Linear and Quasilinear Parabolic Problems: Volume II: Function Spaces [1st ed.] 978-3-030-11762-7;978-3-030-11763-4

Table of contents :
Front Matter ....Pages i-xvi
Auxiliary Material (Herbert Amann)....Pages 3-73
Function Spaces (Herbert Amann)....Pages 75-280
Traces and Boundary Operators (Herbert Amann)....Pages 281-368
Back Matter ....Pages 369-464

Citation preview

Monographs in Mathematics 106

Herbert Amann

Linear and Quasilinear Parabolic Problems Volume II: Function Spaces

Monographs in Mathematics Vol. 106

Series Editors: Herbert Amann Universität Zürich, Zürich, Switzerland Jean-Pierre Bourguignon IHES, Bures-sur-Yvette, France William Y. C. Chen Nankai University, Tianjin, China Associate Editors: Huzihiro Araki, Kyoto University, Kyoto, Japan John Ball, Heriot-Watt University, Edinburgh, UK Franco Brezzi, Università degli Studi di Pavia, Pavia, Italy Kung Ching Chang, Peking University, Beijing, China Nigel Hitchin, University of Oxford, Oxford, UK Helmut Hofer, Courant Institute of Mathematical Sciences, New York, USA Horst Knörrer, ETH Zürich, Zürich, Switzerland Don Zagier, Max-Planck-Institute, Bonn, Germany The foundations of this outstanding book series were laid in 1944. Until the end of the 1970s, a total of 77 volumes appeared, including works of such distinguished mathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. The series came to its name and present appearance in the 1980s. In keeping its well-established tradition, only monographs of excellent quality are published in this collection. Comprehensive, in-depth treatments of areas of current interest are presented to a readership ranging from graduate students to professional mathematicians. Concrete examples and applications both within and beyond the immediate domain of mathematics illustrate the import and consequences of the theory under discussion.

More information about this series at http://www.springer.com/series/4843

Herbert Amann

Linear and Quasilinear Parabolic Problems Volume II: Function Spaces

Herbert Amann Institut für Mathematik Universität Zürich Zürich, Switzerland

ISSN 1017-0480 ISSN 2296-4886 (electronic) Monographs in Mathematics ISBN 978-3-030-11762-7 ISBN 978-3-030-11763-4 (eBook) https://doi.org/10.1007/978-3-030-11763-4 Library of Congress Control Number: 2019932606 Mathematics Subject Classification (2010): 46E35, 46E40, 46F05 © Springer Nature Switzerland AG 2019 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, express 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

Preface

Nil humani quin corrigi possit 1 L. Naumann Systematik der Kochkunst Dresden 1887

In this volume we present a systematic and detailed exposition of the theory of function spaces in an Euclidean setting. Particular emphasis is put on Besov and Bessel potential spaces which form the frame for the study of parabolic differential equations in the next volume. The presentation includes several new features which lets it stand out from other accounts. First, it consistently develops anisotropic spaces. Second, it expounds the whole theory for functions and distributions taking their values in Banach spaces on which we impose only the necessary restrictions. Thus none in the case of Besov spaces, except for reflexivity assumptions in duality theorems. Third, the theory is set forth for spaces whose elements are defined on rectangular corners of Euclidean spaces. By this we pave the way for the investigation of function spaces on Riemannian manifolds, possibly possessing corners and other singularities. This is also put on hold for the third volume. Our approach builds basically on two cornerstones: on Fourier analysis and multiplier theorems, and on extension-restriction techniques. By this we can give a unified presentation incorporating, in particular, Sobolev–Slobodeckii and H¨older space scales. The rather detailed study of these spaces, which are of great importance for the investigation of differential equations, is a further characteristic trait of our treatise. This volume consists of three chapters and an appendix. The first chapter, which is of rather technical nature, collects preparatory material. It supplies a firm basis for the main text which covers Chapters VII and VIII. The first one thereof contains a systematic treatment of anisotropic vector-valued function spaces on corners. In the second one we give a detailed and unified account of trace and boundary operators. For the reader’s convenience, in the appendix we include a downscaled version of L. Schwartz’ theory of vector-valued distributions by admitting only Banach spaces as targets. Particular weight is given on tensor products and convolutions since, in the main text, we make use of such results. It should be mentioned that, already in 2003, I had put a preliminary, slightly more comprehensive version of this appendix on my homepage. 1 There

is nothing on earth that could not be improved.

v

vi

Preface

In essence, this volume forms a profound expansion and amelioration of my earlier lecture notes ‘Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces’ [Ama09]. Besides of adding much more material, I have corrected numerous flaws and imprecisions which observing readers have brought to my attention. Once more, I could rely on the help of Pavol Quittner, Gieri Simonett, and Christoph Walker. They read critically and carefully large parts of the first draft, pointed out plenty of mistakes and misprints, and suggested very advantageous changes and improvements. Sincere thanks are given to all of them for their generous support. Last but not least, I could again experience the immensely valuable support of my wife Gisela who transformed countless barely readable preliminary versions and revisions into TEX files and provided this perfect layout on hand. I am more than deeply grateful to her.

Z¨ urich, January 2019

Herbert Amann

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter VI

Auxiliary Material

1

Restriction-Extension Pairs

1.1

1.3 1.4

Smooth Functions on Corners . . . . Corners . . . . . . . . . . . . . . . Restriction-Extension Operators . Approximation by Test Functions Tempered Distributions on Corners Duality Formulas . . . . . . . . . The Main Theorem . . . . . . . . Duality . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . .

2

Sequence Spaces

2.1

2.4

Duality of Sequence Spaces . . Definitions and Embeddings Duality Pairings . . . . . . . Weighted Sequence Spaces . . Image Spaces . . . . . . . . . Embeddings and Duality . . Interpolation . . . . . . . . . . Unweighted Spaces . . . . . Weighted Spaces . . . . . . . Notes . . . . . . . . . . . . . .

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3

Anisotropy

3.1

Anisotropic Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . Weight Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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39 42 43 45 47 49 52 52 54 55 57 58 59 61 65 67 68 72

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Classical Spaces

1.1

1.3 1.4 1.5 1.6 1.7

Bounded Continuous Functions . . . . . . . . . . . . Banach Spaces of Bounded Continuous Functions Vector Measures . . . . . . . . . . . . . . . . . . . Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . Regular Distributions . . . . . . . . . . . . . . . . Basic Definitions . . . . . . . . . . . . . . . . . . . Restrictions and Extensions . . . . . . . . . . . . . . Distributional Derivatives . . . . . . . . . . . . . . . Reflexivity . . . . . . . . . . . . . . . . . . . . . . . Embeddings . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

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78 78 82 84 84 84 86 90 93 96 98

2

Besov Spaces

2.1

The Definition . . . . . . . . . . . . . . . . Preliminary Estimates . . . . . . . . . . . A Retraction-Coretraction Pair . . . . . The Final Definition . . . . . . . . . . . . Embedding Theorems . . . . . . . . . . . . Little Besov Spaces . . . . . . . . . . . . Embeddings With Varying Target Spaces Duality . . . . . . . . . . . . . . . . . . . . Fourier Multiplier Theorems . . . . . . . .

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99 100 102 104 104 107 109 109 112

3.4

3.5

3.6 3.7

Quasinorms . . . . . . . . . . . . . . . . . . . . . . Parametric Augmentations . . . . . . . . . . . . . Augmented Quasinorms . . . . . . . . . . . . . . Positive Homogeneity . . . . . . . . . . . . . . . Differentiating Inverses . . . . . . . . . . . . . . Slowly Increasing Functions . . . . . . . . . . . Fourier Multipliers and Multiplier Spaces . . . . . Elementary Fourier Multiplier Theorems . . . . Fourier Multiplier Spaces . . . . . . . . . . . . . Resolvent Estimates . . . . . . . . . . . . . . . . Multiplier Estimates . . . . . . . . . . . . . . . . . Resolvent Estimates for Homogeneous Symbols . Functions of Homogeneous Symbols . . . . . . . Dunford Integral Representations . . . . . . . . Powers and Exponentials . . . . . . . . . . . . . Dyadic Partitions of Unity . . . . . . . . . . . . . Preliminary Fourier Multiplier Theorems . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter VII

1.2

2.2

2.3 2.4

Function Spaces

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115 115 117 118 119 120 121 123 124 125 125 126 127 128 128 131

3

Intrinsic Norms, Slobodeckii and H¨older Spaces

3.1 3.2

Commuting Semigroups . . . . . . . . . . . . . . . Semigroups and Interpolation . . . . . . . . . . . . Preliminary Estimates . . . . . . . . . . . . . . . Renorming Intersections of Interpolation Spaces Translation Semigroups . . . . . . . . . . . . . . . Renorming Besov Spaces . . . . . . . . . . . . . . Intersection-Space Characterizations . . . . . . . . Intersection Space Representations . . . . . . . . Equivalent Norms . . . . . . . . . . . . . . . . . Nikol0 ski˘ı Spaces . . . . . . . . . . . . . . . . . . Besov–Slobodeckii and Besov–H¨older Spaces . . . Mixed Intersections . . . . . . . . . . . . . . . . Slobodeckii, H¨older, and Little H¨older Spaces . Little H¨older Spaces . . . . . . . . . . . . . . . . . Very Little H¨older Spaces . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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133 137 138 141 146 149 151 152 153 154 155 155 157 159 164 165

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168 171 173 174 176 176 178

2.6

2.7

2.8 2.9

3.3 3.4 3.5

3.6

3.7 3.8

Operators of Positive Type . . . . . . . . . . Resolvent Estimates . . . . . . . . . . . . . A Representation Theorem . . . . . . . . . Bounded Imaginary Powers . . . . . . . . . Interpolation-Extrapolation Scales . . . . . Renorming by Derivatives . . . . . . . . . . . Equivalent Norms . . . . . . . . . . . . . . Sandwich Theorems . . . . . . . . . . . . . Sobolev Embeddings . . . . . . . . . . . . Interpolation . . . . . . . . . . . . . . . . . . Real and Complex Interpolation . . . . . . Interpolation with Different Target Spaces Embeddings of Intersection Spaces . . . . . Interpolation of Classical Spaces . . . . . . Besov Spaces on Corners . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . .

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4

Bessel Potential Spaces

4.1 4.2 4.3 4.4 4.5

Basic Facts, Embeddings, and Real Interpolation A Marcinkiewicz Multiplier Theorem . . . . . . . Renorming by Derivatives . . . . . . . . . . . . . Duality . . . . . . . . . . . . . . . . . . . . . . . Complex Interpolation . . . . . . . . . . . . . . . A Holomorphic Semigroup . . . . . . . . . . . Interpolation with Different Target Spaces . .

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Contents

4.6 4.7

Intersection-Space Characterizations . . . . . . . . . . . . . . . . . . 182 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5

Triebel–Lizorkin Spaces

5.1

5.8

Maximal Inequalities . . . . . . . . . . . . . Preliminary Estimates for Sequences . . Estimates for a Single Function . . . . . Definition and Basic Embeddings . . . . . . Equivalent Norms . . . . . . . . . . . . . Embeddings . . . . . . . . . . . . . . . . Completeness . . . . . . . . . . . . . . . Fourier Multiplier Theorems . . . . . . . . Interpolation . . . . . . . . . . . . . . . . . Renorming by Derivatives . . . . . . . . . . Sandwich Theorems . . . . . . . . . . . . Sobolev Embeddings and Related Results . Multiplicative Inequalities . . . . . . . . Optimal Sobolev-Type Embeddings . . . Sharp Embeddings of Intersection Spaces Gagliardo–Nirenberg Type Estimates . . . Nonhomogeneous Inequalities . . . . . . Homogeneous Estimates . . . . . . . . . Isotropic Multiplicative Inequalities . . . Sobolev Inequality . . . . . . . . . . . . . Parabolic Estimates . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . .

