach spaces of distributions of Wiener’s type and interpolation, in: Ed. P. Butzer, B. Sz. Nagy and E. Görlich (Eds), Proc. Conf. Oberwolfach, Functional Analysis

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The Gabor Wave Front Set of Compactly Supported Distributions Patrik Wahlberg

Dedicated to Luigi Rodino on the occasion of his 70th birthday.

Abstract We show that the Gabor wave front set of a compactly supported distribution equals zero times the projection on the second variable of the classical wave front set. Keywords Gabor wave front set · Compactly supported distributions

1 Introduction The Gabor wave front set of tempered distributions was introduced by Hörmander 1991 [8]. The idea was to measure singularities of tempered distributions both in terms of smoothness and decay at infinity comprehensively. Using the short-time Fourier transform the Gabor wave front set can be described as the directions in phase space T ∗ Rd where a distribution u ∈ S  (Rd ) does not decay rapidly. The Gabor wave front set of u ∈ S  (Rd ) is empty exactly if u ∈ S (Rd ). The Gabor wave front set behaves differently than the classical C ∞ wave front set, also introduced by Hörmander 1971 (see [7, Chapter 8]). It is for example translation invariant. The Gabor wave front set is adapted to the Shubin calculus of pseudodifferential operators [16] where symbols have isotropic behavior in phase space. With respect to this calculus, the corresponding notion of characteristic set, and the Gabor wave front set, pseudodifferential operators are microlocal and microelliptic, similar to pseudodifferential operators with Hörmander symbols in their natural context.

P. Wahlberg () Department of Mathematics, Linnæus University, Växjö, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-36138-9_27

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The main result of this note concerns the Gabor wave front set of compactly supported distributions. We show (see Corollary 1) that for u ∈ E  (Rd ) we have WFG (u) = {0} × π2 WF(u)

(1)

where WFG denotes the Gabor wave front set, WF denotes the classical wave front set, and π2 is the projection on the covariable (second) phase space Rd coordinate. By [7, Theorem 8.1.3], π2 WF(u) = (u). The symbol (u) denotes the cone complement of the space of directions in Rd in an open conic neighborhood of which the Fourier transform of u ∈ E  (Rd ) decays rapidly. The equality (1) thus describes exactly the Gabor wave front set of u ∈ E  (Rd ) in terms of known ingredients in terms of WF(u): The space coordinate is zero and the frequency directions are exactly the “irregular” frequency directions of u. In the literature there are several concepts of global wave front sets apart from the Gabor wave front set. There is a parametrized version [17] and a Gelfand–Shilov version [2] of the same idea. Melrose [12] introduced the scattering wave front set which was used by Coriasco and Maniccia [3] for propagation of singularities, cf. also [4]. Cappiello [1] has studied the corresponding concept in a Gelfand–Shilov framework. The paper is organized as follows. Section 2 contains notation and background, in Sect. 3 we prove the main results, and finally in Sect. 4 we discuss how the results from Sect. 3 can be applied to propagation of singularities for certain Schrödinger type evolution equations.

2 Preliminaries An open ball of radius r > 0 and center at the origin is denoted Br . The unit sphere in Rd is denoted Sd−1 . We write f (x)  g(x) provided there exists C > 0 such that f (x)  Cg(x) for+all x in the domain of f and g. The Japanese bracket on Rd is defined by x = 1 + |x|2 . Peetre’s inequality is x + ys  xs y|s| ,

s ∈ R.

(2)

The Fourier transform on the Schwartz space S (Rd ) is normalized as F f (ξ ) = f(ξ ) =



f (x)e−ix,ξ  dx, Rd

ξ ∈ Rd ,

f ∈ S (Rd ),

where ·, · is the inner product on Rd , and extended to its dual, the tempered distributions S  (Rd ). The inner product (·, ·) on L2 (Rd ) × L2 (Rd ) is conjugate linear in the second argument. We use this notation also for the conjugate linear action of S  (Rd ) on S (Rd ).

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Some notions of time-frequency analysis and pseudodifferential operators on Rd are recalled [5–7, 11, 13, 16]. Let u ∈ S  (Rd ) and ψ ∈ S (Rd ) \ {0}. The shorttime Fourier transform (STFT) Vψ u of u with respect to the window function ψ, is defined as Vψ u : R2d → C,

z → Vψ u(x, ξ ) = (u, +(z)ψ),

where +(z) = Mξ Tx , z = (x, ξ ) ∈ R2d is the time-frequency shift composed of the translation operator Tx ψ(y) = ψ(y − x) and the modulation operator Mξ ψ(y) = eiy,ξ  ψ(y). We have Vψ u ∈ C ∞ (R2d ) and by [6, Theorem 11.2.3] there exists m 0 such that |Vψ u(z)|  zm ,

z ∈ R2d .

(3)

The order of growth m does not depend on ψ ∈ S (Rd ) \ 0. Let ψ ∈ S (Rd ) satisfy ψ L2 = 1. The Moyal identity (u, g) = (2π )−d

 R2d

Vψ u(z) Vψ g(z) dz,

g ∈ S (Rd ),

u ∈ S  (Rd ),

is sometimes written u = (2π )−d

 R2d

Vψ u(x, ξ ) Mξ Tx ψ dx dξ,

u ∈ S  (Rd ),

(4)

with action understood to take place under the integral. In this form it is an inversion formula for the STFT. Let a ∈ C ∞ (R2d ) and m ∈ R. Then a is a Shubin symbol of order m, denoted a ∈  m , if for all α, β ∈ Nd there exists a constant Cαβ > 0 such that β

|∂xα ∂ξ a(x, ξ )|  Cαβ (x, ξ )m−|α|−|β| ,

x, ξ ∈ Rd .

(5)

The Shubin symbols form a Fréchet space where the seminorms are given by the smallest possible constants in (5). For a ∈  m a pseudodifferential operator in the Weyl quantization is defined by a w (x, D)u(x) = (2π )−d



eix−y,ξ  a R2d



x+y ,ξ 2

u(y) dy dξ,

u ∈ S (Rd ),

when m < −d. The definition extends to general m ∈ R if the integral is viewed as an oscillatory integral. The operator a w (x, D) then acts continuously on S (Rd ) and extends by duality to a continuous operator on S  (Rd ). By Schwartz’s kernel theorem the Weyl quantization procedure may be extended to a weak formulation

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which yields operators a w (x, D) : S (Rd ) → S  (Rd ), even if a is only an element of S  (R2d ). The phase space T ∗ Rd is a symplectic vector space equipped with the canonical symplectic form σ ((x, ξ ), (x  , ξ  )) = x  , ξ  − x, ξ  ,

(x, ξ ), (x  , ξ  ) ∈ T ∗ Rd .

The real symplectic group Sp(d, R) is the set of matrices in GL(2d, R) that leaves σ invariant. To each χ ∈ Sp(d, R) is associated an operator μ(χ ) which is unitary on L2 (Rd ), and determined up to a complex factor of modulus one, such that μ(χ )−1 a w (x, D) μ(χ ) = (a ◦ χ )w (x, D),

a ∈ S  (R2d )

(cf. [5, 7]). The operators μ(χ ) are homeomorphisms on S and on S  , and are called metaplectic operators. The metaplectic representation is the mapping Sp(d, R) - χ → μ(χ ) which is a homomorphism modulo sign μ(χ1 )μ(χ2 ) = ±μ(χ1 χ2 ),

χ1 , χ2 ∈ Sp(d, R).

Two ways to overcome the sign ambiguity are to pass to a double-valued representation [5], or to a representation of the so called two-fold covering group of Sp(d, R). The latter group is called the metaplectic group Mp(d, R). The two-toone projection π : Mp(d, R) → Sp(d, R) is μ(χ ) → χ whose kernel is {±1}. The sign ambiguity may be fixed (hence it is possible to choose a section of π ) along a continuous path R - t → χt ∈ Sp(d, R). This involves the so called Maslov factor [10].

3 The Gabor and the Classical Wave Front Sets First we define the Gabor wave front set WFG introduced in [8] and further elaborated in [14]. Definition 1 Let ϕ ∈ S (Rd ) \ 0, u ∈ S  (Rd ) and z0 ∈ T ∗ Rd \ 0. Then z0 ∈ / WFG (u) if there exists an open conic set  ⊆ T ∗ Rd \ 0 such that z0 ∈  and supzN |Vϕ u(z)| < ∞,

N 0.