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188 189 193 196 197 198 199 201 202 203 203 204 204 206 207 208 208 212 215 216 218 219

6

Point-Wise Multiplications

6.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . Continuity of Derivatives . . . . . . . . . . . . . Point-Wise Products . . . . . . . . . . . . . . . . Multiplications in Classical Spaces . . . . . . . . . Spaces of Bounded Continuous Functions . . . . Sobolev Spaces . . . . . . . . . . . . . . . . . . . Spaces of Negative Order . . . . . . . . . . . . . Multiplications in Besov Spaces of Positive Order . Multiplications in Besov Spaces of Negative Order The Reflexive Case . . . . . . . . . . . . . . . . The Non-Reflexive Case . . . . . . . . . . . . . . Multiplications in Bessel Potential Spaces . . . . . Space-Dependent Bilinear Maps . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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5.3 5.4 5.5 5.6

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6.3 6.4

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7

Compactness

7.1

7.5

Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Sets in BUC . . . . . . . . . . . . . . . . . . . . . . . . Compact Sets in Lq . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Embeddings of Besov Spaces . . . . . . . . . . . . . . . Compact Embeddings of H¨older, Sobolev–Slobodeckii, and Bessel Potential Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function Spaces on Intervals . . . . . . . . . . . . . . . . . . . . . . Classical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Retraction-Coretraction Theorem . . . . . . . . . . . . . . . . . Interpolations and Embeddings . . . . . . . . . . . . . . . . . . . The Rellich–Kondrachov Theorem . . . . . . . . . . . . . . . . . . Aubin–Lions Type Theorems . . . . . . . . . . . . . . . . . . . . . . The General Result . . . . . . . . . . . . . . . . . . . . . . . . . . Limit Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Parameter-Dependent Spaces

8.1 8.2 8.3 8.4 8.5

Sobolev Spaces and Bounded Continuous Functions Besov and Bessel Potential Spaces . . . . . . . . . . Intersection-Space Characterizations . . . . . . . . . Fourier Multipliers . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

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282 286 299 302 303 304 304 310 310 312 314 317 320

Chapter VIII Traces and Boundary Operators 1

Traces

1.1 1.2 1.3

Trace Operators . . . . . . . . . . . . . . . . . . . . . . . . The Retraction Theorem . . . . . . . . . . . . . . . . . . . Traces on Half-Spaces . . . . . . . . . . . . . . . . . . . . . Parameter-Dependence . . . . . . . . . . . . . . . . . . . General Besov spaces . . . . . . . . . . . . . . . . . . . . Spaces of Vanishing Traces . . . . . . . . . . . . . . . . . . Sobolev–Slobodeckii Spaces . . . . . . . . . . . . . . . . . Weighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . Weighted Lebesgue Spaces . . . . . . . . . . . . . . . . . Hardy Inequalities . . . . . . . . . . . . . . . . . . . . . . Weighted Space Characterizations of Sobolev–Slobodeckii Further Characterizations of Spaces with Vanishing Traces General Besov Spaces . . . . . . . . . . . . . . . . . . . .

1.4 1.5

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1.8

1.9

Contents

Bessel Potential Spaces . . . . . . . . . . . . . . . . . H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . Representation Theorems for Spaces of Negative Order Spaces of Mildly Negative Order . . . . . . . . . . . . Spaces of Strongly Negative Order . . . . . . . . . . . Duality of Sums and Intersections . . . . . . . . . . . Weighted Space Representations . . . . . . . . . . . . Traces for Corners . . . . . . . . . . . . . . . . . . . . . Traces on a Single Face . . . . . . . . . . . . . . . . . Vanishing Traces on Corners . . . . . . . . . . . . . . Faces of Higher Codimensions . . . . . . . . . . . . . Compatibility Conditions . . . . . . . . . . . . . . . . The Retraction Theorem for Corners . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Boundary Operators

2.1

Boundary Operators on Half-Spaces . . . . . . . . . Normal Boundary Operators . . . . . . . . . . . . Systems of Boundary Operators . . . . . . . . . . . The Boundary Operator Retraction Theorem . . . Embeddings with Boundary Conditions . . . . . . Transmission Operators . . . . . . . . . . . . . . . . Patching Together Half-Spaces . . . . . . . . . . . Interpolation With Boundary Conditions . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . The Main Theorem . . . . . . . . . . . . . . . . . Generalizations . . . . . . . . . . . . . . . . . . . Complex Interpolation of Bessel Potential Spaces Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

2.3 2.4

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322 323 324 324 326 327 330 330 331 332 332 334 335 339

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343 344 347 348 350 353 357 358 358 360 365 367 368

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370 370 371 372 372 373 373 374 374 374

Appendix Vector-Valued Distributions 1

Tensor Products and Convolutions

1.1

Locally Convex Topologies . . . . . . . . The Uniform Boundedness Principle . . Hypocontinuity . . . . . . . . . . . . . Montel Spaces . . . . . . . . . . . . . . Strict Inductive Limits . . . . . . . . . Smooth Functions . . . . . . . . . . . . Test Functions . . . . . . . . . . . . . . Rapidly Decreasing Smooth Functions . Slowly Increasing Smooth Functions . . Spaces of Vector-Valued Distributions .

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Contents

1.2

1.3

1.4

1.5 1.6

1.7

1.8

1.9

Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolutions of Distributions and Test Functions . . . . . . . Translation-Invariant Operators . . . . . . . . . . . . . . . . . Convolutions of Two Distributions . . . . . . . . . . . . . . . . Elementary Properties of Convolutions . . . . . . . . . . . . . Convolutions of Temperate Distributions . . . . . . . . . . . . Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplications . . . . . . . . . . . . . . . . . . . . . . . . . . . Leibniz’ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation by Test Functions . . . . . . . . . . . . . . . . Density by Iteration . . . . . . . . . . . . . . . . . . . . . . . . Approximation by Tensor Products . . . . . . . . . . . . . . . Approximation by Polynomials . . . . . . . . . . . . . . . . . . Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Tensor Products and the Kernel Theorem . . . . . . Algebraic Tensor Products . . . . . . . . . . . . . . . . . . . . Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . Projective Tensor Products . . . . . . . . . . . . . . . . . . . . Nuclear Maps and Spaces . . . . . . . . . . . . . . . . . . . . . Projective Tensor Products and Maps of Finite Rank . . . . . Approximation by Maps of Finite Rank . . . . . . . . . . . . . Completeness of Spaces of Linear Operators . . . . . . . . . . The Abstract Kernel Theorem . . . . . . . . . . . . . . . . . . Tensor Product Characterizations of Some Distribution Spaces Extending Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . General Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . Point-Wise Multiplication . . . . . . . . . . . . . . . . . . . . . . A Characterization of OM . . . . . . . . . . . . . . . . . . . . The General Theorem . . . . . . . . . . . . . . . . . . . . . . . Basic Properties of Multiplications . . . . . . . . . . . . . . . . Scalar Products and Duality Pairings . . . . . . . . . . . . . . . Parseval’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . Duality Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . Tensor Products of Distributions and Kernel Theorems . . . . . Approximation by Tensor Products . . . . . . . . . . . . . . . Tensor Products of Distributions . . . . . . . . . . . . . . . . . Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Tensor Products of Distributions . . . . . . . . . . Kernel Theorems . . . . . . . . . . . . . . . . . . . . . . . . . Convolutions of Vector-Valued Distributions . . . . . . . . . . . . The Basic Theorem . . . . . . . . . . . . . . . . . . . . . . . . Lp Functions with Compact Supports . . . . . . . . . . . . . . Convolutions of Regular Distributions . . . . . . . . . . . . . .

xiii

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375 375 378 378 380 381 383 383 384 385 387 388 389 390 390 390 392 393 393 394 395 397 398 398 400 405 405 406 407 409 413 414 415 416 416 416 419 421 425 427 428 428 430 430

xiv

Contents

Tensor Products and Convolutions . . . . . . . . . . Basic Properties . . . . . . . . . . . . . . . . . . . . Convolution Algebras . . . . . . . . . . . . . . . . . Convolutions of Bounded and Integrable Functions The Convolution Theorem . . . . . . . . . . . . . . 2

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431 433 436 437 439

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441 442 444 444 446

Vector Measures and the Riesz Representation Theorem Measures of Bounded Variation . . . . . . Integrals with Respect to Vector Measures Vector Measures as Distributions . . . . . Convolutions Involving Vector Measures . The Riesz Representation Theorem . . . .

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Notations and Conventions We use the notations and conventions introduced on pages 1–7 of Volume I, with the following exceptions, applied in the main body of this volume (but not in the appendix): • All abstract vector spaces are over the complex field. • K does not stand for either the real or the complex number field. Instead, it symbolizes corners which are introduced in Subsection VI.1.1. For the reader’s convenience, we reproduce here some of the very basic notation established in Volume I. Let X and Y be nonempty sets. Then Y X is the set of all maps u : X → Y . If A is a subset of X, the characteristic function of A (in X) is denoted by χA . We also put 1 := 1X := χX . The set of all x ∈ X in which definitions and relations hold is often denoted by [ . . . ], where . . . stand for the definitions and relations. For example, if u ∈ RX then © ª [u ≥ 0] := x ∈ X ; u(x) ≥ 0 etc. Suppose X and Y are Hausdorff topological spaces. Then C(X, Y ) is the set of all continuous maps in Y X , endowed with the compact-open topology. We write X ,→ Y

or

i : X ,→ Y ,

if X is continuously injected in Y , that is, X ⊂ Y and the natural injection i: X→Y ,

x 7→ x d

d

is continuous. If X is a dense subset of Y , we write X ⊂ Y . Thus X ,→ Y means that X is densely and continuously injected in Y . We often write 1X , or simply 1, for the identity mapping, idX : X → X, x 7→ x, if no confusion seems likely. Given a subset M of a vector space, we put q M := M \{0} . Assume X and Y are topological vector spaces. Then L(X, Y ) is the vector space of all continuous linear maps from X in Y , and L(X) := L(X, X) . In this case X ,→ Y means also that X is a vector subspace of Y , that is, i belongs to L(X, Y ). Moreover, © ª Lis(X, Y ) := T ∈ L(X, Y ) ; T is bijective and T −1 ∈ L(Y, X) 1

2

Notations and Conventions

is the set of all topological linear (toplinear) isomorphisms from X onto Y , and Laut(X) := Lis(X, X) is the group of all toplinear automorphisms of X, the general linear group, also denoted by GL(X). By a locally convex space (LCS) we mean a Hausdorff locally convex topological vector space. If X and Y are LCSs, then L(X, Y ) is equipped with the bounded convergence topology. Throughout this volume, unless specified otherwise, c denotes constants ≥ 1 which may be different from occurrence to occurrence, but are always independent of the free variables appearing at a given place.

Chapter VI

Auxiliary Material Our in-depth study of vector-valued function spaces is based on a considerable amount of technical tools. Among them stand out extension and restriction theorems and Fourier multiplier estimates. In the first section of this chapter we study smooth Banach-space-valued functions and tempered distributions on rectangular Euclidean corners. We establish a restriction-extension formalism which plays a fundamental role throughout the whole volume. On its basis we can –– in the next chapter –– introduce function spaces on corners and transfer their properties, derived by Fourier analytic methods on the ‘full’ Euclidean space, to the corner setting. Section 2 essentially summarizes the more or less well-known theory of sequence spaces. We emphasize duality and interpolation properties. That section provides a firm basis for many later considerations, not only in the following chapters, but in the next volume too. Section 3 is the most technical one. First, we introduce weight systems and related quasinorms, as well as parameter-dependent versions thereof. Next we launch Fourier multiplier spaces and derive multiplier estimates, for homogeneous functions in particular. These results are of use not only at various later places, but in the next volume also. Preparing for the Fourier analytic approach to Besov spaces, we investigate in the last subsection to some extent dyadic properties of unity. The reader is strongly advised to browse with some care –– even at a first reading –– at least through the beginnings of Sections 1 and 3 to grasp the definitions of r-e pairs and weight systems.

© Springer Nature Switzerland AG 2019 H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4_6

3

4

1

VI Auxiliary Material

Restriction-Extension Pairs

The theory of function spaces on Rd is most easily established by means of Fourier analysis. Afterwards it is important to be able to transfer the so derived results to half-spaces or, more generally, to corners. In this section we develop the technical tools for this approach. In the first subsection we introduce the concepts of (model) corners and the basic operators which allow an efficient handling of restrictions and extensions of smooth functions to and from corners, respectively. These operators are, in the second subsection, extended to spaces of tempered distributions. This leads to the main result of this section, namely Theorem 1.2.3. In the last subsection it is shown that spaces of tempered distributions on corners can be characterized by duality, similarly as for scalar-valued distributions on Euclidean spaces. However, the results of this section apply to functions and distributions with values in arbitrary Banach spaces. The simple but important concept underlying this approach, and being of great value in later sections also, is that of retractions and corresponding coretractions. These notions have already been introduced in Subsection I.2.3. There some useful general facts have been established as well. To facilitate the exposition and to provide the necessary precision we now introduce the following more precise manner of speaking: Let X and Y be LCSs, r ∈ L(X , Y) a retraction, and rc ∈ L(Y, X ) a coretraction. Then we express this by saying: (r, rc ) is a retraction-coretraction pair (an r-c pair, for short) for (X , Y). Recall that this means that rc is a right inverse for r.

1.1

Smooth Functions on Corners

In this subsection we introduce the concept of corners and establish the basic restriction-extension pair for rapidly decaying smooth functions. Corners Suppose 1 ≤ k ≤ d. A standard k -corner in Rd is a Cartesian product subset1 K = Kdk = I1 × · · · × Ik × Rd−k © ª of Rd such that Ij ∈ R+ , (0, ∞) for 1 ≤ j ≤ k. We denote by J = JK the subset of {1, . . . , k} such that Ij = R+ iff j ∈ J, and J ∗ := {1, . . . , k}\J. We also set Ij∗ := (0, ∞) if j ∈ J, and Ij∗ := R+ if j ∈ J ∗ . Then K∗ = I1∗ × · · · × Ik∗ × Rd−k is the adjoint corner of K. Note K∗∗ = K. Moreover, K = K iff J = {1, . . . , k}, and 1 K should not be confused with the same notation used in volume I to denote either the real or the complex field. In this volume K always denotes a corner.

VI.1 Restriction-Extension Pairs

5

˚ We call J type of K and write more precisely K = K(J) to express then K∗ = K. the fact that K is of type J. Let K be a standard k-corner of type J and 1 ≤ j ≤ k. Then ∂j K := I1 × · · · × Ij−1 × {0} × Ij+1 × · · · × Ik × Rd−k is the j -face of K. Observe that ∂j K ⊂ K iff j ∈ J, and ∂j K ∩ K = ∅ if j ∈ J ∗ . x1

x

2

On the left there is the standard 3-corner K of type {1, 3} in R3 . The lightly shaded face ∂2 K does not belong to K.

x3

© ª For I ∈ R, ±R+ , ±(0, ∞) we put I × ∂j K := I1 × · · · × Ij−1 × I × Ij+1 × · · · × Ik × Rd−k . j

Deviating from definition (III.4.1.4), it is now more convenient to put qk,m (u) := max sup hxik |∂ α u(x)|E , |α|≤m x∈Rd

k, m ∈ N ,

(1.1.1)

where hxi := (1 + |x|2 )1/2 . It is a family of seminorms generating the topology of the Fr´echet space S(Rd , E), the space of smooth and rapidly decreasing E-valued functions on Rd . We denote by S(K, E) the Fr´echet space of smooth2 and rapidly decreasing E-valued functions on K, whose topology is generated by the family © K ª K of seminorms qk,m ; k, m , where qk,m is obtained from (1.1.1) by replacing Rd by K. Then, by definition, S(K, E) is the closed linear subspace of S(K, E) consisting of all u satisfying ∂ α u|xj =0 = 0 , α ∈ Nd , j ∈ J ∗ .

(1.1.2)

Let Sd be the group of permutations of {1, . . . , d}. We define a linear operation of Sd on Rd by setting sx := (xs(1) , . . . , xs(d) ) for s ∈ Sd and x ∈ Rd . It induces a linear representation { Us ; s ∈ Sd } of Sd on L1,loc (Rd , E) by defining Us u(x) := u(s−1 x) , 2 Differentiable

a.a. x ∈ Rd ,

u ∈ L1,loc (Rd , E) .

at x ∈ ∂j K means ‘differentiable from the right’.

6

VI Auxiliary Material

It is obvious that it restricts to a continuous linear representation on S(Rd , E). In fact, it follows from (1.1.1) that Us S(Rd , E) = S(Rd , E) ,

s ∈ Sd .

(1.1.3)

A subset K of Rd is a (general) k -corner in Rd if there exist a standard b in Rd and a permutation s ∈ Sd such that K = sK b := { sx ; x ∈ K b }. k-corner K ∗ ∗ b Then K := sK , and b E) . S(K, E) := Us S(K,

(1.1.4)

Restriction-Extension Operators The subsequent constructions are based on the following technical lemma. ¡ ¢ 1.1.1 Lemma There exists h ∈ C ∞ (0, ∞), R satisfying Z ∞ Z ∞ s ` t |h(t)| dt < ∞ , (−1) t` h(t) dt = 1 , h(1/t) = −th(t) 0

(1.1.5)

0

for s ∈ R, ` ∈ Z, and t > 0. Proof We denote by C\R+ → C, z 7→ z 1/4 the branch of z 1/4 which satisfies (x + i 0)1/4 = x1/4 for x ≥ 0. Then (x − i 0)1/4 = ix1/4 , x ≥ 0. Put ¡ ¢ f (z) := (1 + z)−1 exp −(1 − i)z 1/4 − (1 + i)z −1/4 , z ∈ C\R+ . Then f (x + i 0) = (1 + x)−1 e−(x

1/4

+x−1/4 )

¡ ¢ cos(x1/4 − x−1/4 ) + i sin(x1/4 − x−1/4 )

and f (x − i0) = f (x + i 0) for x ∈ R+ . Let Γ be a piece-wise smooth path in C\R+ running from ∞ − i 0 to ∞ + i0 such that −1 is to its left. Then, by Cauchy’s theorem, Z Z ∞ 1/4 −1/4 ) z ` f (z) dz = 2i (1 + x)−1 x` e−(x +x sin(x1/4 − x−1/4 ) dx Γ

0

for ` ∈ Z. Since z k f (z) → 0 for k ∈ N as |z| → ∞, we can apply the residue theorem to deduce Z √ z ` f (z) dz = 2πi Res(z ` f, −1) = 2πi (−1)` e−2 2 , Γ

thanks to (−1)1/4 = (1 + i ) h(t) := π −1 e2

±√

√

2

2. Thus, putting

(1 + t)−1 e−(t

1/4

+t−1/4 )

sin(t1/4 − t−1/4 )

for t > 0, we see that the second claim is true. The other assertions are now clear. ¥

VI.1 Restriction-Extension Pairs

7

Now we suppose that K = I1 × · · · × Ik × Rd−k is a standard k-corner in Rd of type J. We set xbi := (x1 , . . . , xi−1 , xi+1 , . . . , xd ) and

¡ i b¢ x ; xi := x = (x1 , . . . , xd ) .

Then, given 1 ≤ i ≤ k, ` ∈ Z, and x ∈ (−∞, 0) × ∂i K, i

Z ε`i u(x) := (−1)`

∞

¡ ¢ t` h(t)u −txi ; xbi dt ,

u ∈ S(K, E) ,

(1.1.6)

0

and εi := ε0i , where h is as in Lemma 1.1.1. Note ∂i (ε`i u)(x) = ε`+1 ∂i u(x) , i

∂j (ε`i u)(x) = ε`i ∂j u(x) ,

j 6= i ,

(1.1.7)

for x ∈ (−∞, 0) × ∂i K. i

For Kdk let Kk,k−1 = Kdk,k−1 := I1 × · · · × Ik−1 × Rd−k+1 ,

K1,0 := Rd .

Then ¡ ¢ K ∪ (R\Ik ) × ∂k K = Kk,k−1

(1.1.8)

k

and ∗ (K∗ )k,k−1 = I1∗ × · · · × Ik−1 × Rd−k−1 = (Kk,k−1 )∗ =: K∗k,k−1 .

Suppose k ∈ J. Given ` ∈ N and u ∈ S(K, E),   u(x) , x∈K, E`k u(x) := `  εk u(x) , x ∈ (−∞, 0) × ∂k K ,

(1.1.9)

k

and Ek := E0k . It follows from Lemma 1.1.1 that ¡ ¢ E`k ∈ L S(K, E), S(Kk,k−1 , E) .

(1.1.10)

The operator of point-wise restriction Rk is defined by Rk u(x) := u(x) ,

x∈K,

u ∈ S(Kk,k−1 , E) ,

and satisfies ¡ ¢ Rk ∈ L S(Kk,k−1 , E), S(K, E) .

(1.1.11)

¡ ¢ It is clear that (Rk , E`k ) is an r-c pair for S(Kk,k−1 , E), S(K, E) if k ∈ J.

8

VI Auxiliary Material

Let k ∈ J ∗ . Then the trivial extension operator ˚ Ek is given for u ∈ S(K, E) by ( u(x) , x ∈ K , ˚ Ek u(x) := (1.1.12) 0, x ∈ (−R+ ) × ∂k K . k

It follows from (1.1.2) that ¡ ¢ ˚ Ek ∈ L S(K, E), S(Kk,k−1 , E) .

(1.1.13)

Note that εk is also well-defined if u ∈ S(Kk,k−1 , E) and x ∈ K. Hence ˚ k u(x) := u(x) − εk u(x) , R

x∈K,

(1.1.14)

is meaningful for u ∈ S(Kk,k−1 , E). By means of Lemma 1.1.1 it is verified that ˚ k u|xk =0 = 0 , ∂k` R

`∈N.