(6)

z∈

This means that Vϕ u decays rapidly (super-polynomially) in . The condition (6) is independent of ϕ ∈ S (Rd ) \ 0, in the sense that super-polynomial decay will hold also for Vψ u if ψ ∈ S (Rd ) \ 0, in a possibly smaller cone containing z0 . The Gabor wave front set is a closed conic subset of T ∗ Rd \ 0. By [8, Proposition 2.2] it

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is symplectically invariant in the sense of WFG (μ(χ )u) = χ WFG (u),

u ∈ S  (Rd ).

χ ∈ Sp(d, R),

(7)

The Gabor wave front set is naturally connected to the definition of the C ∞ wave front set [7, Chapter 8], often called just the wave front set and denoted WF. A point in the phase space (x0 , ξ0 ) ∈ T ∗ Rd such that ξ0 = 0 satisfies (x0 , ξ0 ) ∈ / WF(u) if there exists ϕ ∈ Cc∞ (Rd ) such that ϕ(0) = 0, an open conical set 2 ⊆ Rd \ 0 such that ξ0 ∈ 2 , and sup ξ N |Vϕ u(x0 , ξ )| < ∞,

ξ ∈2

N 0.

The difference compared to WFG (u) is that the C ∞ wave front set WF(u) is defined in terms of super-polynomial decay in the frequency variable, for x0 fixed, instead of super-polynomial decay in an open cone in the phase space T ∗ Rd containing the point of interest. Pseudodifferential operators with Shubin symbols are microlocal with respect to the Gabor wave front set. In fact we have by [8, Proposition 2.5] WFG (a w (x, D)u) ⊆ WFG (u) provided a ∈  m and u ∈ S  (Rd ). In the next result we relate the Gabor wave front set with the C ∞ wave front set for a tempered distribution. We use the notation π2 (x, ξ ) = ξ for the projection π2 : T ∗ Rd → Rd onto the covariable. Proposition 1 If u ∈ S  (Rd ) then {0} × π2 WF(u) ⊆ WFG (u). / WFG (u). Let ϕ ∈ Cc∞ (Rd ) satisfy ϕ(0) = 0. Proof Suppose |ξ0 | = 1 and (0, ξ0 ) ∈ There exists an open conic set  ⊆ T ∗ Rd \ 0 such that (0, ξ0 ) ∈  and sup (x, ξ )N |Vϕ u(x, ξ )| < ∞,

N 0.

(x,ξ )∈

We have to show that (x0 , ξ0 ) ∈ / WF(u) for all x0 ∈ Rd . Let x0 ∈ Rd . Define for ε > 0 the open conic set containing ξ0 2,ε

 = ξ ∈ Rd \ 0 :





ξ d



|ξ | − ξ0 < ε ⊆ R \ 0.

We claim that there exists ε > 0 such that ({x0 } × 2,ε ) \ B1/ε ⊆ 

(8)

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holds. To prove this we assume for a contradiction that there exists ξn ∈ 2,1/n such that |(x0 , ξn )| n and (x0 , ξn ) ∈ /  for alln ∈ N. Since  is conic we have |ξn |−1 (x0 , ξn ) ∈ / . The sequence |ξn |−1 (x0 , ξn ) n ⊆ R2d is bounded so we get for a subsequence 

ξnj x0 , |ξnj | |ξnj |

→ (x, ξ ),

j → ∞,

where (x, ξ ) ∈ /  due to  being open, and |ξ | = 1. From |ξn | n − |x0 | we may conclude that x = 0. From ξn ∈ 2,1/n for all n ∈ N we obtain ξ = ξ0 , which gives (0, ξ0 ) ∈ / . This is a contradiction which shows that (8) must hold for some ε > 0. Thus we have for N 0 arbitrary sup ξ ∈2,ε ,

|ξ |ε−1 +|x

0|

ξ N |Vϕ u(x0 , ξ )| 

sup ξ ∈2,ε ,

|ξ |ε−1 +|x

0|

(x0 , ξ )N |Vϕ u(x0 , ξ )|

 sup (x, ξ ) |Vϕ u(x, ξ )| < ∞ N

(x,ξ )∈

which shows that (x0 , ξ0 ) ∈ / WF(u).