(1.1.15)

This implies ¡ ¢ ˚ k ∈ L S(Kk,k−1 , E), S(K, E) . R (1.1.16) ¡ ¢ ˚k , ˚ It is obvious that (R Ek ) is an r-c pair for S(Kk,k−1 , E), S(K, E) if k ∈ J ∗ . We set ( (Rk , Ek ) , if k ∈ J , ¡ ¢ e k, E e k := R ˚k , ˚ (R Ek ) , if k ∈ J ∗ . Then (1.1.8), (1.1.11), and (1.1.16) imply ¡ ¢ ek ◦ R e k−1 ◦ · · · ◦ R e 1 ∈ L S(Rd , E), S(K, E) . RK := R

(1.1.17)

Similarly, we get from (1.1.8), (1.1.10), and (1.1.13) ¡ ¢ e1 ◦ E e2 ◦ · · · ◦ E e k ∈ L S(K, E), S(Rd , E) . EK := E (1.1.18) ¡ ¢ Clearly, (RK , EK ) is an r-c pair for S(Rd , E)), S(K, E) . Now we drop the assumption that K be a standard corner, that is, we assume •

K is a k-corner in Rd .

b in Rd and s ∈ Sd such that K = sK. b We Thus there exist a standard k-corner K put RK := Us ◦ RK ◦ Us−1 ,

EK := Us ◦ EK ◦ Us−1 .

It follows from, (1.1.3), (1.1.4), (1.1.17), and (1.1.18) that ¡ ¢ ¡ ¢ RK ∈ L S(Rd , E), S(K, E) , EK ∈ L S(K, E), S(Rd , E) .

(1.1.19)

(1.1.20)

VI.1 Restriction-Extension Pairs

9

Moreover, (RK , EK ) is an r-c pair. We express (1.1.20) and the latter fact by saying: ¡ ¢ (RK , EK ) is a restriction-extension pair, an r-e pair, for S(Rd , E), S(K, E) . b 0 is a standard k-corner in Rd such that K = s0 K b 0 . Set Suppose s0 ∈ Sd and K −1 0 0 b b s1 x := s (s x) so that s1 K = K. Then Us0 ◦ RK0 ◦ Us−1 = Us ◦ Us1 ◦ RK0 ◦ Us−1 ◦ Us−1 = Us ◦ RK ◦ Us−1 = RK . 0 1 Similarly, Us0 ◦ EK0 ◦ Us−1 = Us ◦ EK ◦ Us−1 = EK . Thus (RK , EK ) is well-defined by 0 (1.1.19) independently of the particular representation of K. Approximation by Test Functions The following simple observation will repeatedly be of use. 1.1.2 Lemma Let X and Y be LCSs and (r, rc ) an r-c pair for (X , Y). Suppose X0 is a dense subset of X . Then r(X0 ) is dense in Y. Proof Suppose y ∈ Y and U is a neighborhood of y in Y. Then, since r is surjective, r−1 (U ) is a neighborhood of r−1 (y) in X . Hence there exists x ∈ X0 ∩ r−1 (y) by the density of X0 in X. Then r(x) ∈ U ∩ r(X0 ). This proves the claim. ¥ It is¡ the purpose of the ¢following considerations to extend (RK , EK ) to an r-e pair for S 0 (Rd , E), S 0 (K, E) . For this we need some preparation. We denote by D(K, E) the subspace of S(K, E) consisting of all functions with compact support, endowed with the usual LF topology (analogous to the definition of D(Rd , E)). It is the space of E-valued test functions on K. 1.1.3 Lemma (i) Let K be a standard k-corner. Set Ij := R for k + 1 ≤ j ≤ d. Then d−1 O

d

S(Ii ) ⊗ S(Id , E) ⊂ S(K, E) .

i=1 d

(ii) If K is a corner in Rd , then D(K, E) ,→ S(K, E). Proof (1) It is known (e.g., Corollary 1.8.2 in the Appendix or Theorem 3.9.2 in [Tre67]) that d

S(R)⊗(d−1) ⊗ S(R, E) ⊂ S(Rd , E) .

(1.1.21)

Suppose u ∈ S(K, E). Then EK u ∈ S(Rd , E). Hence there exists a sequence (uj ) in S(R)⊗(d−1) ⊗ S(R, E) with uj → EK u in S(Rd , E). It follows RK uj → RK EK u = u

10

VI Auxiliary Material

in S(K, E). Given v := v1 ⊗ · · · ⊗ vd in S(Rd )⊗(d−1) ⊗ S(R, E), it is clear that Nd−1 Nd−1 RK v belongs to i=1 S(Ii ) ⊗ S(Id , E). Hence RK uj ∈ i=1 S(Ii ) ⊗ S(Id , E) for j ∈ N. This proves (i). (2) We fix ϕ ∈ D(R) with ϕ(x) = 1 for |x| ≤ 1 and put ϕr (x) := ϕ(x/r) for x ∈ R and r > 0. Suppose u ∈ S(R+ , E). Then ϕr u ∈ D(R+ , E) for r > 0. Note that hxik ≤ hxik+1 /r for |x| ≥ r. Moreover, ∂ α (ϕr − 1) vanishes for |x| < r and α ∈ N. Hence, by Leibniz’ rule, suphxik |∂ m (ϕr u − u)|E (x) ≤ c(m) x>0

m X

r−j suphxik |∂ m−j u|E (x)

j=0

x>r

≤ c(m)r−1 qk+1,m (u) for k, m ∈ N and r > 1. Thus ϕr u → u in S(R+ , E) as r → ∞. This shows that D(R+ , E) is dense in S(R+ , E). (3) Let τy be the right translation, that is, (τy u)(x) = u(x − y) ,

u: R→E ,

x∈R,

y ∈ R+ .

Then { τy ; y ≥ 0 } is a strongly continuous semigroup on S(R, E) (cf. the proof of Theorem VII.3.3.1 below). ¡ We set I := (0, ¢ ∞) and ρy := RI ◦ τy ◦ EI . Since (RI , EI ) is an r-e pair for S(R, E), S(I, E) , it follows that { ρy ; y ≥ 0 } is a strongly continuous semigroup on S(I, E). Note that supp(ρy u) ⊂ [y, ∞) for y > 0 and u ∈ S(I, E). Hence ϕr (ρy u) ∈ D(I, E). It follows from u − ϕr (ρy u) = u − ρy u − (ϕr − 1)ρy u , by choosing first y sufficiently small and then r sufficiently large, that each element u of S(I, E) can be arbitrarily closely approximated in S(I, E) by elements of D(I, E). Hence D(I, E) is dense in S(I, E). (4) Clearly, D(K, E) ,→ S(K, E). We deduce from steps (2) and (3) and the Nd−1 density of D(R, E) in S(R, E) that each u ∈ i=1 S(Ii ) ⊗ S(Id , E) can be arbiNd−1 trarily closely approximated in S(Rd , E) by elements of i=1 D(Ii ) × D(Id , E). Since each such element belongs to D(K, E), we see from (1.1.21) that D(K, E) is dense in S(K, E). This proves (ii). ¥

1.2

Tempered Distributions on Corners

Throughout this subsection •

K is a corner in Rd .

As already mentioned, we now extend the r-e pair (RK , EK ) to distributions. In analogy to the full-space case, the space of tempered distributions on K is defined

VI.1 Restriction-Extension Pairs

11

by ¡ ¢ S 0 (K, E) := L S(K∗ ), E . Note that3

(1.2.1)

Z (u, ϕ) 7→ hu, ϕiK :=

uϕ dx

(1.2.2)

K

is a continuous bilinear map S(K, E) × S(K∗ ) → E. It follows from (1.1.2), partial integration, and Fubini’s theorem that h∂ α u, ϕiK = (−1)|α| hu, ∂ α ϕiK ,

α ∈ Nd .

(1.2.3)

Similarly as in the classical case, we define Tu ∈ S 0 (K, E) for u ∈ S(K, E) by ϕ ∈ S(K∗ ) .

Tu ϕ := hu, ϕiK ,

Since u 7→ Tu is injective we can identify u ∈ S(K, E) with Tu ∈ S 0 (K, E) so that S(K, E) ,→ S 0 (K, E) .

(1.2.4)

We use the notation hu, ¡ϕiK also for the¢ value of u ∈ S 0 (K, E) on ϕ ∈ S(K∗ ) and call h·, ·iK E -valued S 0 (K, E), S(K∗ ) duality pairing induced by (1.2.2). The (K)-distributional derivative ∂ α u of u ∈ S 0 (K, E) is then defined for α ∈ Nd by h∂ α u, ϕiK := (−1)|α| hu, ∂ α ϕiK ,

ϕ ∈ S(K∗ ) .

(1.2.5)

By (1.2.3) and (1.2.4) it extends the classical derivative on K. Furthermore, ¡ ¢ ¡ ¢ ∂ α ∈ L S(K, E) ∩ L S 0 (K, E) . (1.2.6) It should be observed that ∂ α u is not the distributional derivative of u in the sense of Schwartz distributions if K 6= K, since S(K∗ ) is then substantially larger than ˚ the space of test functions D(K). In the following, we shall often use pull-back operators. For this reason we collect in the next remarks some of their basic properties. 1.2.1 Remarks Let X and Y be LCSs. (a) Suppose f ∈ L(X , Y). Then the pull-back f ∗ u of u ∈ L(Y, E) is defined by f ∗ u := u ◦ f . Thus ¡ ¢ f ∈ L(X , Y) =⇒ f ∗ ∈ L L(Y, E), L(X , E) . If Z is a further LCS and g ∈ L(Y, Z), then (g ◦ f )∗ = f ∗ ◦ g ∗ , 3 uϕ

:= ϕu.

(idX )∗ = idL(X ,E) .

12

VI Auxiliary Material

Consequently, f ∈ Lis(X , Y)

=⇒

¡ ¢ f ∗ ∈ Lis L(Y, E), L(X , E) ,

and (f ∗ )−1 = (f −1 )∗ . (b) Let i : X ,→ Y and u ∈ L(Y, E). Then i∗ u = u ◦ i = u|X , considered as a d

continuous linear map from X into E. If X ,→ Y, then i∗ is injective. Hence it is justified to identify L(Y, E) with a linear subspace of L(X , E) by identifying u ∈ L(Y, E) with u |X ∈ L(X , E). It follows d

X ,→ Y

=⇒

L(Y, E) ,→ L(X , E) .

(1.2.7)

¡ c ∗ ∗¢ c (c) Let (r, ¡ r ) be an r-c pair ¢ for (X , Y). Then (a) implies that (r ) , r is an r-c pair for L(X , E), L(Y, E) . (d) Since L(X , C) = X 0 we get f ∗ = f 0 ∈ L(Y 0 , X 0 ) for f ∈ L(X , Y). Thus (1.2.7) generalizes d

X ,→ Y (cf. Proposition V.1.4.8).

=⇒

Y 0 ,→ X 0

¥

Duality Formulas It follows from Lemma 1.1.3(ii) and (1.2.7) that ¡ ¢ S 0 (K, E) ,→ D0 (K, E) := L D(K∗ ), E . In particular, S 0 (K, E) ,→ D0 (K, E) which shows that ˚ each u ∈ S 0 (K, E) is an E-valued Schwartz distribution on K.

(1.2.8)

1.2.2 Lemma It holds hEK u, ϕiRd = hu, RK∗ ϕiK ,

u ∈ S(K, E) ,

ϕ ∈ S(Rd ) ,

hRK u, ϕiK = hu, EK∗ ϕiRd ,

u ∈ S(Rd , E) ,

ϕ ∈ S(K∗ ) .

and

Proof (1) Let K be a standard k-corner of type J. Suppose k ∈ J, u ∈ S(K, E), and ϕ ∈ S(K∗k,k−1 ). Then, by (1.1.9), Z Z hEK u, ϕiKk,k−1 = ϕu dx + ϕεk u dx . (1.2.9) K

(−∞,0)×∂k K k

VI.1 Restriction-Extension Pairs

13

¡c ¡ c ¢¢ k 7→ u y; x k We write u(y; ·) := x etc. Then, by Lemma 1.1.1 and Fubini’s theorem, Z 0 Z 0 Z ∞ ϕ(y; ·)εk u(y; ·) dy = ϕ(y; ·) h(t)u(−ty; ·) dt dy −∞ −∞ 0 Z ∞Z ∞ = h(t)u(ty; ·)ϕ(−y; ·) dt dy y7→−y 0 0 Z ∞Z ∞ = h(t)u(z; ·)ϕ(−z/t; ·) dz dt/t y7→z:=ty 0 0 Z ∞Z ∞ = h(1/s)u(z; ·)ϕ(−sz; ·) dz ds/s t7→s:=1/t

=

h(1/s)=−sh(s)

=

Z

0 ∞

Z

0 ∞

−

u(z; ·)h(s)ϕ(−sz; ·) dz ds Z

0

0 ∞

−

u(z; ·)(εk ϕ)(z; ·) dz . 0

By integrating this relation over ∂k K we get from (1.2.9) and (1.1.14) Z Z ˚ k ϕiK . hEk u, ϕiKk,k−1 = uϕ dx − uεk ϕ dx = hu, R K

K

(2) Let K be as in (1) and assume k ∈ J ∗ . Then we obtain by (1.1.12) Z ˚ hEk u, ϕiKk,k−1 = uϕ dx = hu, Rk ϕiK K

S(K∗k,k−1 )

for u ∈ S(K, E) and ϕ ∈ since Rk ϕ is the point-wise restriction. (3) From (1) and (2) and the definition of (RK , EK ) we find hEK u, ϕiRd = hu, RK∗ ϕiK ,

u ∈ S(K, E) ,

ϕ ∈ S(Rd ) ,

if K is a standard corner. b where K b is a standard corner and s ∈ Sd . Then, (4) Now suppose K = sK x 7→ sx being an isometry, Z Z −1 hUs u, ϕiK = u(s x)ϕ(x) dx = u(y)ϕ(sy) dy = hu, Us−1 ϕiK K

K

and, since sRd = Rd , we infer from (1.1.3), (1.1.4), (1.1.19), and step (3) hEK u, ϕiRd = hUs ◦ EK ◦ Us−1 u, ϕiRd = hEK ◦ Us−1 u, Us−1 ϕiRd = hUs−1 u, RK∗ ◦ Us−1 ϕiK = hu, Us ◦ RK∗ ◦ Us−1 ϕiK = hu, RK∗ ϕiK for u ∈ S(K, E) and ϕ ∈ S(Rd ). This proves the first assertion. It is obvious that the above arguments give the second claim as well. ¥

14

VI Auxiliary Material

¡ ¢ By (1.1.20) we know that (RK∗ , EK∗ ) is an r-c pair for S(Rd ), S(K∗ ) . Hence, by (1.2.1) and Remark 1.2.1(c), ¡ ¢ ¡ ¢ (E∗K∗ , R∗K∗ ) := (EK∗ )∗ , (RK∗ )∗ is an r-c pair for S 0 (Rd , E), S 0 (K, E) . (1.2.10) Lemma 1.2.2 shows that

and

¡ ¢ E∗K∗ ∈ L S 0 (Rd , E), S 0 (K, E) ¡ ¢ is an extension of RK ∈ L S(Rd , E), S(K, E)

(1.2.11)

¡ ¢ R∗K∗ ∈ L S 0 (K, E), S 0 (Rd , E) ¡ ¢ is an extension of EK ∈ L S(K, E), S(Rd , E) .

(1.2.12)

d

It is known (cf. Theorem 1.3.6 in the Appendix) that S(Rd , E) ,→ S 0 (Rd , E). Hence it follows from (1.2.4), (1.2.11), and (1.2.12) that the following diagrams4 RK

S(Rd , E)

¤¡

-

S(K, E)

¤¡

d

?

E∗K∗

S 0 (Rd , E)

S(K, E)

¤¡

?

-

EK

S(Rd , E)

¤¡

d

?

S 0 (K, E)

-

S 0 (K, E)

R∗K∗

-

? S 0 (Rd , E)

are commuting. Moreover, (1.2.10) and Lemma 1.1.2 imply that S(K, E) is also dense in S 0 (K, E). From this we infer that (E∗K∗ , R∗K∗ ) is uniquely determined by (RK , EK ). For this reason we write again (RK , EK ) for (E∗K∗ , R∗K∗ ) without fearing and say ¡ 0 confusion ¢ that (RK , EK ) is a restriction-extension pair (an r-e pair) for S (Rd , E), S 0 (K, E) . The Main Theorem The above facts are summarized in part (i) of the following basic restrictionextension theorem for smooth functions and tempered distributions. 1.2.3 Theorem Let K be a corner in Rd . ¡ ¢ (i) (RK , EK ) is an r-e pair for S 0 (Rd , E), S 0 (K, E) and the diagrams S(Rd , E)

RK

¤¡

-

d

? S 0 (Rd , E)

S(K, E)

¤¡

d

RK

-

? S 0 (K, E)

S(K, E)

EK

¤¡

-

S(Rd , E)

¤¡

d

? S 0 (K, E)

d

EK

-

? S 0 (Rd , E)

are commuting. 4 In

all diagrams occurring in this volume, arrows represent continuous linear maps.

VI.1 Restriction-Extension Pairs

15

(ii) RK u is the restriction of u ∈ S 0 (Rd , E) to S 0 (K, E) in the sense of Schwartz distributions. (iii) ∂ α ◦ RK = RK ◦ ∂ α , α ∈ Nd . Proof

(1) Claim (i) has been proved by the above considerations. ∗ 0 d ˚ ˚ (2) Since RK u = (E˚ K ) u for u ∈ S (R , E) and E˚ K = E1 ◦ · · · ◦ Ek if K is a standard k-corner, we get ˚ , ϕ ∈ S(K)

u ∈ S 0 (Rd , E) ,

hRK u, ϕiK = hu, ϕiK ,

(1.2.13)

from (1.1.12). It follows from the continuity of Us on S(K) and the density assertions of part (i) that hUs u, ϕiK = hu, Us−1 ϕiK ,

b E) , u ∈ S 0 (K,

ϕ ∈ S(K) .

From this and (1.2.13) we infer that the latter holds for arbitrary corners, which proves (ii). (3) Using (1.2.3), (1.2.5), (ii), and the continuity of ∂ α on S(K, E) and S (K, E) we get 0

h∂ α RK u, ϕiK = (−1)|α| hRK u, ∂ α ϕiK = (−1)|α| hu, ∂ α ϕiK = h∂ α u, ϕiK = hRK ∂ α u, ϕiK ˚ This implies (iii). for u ∈ S 0 (Rd , E) and ϕ ∈ S(K).

¥

1.2.4 Remarks (a) The r-e pair (RK , EK ) is universal in the following sense: sup1 1 pose 1 is a Banach ¢space with E1 ,→ E and let (RK , EK ) be the r-e pair for ¡ 0 E d 0 S (R , E1 ), S (K, E1 ) constructed above. Then the diagrams S 0 (Rd , E1 )

¤£

-

R1K

?

S 0 (K, E1 )

S 0 (Rd , E) RK

¤£

-

?

S 0 (K, E)

S 0 (K, E1 )

¤£

-

E1K

?

S 0 (Rd , E1 )

S 0 (K, E) EK

¤£

-

?

S 0 (Rd , E)

are commuting. Furthermore, the particular form of (RK , EK ) is independent of the choice of E. (b) Let X ∈ {S, S 0 }. Then RK u = RK u for u ∈ EK X (K, E). Proof (1) Suppose X = S and k ∈ J ∗ . Let u = ˚ Ek v for some v ∈ S(K, E). Then ¡ ck ¢ = 0) for xk ≤ 0. Hence definition (1.1.14) εk u(x) = 0 for x ∈ K, since u xk ; x ˚ k u(x) = u(x) = Rk u(x) for u(x) ∈ K. From this we get the claim in the implies R present case, due to RK = Rk ◦ Rk−1 ◦ · · · ◦ R1 if K is a standard k-corner.

16

VI Auxiliary Material

(2) If X = S 0 , then the assertion follows from (1) by continuous extension based on Theorem 1.2.3. ¥ The following completeness assertion will be of use in later sections. 1.2.5 Proposition Let K be a corner in Rd and X ∈ {Rd , K}. Then S 0 (X, E) is a complete LCS. Proof (1) Suppose X = Rd . It is well-known that S(Rd ) and S 0 (Rd ) are complete and reflexive (e.g., Theorem 1.1.2 in the Appendix). Hence Lemma 1.4.7 ibidem ¡ ¢ ¡¡ ¢0 ¢ guarantees that S 0 (Rd , E) = L S(Rd ), E = L S 0 (Rd ) , E is complete. (2) Assume X = K. It follows from Theorem 1.2.3(i) and Lemma I.2.3.1 that S 0 (K, E) is isomorphic to a closed linear subspace of S 0 (Rd , E). Since the latter space is complete by (1), S 0 (K, E) is also complete. ¥

1.3

Duality

The theorem proved in this subsection shows that S 0 (K, E) can be characterized by vector-valued duality. It is the basis for the duality theory of vector-valued Besov and Bessel potential spaces. For the sake of a uniform presentation we put (Rd )∗ := Rd in what follows. 1.3.1 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. Then S 0 (X, E 0 ) = S(X∗ , E)0 with respect to the unique duality pairing satisfying Z ­ 0 ® 0 hu , uiS(X∗ ,E) = u (x), u(x) E dx

(1.3.1)

(1.3.2)

X∗

for u0 ∈ S(X, E 0 ) and u ∈ S(X∗ , E). Proof (1) Suppose X = Rd . Then the assertion follows from L. Schwartz’ theory of vector-valued distributions [Schw57b]. In fact, h·, ·iS(Rd ,E) is uniquely deter¡ ¢ ¡ ¢ mined by its values on (ϕ0 ⊗ e0 , ϕ ⊗ e) ∈ D(Rd ) ⊗ E 0 × D(Rd ) ⊗ E and it is a hypocontinuous bilinear form on S 0 (Rd , E) × S(Rd , E). More precisely, the claim is implied by Corollary 1.4.10 and Theorem 1.7.5 of the Appendix and the density of D in S. (2) Assume X = K. It is a consequence of Theorem 1.2.3 and Lemma I.2.3.1 that EK∗ S(K∗ , E) is a closed linear subspace of S(Rd , E) and ¡ ¢ EK∗ ∈ Lis S(K∗ , E), EK∗ S(K∗ , E) . (1.3.3)