 

Proposition 2 If u ∈ E  (Rd ) + S (Rd ) then WFG (u) ⊆ {0} × π2 WF(u).

(9)

Proof We may assume u ∈ E  (Rd ). We start with the less precise inclusion WFG (u) ⊆ {0} × Rd \ 0.

(10)

/ {0} × Rd \ 0. Then x0 = 0 so for some C > 0 we have Suppose 0 = (x0 , ξ0 ) ∈ (x0 , ξ0 ) ∈  = {(x, ξ ) ∈ T ∗ Rd \ 0 : |ξ | < C|x|} ⊆ T ∗ Rd \ 0 which is an open conic set. If we pick ϕ ∈ Cc∞ (Rd ) we have Vϕ u(x, ξ ) = 0 if |x| A for A > 0 sufficiently large due to u ∈ E  (Rd ). Since we have the bound (3) for some m 0, we obtain for any N 0 sup (x, ξ )N |Vϕ u(x, ξ )| = (x,ξ )∈



sup

(x, ξ )N |Vϕ u(x, ξ )|

sup

xN +m < ∞.

(x,ξ )∈, |x|A

(11)

(x,ξ )∈, |x|A

It follows that (x0 , ξ0 ) ∈ / WFG (u), which proves the inclusion (10). In order to show the sharper inclusion (9), suppose 0 = (x0 , ξ0 ) ∈ / {0} × π2 WF(u). Then either x0 = 0 or ξ0 ∈ / π2 WF(u). If x0 = 0 then by (10) we have / WFG (u). Therefore we may assume that x0 = 0 and ξ0 ∈ / π2 WF(u), and (x0 , ξ0 ) ∈ our goal is to show (0, ξ0 ) ∈ / WFG (u), which will prove (9).

Gabor Wave Front Set of Compactly Supported Distributions

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By [7, Proposition 8.1.3] we have π2 WF(u) = (u), where (u) ⊆ Rd \ 0 is a / (u) if η ∈ 2 closed conic set defined as follows. A point η ∈ Rd \ 0 satisfies η ∈ where 2 ⊆ Rd \ 0 is open and conic, and u(ξ )| < ∞, sup ξ N |

ξ ∈2

N 0.

(12)

Thus we have ξ0 ∈ / (u), so there exists an open conic set 2 ⊆ Rd \ 0 such that ξ0 ∈ 2 , and (12) holds. Let 2 ⊆ Rd \ 0 be an open conic set such that ξ0 ∈ 2 and 2 ∩ Sd−1 ⊆ 2 . We have −d  u ∗ T; Vϕ u(x, ξ ) = uT x ϕ(ξ ) = (2π )  x ϕ(ξ )

which gives u| ∗ |g|(ξ ), |Vϕ u(x, ξ )|  |

x, ξ ∈ Rd ,

(13)

where g(ξ ) =  ϕ (−ξ ) ∈ S (Rd ). By the Paley–Wiener–Schwartz theorem we have for some m 0 | u(ξ )|  ξ m ,

ξ ∈ Rd .

(14)

Define the open conic set  = {(x, ξ ) ∈ T ∗ Rd \ 0 : |x| < |ξ |, ξ ∈ 2 }. / WFG (u) it thus suffices to show Then (0, ξ0 ) ∈ . To prove (0, ξ0 ) ∈ sup (x, ξ )N |Vϕ u(x, ξ )| < ∞,

N 0.

(x,ξ )∈

In turn, these estimates will by (13) follow from the estimates u| ∗ |g|)(ξ ) < ∞, sup ξ N (|

N 0.

ξ ∈2

(15)

Let ε > 0 and split the convolution integral as  | u| ∗ |g|(ξ ) =

ηεξ 

 | u(ξ − η)| |g(η)| dη +

η>εξ 

| u(ξ − η)| |g(η)| dη.

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If ξ ∈ 2 , |ξ | 1 and η  εξ  for ε > 0 sufficiently small, then ξ − η ∈ 2 . Using (2) and (12) we thus obtain if ξ ∈ 2 and |ξ | 1 for any N 0 

 ηεξ 

| u(ξ − η)| |g(η)| dη 

ηεξ 

 ξ −N

ξ − η−N |g(η)| dη



Rd

ηN |g(η)| dη

(16)

 ξ −N since g ∈ S (Rd ). The remaining integral we estimate using (14). We thus have for any ξ ∈ Rd and any N 0 

 η>εξ 

| u(ξ − η)| |g(η)| dη 

η>εξ  −N



 ξ 

 ξ −N −N

 ξ 

ξ − ηm |g(η)| dη

η>εξ 

ξ m+N ηm |g(η)| dη (17)



η>εξ 

η2m+N |g(η)| dη

.