VI.1 Restriction-Extension Pairs

17

The same is true if EK∗ and S(K∗ , E) are replaced by EK and S 0 (K, E 0 ), respectively. Hence hu0 , uiS(K∗ ,E) := hEK u0 , EK∗ uiS(Rd ,E) ,

u0 ∈ S 0 (K, E 0 ) ,

u ∈ S(K∗ , E) ,

defines a hypocontinuous bilinear form h·, ·iS(K∗ ,E) on S 0 (K, E 0 ) ⊗ S(K∗ , E). (3) Suppose u0 ∈ S(K, E 0 ) and u = ϕ ⊗ e ∈ D(K∗ ) ⊗ E ⊂ S(K∗ ) ⊗ E. Then EK∗ u = (EK∗ ϕ) ⊗ e ∈ S(Rd ) ⊗ E , Lemma 1.2.2, and (1.2.2) imply ­ ® hu0 , uiS(K∗ ,E) = hEK u0 , EK∗ uiS(Rd ,E) = hEK u0 , EK∗ ϕiRd , e E ­ ® ­ ® = hu0 , RK∗ EK∗ ϕiK∗ , e E = hu0 , ϕiK∗ , e E Z Z ­ 0 ® ­ 0 ® = u (x), ϕ(x) ⊗ e E dx = u (x), u(x) E dx . K∗

(1.3.4)

K∗

By Theorem 1.3.6 of the Appendix we know that S(Rd ) ⊗ E is dense in S(Rd , E). Suppose u ∈ EK∗ S(K∗ , E) and (uj ) is a sequence in S(Rd ) ⊗ E converging to u. Since PK∗ := EK∗ RK∗ is a continuous projection in S(Rd , E) onto EK∗ S(K∗ , E), it follows that (PK∗ uj ) is a sequence in EK∗ S(K∗ , E) converging to u. If v=

n X

ϕk ⊗ ek ∈ S(Rd ) ⊗ E ,

k=0

then ψk := RK∗ ϕk ∈ S(K∗ ) implies PK ∗ v =

n X

EK∗ ψk ⊗ ek ∈ EK∗ S(K∗ ) ⊗ E .

k=0

This shows that EK∗ S(K∗ ) ⊗ E is dense in EK∗ S(K∗ , E). Since D(K∗ ) is dense in S(K∗ ) by Lemma 1.1.3, it follows that EK∗ D(K∗ ) ⊗ E is dense in EK∗ S(K∗ , E). By Theorem 1.2.3, S(K, E 0 ) is dense in S 0 (K, E 0 ). These density properties and the hypocontinuity of h·, ·iS(K∗ ,E) imply that the latter bilinear form is uniquely ¡ ¢ determined by its restriction to S(K, E 0 ) × D(K∗ ) ⊗ E . From this and (1.3.4) we see that (1.3.2) applies. (4) Suppose u0 ∈ S 0 (K, E 0 ) and hu0 , uiS(K∗ ,E) = 0 for all u ∈ S(K∗ , E). Then ∗ hu , ϕ ⊗ eiS(K∗ ,E) = 0 for ­all (ϕ, e) ∈ S(K ) × E. It follows, similarly as in step ® ­ ® (3), 0 0 that hu , ϕ ⊗ eiS(K∗ ,E) = hu , ϕiK∗ , e E . Hence we derive from hu0 , ϕiK∗ , e E = 0 for all e ∈ E that hu0 , ϕiK∗ = 0 for all ϕ ∈ S(K∗ ). Thus u0 = 0. We deduce similarly that, given u ∈ S(K∗ , E), hu0 , uiS(K∗ ,E) = 0 for all u ∈ S 0 (K, E 0 ) implies u = 0. Hence h·, ·iS(K∗ ,E) is separating. This proves the theorem. ¥ 0

18

VI Auxiliary Material

1.3.2 Corollary The dual of ¡ ¢ ¡ ¢ RK∗ ∈ L S(Rd , E), S(K∗ , E) equals EK ∈ L S 0 (K, E 0 ), S 0 (Rd , E 0 ) and the dual of ¡ ¢ ¡ ¢ EK∗ ∈ L S(K∗ , E), S(Rd , E) equals RK ∈ L S 0 (Rd , E 0 ), S 0 (K, E 0 ) . Proof Let u ∈ S(K, E 0 ) and v = ϕ ⊗ e ∈ S(Rd ) ⊗ E. Then, by Lemma 1.2.2 and K∗∗ = K, ­ ® ­ ® hEK u, ϕiS(Rd ,E) = hEK u, ϕiRd , e E = hu, RK∗ ϕiK∗ , e E = hu, RK∗ viS(K∗ ,E) . ¡ ¢ d From this, S(K, E 0 ) ,→ S 0 (K, E 0 ), and EK ∈ L S 0 (K, E 0 ), S 0 (Rd , E 0 ) it follows hEK u, viS(Rd ,E) = hu, RK∗ viS(K∗ ,E)

(1.3.5)

for u ∈ S 0 (K, E 0 ) and v ∈ S(Rd ) ⊗ E. By employing the density of S(Rd ) ⊗ E in S(Rd , E), we thus find that (1.3.5) holds for v ∈ S(Rd , E). This implies the first assertion. The second one follows by similar arguments. ¥

1.4

Notes

As mentioned in the introduction to this section, the basic result is Theorem 1.2.3, the restriction-extension theorem for smooth functions and tempered distributions on corners. Of course, its nucleus is the corresponding result for half-spaces, that is, 1-corners. This, in turn, is based on the crucial Lemma 1.1.1 which is taken from [Ham75]. In the case of a closed half-space H := R+ × Rd−1 , the extension operator EH is kind of a continuous version of Seeley’s extension operator [See64]. It has, however, the advantage that it gives rise to the important duality formulas of Corollary 1.3.2. The section at hand is a reworking and amplification of Sections 4.1–4.3 of [Ama09], which was inspired by [Ham75].

VI.2 Sequence Spaces

2

19

Sequence Spaces

By means of localization techniques we will naturally be led to consider spaces of Banach-space-valued sequences. Such localizations occur in connection with Besov spaces, the solvability theory of elliptic and parabolic boundary vale problems, and in the study of function spaces and differential equations on Riemannian manifolds. For this reason we present in this section some of the basic results on sequence spaces for the purpose of easy reference. In particular, we prove a general duality theorem for Banach-space-valued `p -spaces, introduce weighted sequence spaces, and collect their fundamental interpolation properties.

2.1

Duality of Sequence Spaces

With regard to later applications, where concrete enumerations are not convenient, we consider ‘sequences’ on arbitrary countable index sets. Definitions and Embeddings We denote by A a nonempty countable set (an index set) which we give the discrete topology. Thus A is a σ-compact Hausdorff space. Let Xα be for each α ∈ A an LCS. We set X := (Xα ) = (Xα )α∈A and endow Y c(X ) := Xα α

with the product topology, that is, the coarsest locally convex topology for which all projections prβ : c(X ) → Xβ , x = (xα ) 7→ xβ are continuous. By cc (X ) :=

M

Xα

α

we mean the locally convex direct sum. Thus cc (X ) is the linear subspace of c(X ) consisting of all compactly supported sequences equipped with the finest locally convex topology for which all natural injections jβ : Xβ → cc (X ) ,

xβ 7→ (xβ δαβ )α

(2.1.1)

are continuous, where δαβ := 0 if α 6= β and δαα := 1, the Kronecker symbol. If Xα = X for α ∈ A, then we write c(X ) and cc (X ) for c(X ) and cc (X ), respectively. Thus c(X ) = X A is the LCS of all X -valued functions on A with the topology of point-wise convergence. Since every map from a discrete space into a topological space is continuous

20

VI Auxiliary Material

and A is σ-compact, X A = C(A, X ). Moreover, cc (X ) = Cc (A, X ) is the space of compactly supported continuous X -valued functions on A with the obvious LF topology. Let h·, ·iα be the (Xα0 , Xα )-duality pairing. Then, letting X 0 := (Xα0 ), X h ·, ·ii : c(X 0 ) × cc (X ) → C , (x0 , x) 7→ hx0α , xα iα α

is a separating continuous bilinear form, and cc (X )0 = c(X 0 ) (2.1.2) ¡ ¢ with respect to h ·, ·ii, that is, h ·, ·i i is the cc (X )0 , cc (X ) -duality pairing (cf. Corollary 1 in Section IV.4.3 of H.H. Schaefer [Sch71]). Suppose E = (Eα ), where each Eα = (Eα , |·|α ) = (Eα , |·|Eα ) is a Banach space and 1 ≤ r ≤ ∞. Then `r (E) = `r (A, E) is the linear subspace of c(E) consisting of all x = (xα ) such that5 ( ¡P ¢1/r |xα |rα , r 1, then cc (E 0 ) is dense in `r0 (E 0 ). Thus we infer from (2.1.10) that (2.1.14) holds for each y in `r0 (E 0 ). Consequently, by (2.1.9), kTy k(`r (E))0 = kyk`r0 (E 0 ) ,

y ∈ `r0 (E 0 ) .

(2.1.15)

¡ ¢0 Hence y 7→ Ty is a linear isometry from `r0 (E 0 ) into `r (E) if 1 ≤ r < ∞.

24

VI Auxiliary Material

(8) Assume y ∈ `1 (E 0 ) and 0 < ε < 1. Then, given K b A, it follows from (2.1.12) that kTy k(c0 (E))0 =

|hhy, xii| |hhy, PK zii| ≥ ≥ (1 − ε) kPK0 yk`1 (E 0 ) . kPK zk`∞ (E) x∈cc (E) kxk`∞ (E) sup

Consequently,

kTy k(c0 (E))0 ≥ kPK0 yk`1 (E 0 ) .

Since this is true for every K b A, and since limK PK0 y = y in `1 (E 0 ), we get kTy k(c0 (E))0 ≥ kyk`1 (E 0 ) . By H¨older’s inequality |Ty x| = |hhy, xi i| ≤ kyk`1 (E 0 ) kxk`∞ (E) = kyk`1 (E 0 ) kxkc0 (E) ¡ ¢0 for x ∈ c0 (E). Thus Ty | c0 (E) belongs to c0 (E) and kTy k(c0 (E))0 ≤ kyk`1 (E 0 ) ,

y ∈ `1 (E 0 ) .

(2.1.16)

¡ ¢0 Hence y → 7 Ty is a linear isometry from `1 (E 0 ) into c0 (E) . (9) Given K b A, we set P jK := cc (E) → cc (E) , x 7→ α∈K jα xα , where jα : Eα → cc (E) is the canonical injection (2.1.1). ¡ ¢0 Suppose 1 ≤ r < ∞. Let f belong to `r (E) . We define y = (yα ) ∈ c(E 0 ) by yα := f ◦ jα and set yK := PK0 y. Then, by step (2), hyK , xii = hPK0 y, xi i = hy, PK xi i X X = hyα , xα iα = f (jα xα ) = f (jK x) α∈K

(2.1.17)

α∈K

for x ∈ `r (E). Hence |TyK (x)| ≤ kf k(`r (E))0 kjK xk`r (E) = kf k(`r (E))0

³X

|xα |rα

´1/r

α∈K

≤ kf k(`r (E))0 kxk`r (E) . From this, yK ∈ cc (E 0 ) ⊂ `r0 (E 0 ), and (2.1.15) we deduce kyK k`r0 (E 0 ) = kTyK k(`r (E))0 ≤ kf k(`r (E))0 , Thus

y ∈ `r0 (E 0 ) ,

KbA.

kyk`r0 (E 0 ) ≤ kf k(`r (E))0 .