Combining (16) and (17) shows that the estimates (15) hold, which as earlier / WFG (u). This proves the inclusion (9).   explained proves (0, ξ0 ) ∈ Corollary 1 If u ∈ E  (Rd ) + S (Rd ) then WFG (u) = {0} × π2 WF(u) = {0} × (u). The next result is a consequence of [8, Proposition 2.7]. We include a proof here in order to give a self-contained account, and also in order to show an alternative proof technique. Proposition 3 If u ∈ S  (Rd ) and WFG (u) ∩ ({0} × Rd ) = ∅ then u ∈ C ∞ (Rd ) and there exists L 0 such that |∂ α u(x)|  Cα xL+|α| ,

x ∈ Rd ,

α ∈ Nd ,

Cα > 0.

Proof From the assumptions it follows that for some C > 0 we have WFG (u) ⊆  := {(x, ξ ) ∈ T ∗ Rd \ 0 : |ξ | < C|x|}.

(18)

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515

Let ψ ∈ S (Rd ) satisfy ψ L2 = 1. We use the Moyal identity (4) and show that the integral for ∂ α u is absolutely convergent for any α ∈ Nd . Thus we write formally ∂ u(y) = (2π ) α

α 

−d

βα

β

R2d

Vψ u(x, ξ ) (iξ )β eiξ,y ∂ α−β ψ(y − x) dx dξ. (19)

We split the integral in two parts. Since R2d \  ⊆ T ∗ Rd is a closed conic set that does not intersect WFG (u) we have for any N 0 and any k 0







R2d \



Vψ u(x, ξ ) (iξ )β eiξ,y ∂ α−β ψ(y − x) dx dξ





R2d \

 

R2d \

 

R2d \

|Vψ u(x, ξ )| ξ |α| |∂ α−β ψ(y − x)| dx dξ (x, ξ )−N ξ |α| |∂ α−β ψ(y − x)| dx dξ ξ |α|−N xN y − x−k dx dξ



 yk

(20)

R2d

ξ |α|−N xN −k dx dξ

 yk provided N > d + |α| and k > d + N. For the remaining part of the integral we use the estimate (3) for some m 0. Using |ξ | < C|x| when (x, ξ ) ∈ , this gives for any k 0







Vψ u(x, ξ ) (iξ )β eiξ,y ∂ α−β ψ(y − x) dx dξ



   (x, ξ )m ξ |α| |∂ α−β ψ(y − x)| dx dξ |ξ | 0 w

WFG (e−tq

w (x,D)

u) ⊆

   e2tIm F (WFG (u) ∩ S) ∩ S \ 0

Gabor Wave Front Set of Compactly Supported Distributions

517

where the singular space is defined by S=

 2d−1 

  Ker Re F (Im F )j ∩ T ∗ Rd ⊆ T ∗ Rd .

(23)

j =0

Under the additional assumption on the Poisson bracket {q, q} = 0, [15, Corollary 6.3] says that S = Ker(Re F ) and hence WFG (e−tq

w (x,D)

u) ⊆

   e2tIm F (WFG (u) ∩ Ker(Re F )) ∩ Ker(Re F ) \ 0

for u ∈ S  (Rd ) and t > 0. If we combine these results with Proposition 3 we get the following consequence. Corollary 2 Let q be a quadratic form on T ∗ Rd defined by a symmetric matrix Q ∈ C2d×2d , Re Q 0 and F = JQ. If S ∩ ({0} × Rd ) = {0}, which if {q, q} = 0 reads Ker(Re F ) ∩ ({0} × Rd ) = {0}, then for u ∈ S  (Rd ) and t > 0 we have e−tq |∂ α e−tq

w (x,D)

w (x,D)

u(x)|  xLt +|α| ,

u ∈ C ∞ (Rd ), and

x ∈ Rd ,

α ∈ Nd ,

for some Lt 0. Next we specialize to the Cauchy initial value problem for the harmonic oscillator Schrödinger equation 

∂t u(t, x) + i(|x|2 − x )u(t, x) = 0, t 0, x ∈ Rd , u(0, ·) = u0 ∈ S  (Rd ).