VI.2 Sequence Spaces

25

It follows from (2.1.17) that Ty (jK x) = f (jK x) ,

x ∈ `r (E) .

¡ ¢0 Since Ty ∈ `r (E) by step (3), and limK jK x = x in `r (E), we find Ty (x) = f (x) for x ∈ `r (E), that is, Ty = f . This shows that (y 7→ Ty ) is a surjection from ¡ ¢0 `r0 (E 0 ) onto `r (E) . Thus it follows from step (7) that assertion (2.1.7) is true. ¡ ¢0 (10) Assume f ∈ c0 (E) . We define y ∈ c(E 0 ) as in the preceding step. Then (2.1.17) applies for x ∈ c0 (E). From this and step (8) we get, similarly as above, kyK k`1 (E 0 ) = kTyK k(c0 (E))0 ≤ kf k(c0 (E))0 , KbA. Hence y ∈ `1 (E 0 ). Furthermore, since limK jK x = x in c0 (E), we deduce from Ty (jK x) = f (jK x) that Ty (x) = f (x) for all x in c0 (E). Thus Ty | c0 (E) = f . Due to the results of step (8), this implies the validity of (2.1.8). (11) If E is reflexive, then (E 0 )0 = E. Hence the last assertion is obvious.

2.2

¥

Weighted Sequence Spaces

In the study of Besov spaces, weighted sequences play a fundamental role. In this connection the weights are powers of 2, naturally related to dyadic decompositions studied in the next section. In the section at hand, we transpose the preceding results to weighted sequence spaces. Image Spaces Let X and Y be LCSs and ϕ ∈ L(X , Y). The image space ϕX is the image of X in Y under ϕ endowed with the unique locally convex topology for which ϕ, b defined by the commutativity of the diagram ϕ X

@ @ R @

¡

µ ¡¡ ϕ

ϕX ⊂ Y

(2.2.1)

X / ker(ϕ)

is a toplinear isomorphism. Here the unlabeled arrow represents the canonical projection. 2.2.1 Remarks (a) ϕX is an LCS with ϕX ,→ Y, and ϕ is a continuous surjection onto ϕX . Let P be a generating family of seminorms for X and set © ª pb(y) := inf p(x) ; x ∈ ϕ−1 (y) , y ∈ ϕX , p ∈ P .

26

VI Auxiliary Material

Then the family { pb ; p ∈ P } generates the topology of ϕX . If X is a Banach space, then ϕX is a Banach space as well with the quotient norm © ª y 7→ kykϕX := inf kxkX ; x ∈ ϕ−1 (y) . In particular, if ϕ is injective, then kykϕX = kϕ−1 ykX ,

y ∈ ϕX .

Proof This follows from the closedness of ker(ϕ) and standard properties of quotient spaces. ¥ (b) Let X0 and Z be LCSs such that i : X0 ,→ X and Y ,→ Z. We write ϕX0 for (ϕ ◦ i)X0 . Then ϕX0 ,→ ϕX ,→ Z . d

d

If X0 ,→ X , then ϕX0 ,→ ϕX . Proof The first assertion is obvious. The second one follows from (2.2.1) and the continuity of the canonical projection. ¥ (c) If X is a reflexive or separable Banach space, then ϕX is also reflexive or separable, respectively. Proof Quotients of Banach spaces modulo closed linear subspaces possess these properties. ¥ Let a ∈ C\{0}. We identify it with the multiplication operator x 7→ ax for . x ∈ E. Then aE = (E, |·|aE ) and aE = E. In fact, |·|aE = |·|/|a|. Furthermore, given x0 ∈ E 0 , |x0 |(aE)0 = sup

x∈aE

|hx0 , xiE | |hax0 , yiE | = sup = |ax0 |E 0 = |x0 |a−1 E 0 . |x|aE |y| E y∈E

Hence a−1 E 0 = (aE)0 with respect to h·, ·iE .

(2.2.2)

Embeddings and Duality We define weighted sequence spaces as follows: ¡ ¢ for s ∈ R we set 2−s E := (2−sk E)k∈N . Then `sr (E) := `r (2−s E) ,

1≤r≤∞,

cs0 (E) := c0 (2−s E) .

(2.2.3)

VI.2 Sequence Spaces

27

Thus, for x ∈ E N it holds: x ∈ `sr (E) and

x ∈ cs0 (E)

(2ks xk )k ∈ `r (E)

⇐⇒

⇐⇒

2ks |xk |E → 0 as k → ∞ .

Hence, denoting by κs the weighted counting measure which assigns the value 2ks to {k} ⊂ N, `sr (E) = Lr (N, κs , E) . Note

`0r (E) = `r (E) ,

c00 (E) = C0 (N, E) .

In the following theorem, which is for most of its parts a direct consequence of the results of Subsection 2.1, we collect the basic properties of these spaces. 2.2.2 Theorem Suppose s, t ∈ R and 1 ≤ q, r ≤ ∞. (i) It holds: d

d

d

cc (E) ,→ `sq (E) ,→ `sr (E) ,→ cs0 (E) ,→ `s∞ (E) ,

1 ≤ q < r < ∞ , (2.2.4)

and `sq (E) ,→ `tr (E) ,

s>t.

(2.2.5)

(ii) Let E0 and E1 be Banach spaces with E1 ,→ E0 . Then `sr (E1 ) ,→ `sr (E0 ) ,

cs0 (E1 ) ,→ cs0 (E0 ) .

d

If E1 ,→ E0 , then d

`sr (E1 ) ,→ `sr (E0 ) , (iii) With respect to 8 h ·, ·ii, ¡ s ¢0 0 `r (E) = `−s r 0 (E ) ,

d

cs0 (E1 ) ,→ cs0 (E0 ) .

r0,

q ξ ∈ (Rd ) .

(3.1.6)

Note that (3.1.4) implies σs (σt u) = σst u ,

q u : (Rd ) → E .

s, t > 0 ,

(3.1.7)

We use the universal notation σt = σtν in which we do not indicate the image space E. It will always be clear from the context which E is being considered. ¡ ¢ Let G be the multiplicative group9 (0, ∞), q and X an LCS. A linear representation of G on X is a map G → L(X ), t 7→ Tt satisfying Ts Tt = Tst and T1 = 1X . Note that { Tt ; t ∈ G } is a commutative subgroup of the automorphism group Laut(X ), and (Tt )−1 = Tt−1 = T1/t ,

t∈G.

The representation is continuous, resp. strongly continuous, if t 7→ Tt is continuous from G into L(X ), resp. into Ls (X ). This is the case iff the respective continuity holds at t = 1. 3.1.2 Lemma t 7→ σt is a continuous linear representation of G on D(Rd , E) which is strongly continuous on L1,loc (Rd , E). Proof

We simply write D for D(Rd , E) etc.

(1) Recall that D = −−→ lim DK ,

where

DK :=

©

u ∈ C ∞ ; supp(u) ⊂ K

ª

,

KbRd

endowed with the topology induced by the seminorms u 7→ pm,K (u) := max sup |∂ α u|E , |α|≤m K

m∈N,

(cf. Subsection 1.1 of the Appendix). 9 It

can be replaced by any multiplicatively written continuous commutative group.

VI.3 Anisotropy

35

(2) The chain rule implies10 , for k ∈ N, p ∂ α (σt u) = tα ω σt (∂ α u) , u ∈ Ck ,

α qω ≤ k .

(3.1.8)

Suppose u ∈ Dt p K . Then we get from (3.1.8) pm,K (σt u) = max sup |∂ α (σt u)|E |α|≤m K ¡ p ¢ = max tα ω sup |σt (∂ α u)|E ≤ c(t, m)pm,t p K (u) . |α|≤m

(3.1.9)

K

Hence (u 7→ σt u) ∈ L(Dt p K , DK ) . Thus, by DK ,→ D, it follows (u 7→ σt u) ∈ L(Dt p K , D) for K b Rd . Consequently, (u 7→ σt u) ∈ L(D), by the properties of LF spaces. This and (3.1.7) show that t 7→ σt is a linear representation of G on D. (3) Note that |t q ξ − ξ| ≤ max |tνi − 1| |ξ| . 1≤i≤`

Hence, given

11

1

u∈C , Z

1

u(t q ξ) − u(ξ) =

¡ ¢ ∂u ξ + τ (t q ξ − ξ) dτ (t q ξ − ξ)

0 d

implies, for ξ ∈ M ⊂ R , |σt u − u|E (ξ) ≤ max |tνi − 1| |ξ| sup |∂u|L(Rd ,E) , 1≤i≤`

where Mt :=

©

(3.1.10)

Mt

(1 − τ )ξ + τ (t q ξ) ; ξ ∈ M, 0 ≤ τ ≤ 1

ª

.

By (3.1.8), ∂ α (σt u − u) = tα

¢ p ω¡ p σt (∂ α u) − ∂ α u + (tα ω − 1)∂ α u .

From this and (3.1.10) we deduce, for ξ ∈ M and |α| ≤ m, |∂ α (σt u − u)(ξ)|E ≤ tα

pω

max |tνi − 1| |ξ|

1≤i≤`

+ |tα

pω

max sup |∂ β u|E

|β|≤m+1 Mt

− 1| sup |∂ α u| .

(3.1.11)

M

(4) Now suppose M b K b Rd and u ∈ DM . Then u ∈ DK and there exists ε ∈ (0, 1) such that |t − 1| ≤ ε implies Mt ⊂ K. Moreover, 0 < t < 2. Let m ∈ N. 10 We use standard multi-index notation. In particular, |α| = α + · · · + α and 1 d α q ω = α1 ω1 + · · · + αd ωd for α ∈ Nd . (This can cause no confusion since (3.1.3) is not defined on Nd .) Moreover, somewhat inconsistently, the coordinates of multi-indices carry lower indices. 11 ∂ is the Fr´ echet derivative on Rd .

36

VI Auxiliary Material

We infer from (3.1.11) that ¡ ¢ p pm,K (σt u − u) ≤ c(m, K) max |tνi − 1| + |tα ω − 1| pm+1,K (u) 1≤i≤`

for u ∈ DK and |t − 1| ≤ ε. This implies σt u → u in D for t → 1, uniformly with respect to u in bounded sets. Hence σt → 1D in L(D) as t → 1. (5) Let u ∈ L1,loc . By using the transformation rule d(t q ξ) = t|ω| dξ

(3.1.12)

for the Lebesgue measure dξ on Rd , a change of variables gives Z Z Z −|ω| p |σt u(ξ)|E dξ = |u(t ξ)|E dξ = t |u(η)|E dη , t pK t pK K

K b Rd .

This implies σt ∈ L(L1,loc ), uniformly for t in compact subsets of G. d

(6) Recall that D ,→ L1,loc . Hence it follows from step (4) that ¡ ¢ (t 7→ σt ) ∈ C G, L(D, L1,loc ) .