(24)

This problem is a particular case of the general problem (22) with Q = iI2d . When w Re Q = 0 the propagator is the unitary group e−tq (x,D) = μ(e2tIm F ), t ∈ R, on L2 (Rd ) [15], and WFG (e−tq

w (x,D)

u0 ) = e2tIm F WFG (u0 ),

t ∈ R,

u0 ∈ S  (Rd ).

The propagation is exact and time is reversible. This result is a consequence of the metaplectic representation and (7). The quoted results [15, Theorem 6.2 and Corollary 6.3] are not needed.

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For the equation (24) we thus have periodic propagation of the Gabor wave front set:  cos(2t)Id sin(2t)Id −tq w (x,D) 2tJ WFG (e u0 ) = e WFG (u0 ) = WFG (u0 ), t ∈ R − sin(2t)Id cos(2t)Id (25) (cf. [15, Example 7.5]). The following result gives a partial explanation, from the point of view of the Gabor wave front set, of Weinstein’s [18] and Zelditch’s [20] results on the propagation of the C ∞ wave front set for the harmonic oscillator. The result says that a compactly supported initial datum will give a solution that is smooth except for a lattice on the time axis. At the points of the lattice the propagator is the identity or coordinate reflection f (x) → f (−x), which gives precise propagation of singularities by possible sign changes. Note that we do not allow as general potentials as in [18, 20]. Proposition 4 Consider the equation (24) and suppose u0 ∈ E  (Rd ) + S (Rd ). If w t∈ / (π/2)Z then e−tq (x,D) u0 ∈ C ∞ (Rd ) and for some Lt 0 |∂ α e−tq

w (x,D)

u0 (x)|  xLt +|α| ,

x ∈ Rd ,

α ∈ Nd .

(26)

If t ∈ (π/2)Z then WFG (e−tq WF(e

w (x,D)

−tq w (x,D)

u0 ) = (−1)2t/π WFG (u0 ),

(27)

u0 ) = (−1)2t/π WF(u0 ).

(28)

Proof Corollary 1 implies WFG (u0 ) ⊆ {0} × (Rd \ 0). Combined with (25) this means that WFG (e−tq

w (x,D)

u0 ) ∩ ({0} × Rd ) = ∅

unless t ∈ (π/2)Z. By Proposition 3 we then have e−tq (x,D) u0 ∈ C ∞ (Rd ) and the estimates (26) unless t ∈ (π/2)Z. If t = π k/2 for k ∈ Z then cos(2t) = (−1)k and sin(2t) = 0. Thus (25) w yields the following conclusion. If t = π k for k ∈ Z then WFG (e−tq (x,D) u0 ) = w (x,D) −tq WFG (u0 ) whereas if t = π(k + 1/2) for some k ∈ Z then WFG (e u0 ) = −WFG (u0 ). This proves (27). When k = 2t/π ∈ Z we have w

 e

2tJ

=

cos(2t)Id sin(2t)Id − sin(2t)Id cos(2t)Id

= (−1)k I2d

Gabor Wave Front Set of Compactly Supported Distributions

519

and therefore the corresponding metaplectic operator is μ(e2tJ )f (x) = f ((−1)k x). w Thus the propagator e−tq (x,D) = μ(e2tJ ) is the reflection operator f (x) → w k f ((−1) x) when k = 2t/π ∈ Z, which justifies WF(e−tq (x,D) u0 ) = 2t/π (−1) WF(u0 ) when 2t/π ∈ Z, that is, (28).   If t ∈ π(1 + 2Z)/4 then e2tJ = ±J and consequently e−tq

w (x,D)

 = μ(e2tJ ) = μ(±J) =

(2π )−d/2 F (2π )d/2 F −1

see e.g. [2]. When t ∈ π(1 + 2Z)/4 the estimates (26) are thus a consequence of the Paley–Wiener–Schwartz theorem [7, Theorem 7.3.1]. When t ∈ / π Z/4 the estimates (26) reveals that the solution satisfies similar estimates as F E  (Rd ).

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