(3.1.13)

Suppose K b Rd , ε > 0, and u ∈ L1,loc . By (3.1.13), the density of D in L1,loc , and (5) we can find v ∈ D such that Z Z |u − v|E dξ + |σt (u − v)|E dξ ≤ ε/2, , |t − 1| ≤ 1/2 . K

K

Hence we infer from (3.1.13) that there exists δ ∈ (0, 1/2) such that Z Z Z Z |σt u − u|E dξ ≤ |σt (u − v)|E dξ + |u − v|E dξ + |σt v − v|E dξ ≤ ε K

K

K

K

for |t − 1| ≤ δ. This proves that t 7→ σt is strongly continuous on L1,loc . ¡ ¢ Since σt ∈ L D(Rd ) , Remark 1.2.1(a) implies ¡ ¢ (σt )∗ ∈ L D0 (Rd , E) , t>0, where ­

(σt )∗ u, ϕ

® Rd

= hu, σt ϕiRd ,

u ∈ D0 (Rd , E) ,

ϕ ∈ D(Rd ) .

¥

(3.1.14)

(3.1.15)

Clearly, (σs )∗ (σt )∗ = (σst )∗ ,

s, t > 0 .

(3.1.16)

Let u ∈ L1,loc (Rd , E) be a regular distribution. Then σt u ∈ L1,loc (Rd , E) by Lemma 3.1.2. By (3.1.12) and a substitution of variables, Z ­ ® hσt u, ϕiRd = u(t q ξ), ϕ(ξ) E dξ = t−|ω| hu, σ1/t ϕiRd (3.1.17) Rd

for ϕ ∈ D(Rd ).

VI.3 Anisotropy

37

Now we define an action of G on D0 (Rd , E), associated with [`, d, ν], by setting σt u := t−|ω| (σ1/t )∗ u ,

u ∈ D0 (Rd , E) ,

t>0,

that is, hσt u, ϕiRd = t−|ω| hu, σ1/t ϕiRd ,

ϕ ∈ D(Rd ) .

(3.1.18)

It follows from (3.1.14) and (3.1.16) that t 7→ σt is a linear representation of G on D0 (Rd , E). Furthermore, (3.1.17) and (3.1.18) show that it is an extension over D0 (Rd , E) of the point-wise defined representation on L1,loc (Rd , E). 3.1.3 Proposition The map t 7→ σt , associated with [`, d, ν], is a strongly continuous linear representation of G on D0 (Rd , E). If X ∈ { D, S, C, C0 , BC, BUC, L1,loc , Lq with 1 ≤ q ≤ ∞, S 0 } , then X (Rd , E) is an invariant linear subspace of D0 (Rd , E) and t 7→ σt is continuous on X (Rd , E) if X ∈ {D, S} and strongly continuous otherwise, unless X belongs to {BC, BUC, L∞ }. Moreover12 , p (i) ∂ α ◦ σt = tα ω σt ◦ ∂ α on D0 (Rd , E). (ii) kσt ukq = t−|ω|/q kukq , u ∈ Lq (Rd , E), 1 ≤ q ≤ ∞. (iii) F ◦ σt = t−|ω| σ1/t ◦ F and F −1 ◦ σt = t−|ω| σ1/t ◦ F −1 on S 0 (Rd , E). Proof (1) As for the first assertion, it remains to show that t 7→ σt is strongly continuous on D0 (Rd , E). By step (4) of the preceding proof we know that σt ϕ → ϕ in D(Rd ) as t → 1, uniformly for ϕ in bounded sets. This and (3.1.18) show that, given u ∈ D0 (Rd , E), limt→1 hσt u, ϕiE = hu, ϕi uniformly with respect to ϕ in bounded sets of D(Rd ). Hence limt→1 σt u = u in D0 (Rd , E), that is, with respect to the topology of uniform convergence on bounded subsets of D(Rd ). This proves this claim. (2) Suppose u ∈ S(Rd , E). As in the proof of (3.1.9) we find qk,m (σt u) = max sup h·ik |∂ α (σt u)|E |α|≤m Rd

p ω hξik ht q ξik |∂ α u(t q ξ)|E ≤ c(t, k, m)qk,m (u) ht q ξik |α|≤m ξ∈Rd ¡ ¢ for k, m ∈ N. Hence σt ∈ L S(Rd , E) . From (3.1.11) we derive ¡ ¢ p qk,m (σt u − u) ≤ c(t, k, m) max |tνi − 1| + |tα ω − 1| qk+1,m+1 (u) , = max sup tα

1≤i≤`

where t 7→ c(t, k, m) is continuous on G. This shows that σt u¡→ u in S(Rd , E) ¢ as t → 1, uniformly for u in bounded sets, that is, (t 7→ σt ) ∈ C G, L(S(Rd , E)) . This proves the assertion for X = S. 12 F

:= (u 7→ u) is the Fourier transformation on S 0 (Rd , E) (cf. Section III.4.2).

38

VI Auxiliary Material

(3) Let u ∈ S 0 (Rd , E). It follows from S 0 ,→ D0 that (3.1.18) applies for u in S (R , E) and ϕ in S(Rd ). Thus, due to (2), the arguments which we applied to X = D0 yield the claim for X = S 0 also. 0

d

(4) Let X := C. For u ∈ X and K b Rd we set pK (u) := p0,K (u). Then pK (σt u) = pt p K (u) ,

t>0.

Hence σt ∈ L(X ). Since u is uniformly continuous on compact sets, it follows that σt u → u in X as t → 1. (5) Let X ∈ {BC, BUC, L∞ }. Then, clearly, σt u ∈ X for u ∈ X . Moreover, kσt uk∞ = kuk∞ ,

u∈X .

(3.1.19)

This implies σt ∈ L(X ) and also proves (ii) if q = ∞. d

(6) Since D ,→ C0 ,→ BC, we infer from Lemma 3.1.2 and (3.1.19) that σt belongs to L(C0 ). Since t 7→ σt is strongly continuous on D, we get from σt u − u = σt (u − v) + (σt v − v) + (v − u) ,

v∈D ,

and (3.1.19) that t 7→ σt is strongly continuous on C0 (cf. step (6) of the preceding proof). (7) Suppose 1 ≤ q < ∞. Then (ii) follows by a substitution of variables. Hence d

σt ∈ L(Lq ) and kσt kLq is bounded on compact subsets of G. From this and D ,→ Lq we obtain the strong continuity of t 7→ σt on Lq by an approximation argument as in (6). (8) By step (2) of the proof of Lemma 3.1.2 we know that (i) holds on D. Now we get its validity on D0 by the definition of the distributional derivatives. (9) Suppose u ∈ S(Rd ). Then, using once more a substitution of variables, Z F (σt u)(ξ) =

Rd

e−i hξ,xi u(t q x) dx = t−|ω| σ1/t (F u)(ξ)

for ξ ∈ Rd . This proves that the first part of (iii) holds on S(Rd ). Now we get its validity on S 0 (Rd , E) from the definitions of F and σt for tempered distributions. The second part of (iii) is then obtained by applying F −1 from the left and from the right. Since the cases X ∈ {D, L1,loc } are covered by Lemma 3.1.2, the proposition is proved. ¥

VI.3 Anisotropy

39

Throughout the rest of this volume

•

[`, d, ν] is a weight system for Rd and [d, 1, ω] is its non-reduced version.

•

ν = LCM(ν) = ω.

•

3.2

¡ ¢ t 7→ σt is the linear representation of (0, ∞), q on D0 (Rd , E) induced by the anisotropic dilation (3.1.3).

(3.1.20)

Quasinorms

Let X be a nonempty set. For f, g : X → R+ we write f ∼g

⇐⇒

g/c ≤ f ≤ cg

(3.2.1)

for some constant c ≥ 1. Then ∼ is an equivalence relation on (R+ )X . Let z ∈ C. A distribution u ∈ D0 (Rd , E) is positively z-homogeneous (positively homogeneous of degree z) with respect to [`, d, ν] if σt u = t z u ,

t>0.

If u ∈ C(Rd , E) is positively r-homogeneous for some r > 0, then u(0) = 0. A function Q is a ν-quasinorm on Rd if •

¡ q¢ Q ∈ C(Rd , R+ ) ∩ C ∞ (Rd ) with Q(ξ) > 0 for ξ 6= 0.

• •

Q is even and positively 1-homogeneous with respect to [`, d, ν]. Q satisfies the triangle inequality Q(ξ + ξ 0 ) ≤ Q(ξ) + Q(ξ 0 ) ,

ξ, ξ 0 ∈ Rd .

The following examples and observations will be readopted at various later occasions. 3.2.1 Examples (a) Set N(ξ) := Nν (ξ) :=

` ³X

|ξi |2ν/νi

´1/2ν

i=1

Then N is a ν-quasinorm, the natural ν-quasinorm.

,

ξ ∈ Rd .

40

VI Auxiliary Material

Proof

We set d(0) := 0 and d(i) := d1 + · · · + di for 1 ≤ i ≤ `. Then

∂k N(ξ) =

|ξi |2(ν/νi −1) k ξ , νi N2ν−1 (ξ)

d(i − 1) < k ≤ d(i) ,

1≤i≤`.

(3.2.2)

q ¡ q¢ From this and ν/νi ∈ N we obtain N ∈ C ∞ (Rd ) by induction. By means of the elementary inequality, (x + y)1/k ≤ x1/k + y 1/k ,

x, y ≥ 0 ,

k∈N,

we get from Minkowski’s inequality N(ξ + η) =

` ³X

|ξi + ηi |2ν/νi

´1/2ν

i=1

≤

` ³X ¡

|ξi |1/νi + |ηi |1/νi

¢2ν ´1/2ν

≤ N(ξ) + N(η)

i=1

for ξ, η ∈ Rd . The remaining assertions are clear.

¥

(b) We denote by E(ξ), for ξ 6= 0, the unique t > 0 satisfying |t−1 q ξ| = 1 and set E(0) := 0. Then E is a ν-quasinorm on Rd , the Euclidean ν-quasinorm. Proof

(1) It is obvious that E is well-defined and even. From ¯¡ ¯ ¯¡ ¯ ¢−1 ¢ q (t q ξ)¯ = ¯ E(t q ξ) −1 t q ξ ¯ = |E(ξ)−1 q ξ| 1 = ¯ E(t q ξ)

we get E(ξ) = E(t q ξ)/t for t > 0 and ξ 6= 0. Hence E is positively 1-homogeneous q on (Rd ) . (2) Suppose |ξ − η| ≤ 1 and set ν := max{ν1 , . . . , ν` }. Then E(ξ − η) ≤ 1 and, consequently, ` X |ξi − ηi |2 |ξ − η|2 1= ≤ . E(ξ − η)2νi E2ν (ξ − η) i=1 Hence E(ξ − η) ≤ |ξ − η|1/ν for |ξ − η| ≤ 1. Thus the triangle inequality implies E ∈ C(Rd ). q (3) Now we use (3.1.5). Then we get, for k ∈ {1, . . . , d} and h ∈ R, ¯ ¡ ¢ 0 = |E(ξ + hek ) q (ξ + hek )|2 − |E(ξ) q ξ|2 = a(ξ, h) ¯ b(ξ, h) Rd , (3.2.3) where a(ξ, h) := E(ξ + hek ) q (ξ + hek ) + E(ξ) q ξ

(3.2.4)

VI.3 Anisotropy

41

and b(ξ, h) := E(ξ + hek ) q (ξ + hek ) − E(ξ) q ξ = E(ξ + hek ) q ξ − E(ξ) q ξ + E(ξ + hek ) q ek h .

(3.2.5)

Note that E(ξ + hek ) q ξ − E(ξ) q ξ = and

³¡

¢ ´ E(ξ + hek )−ωj − E(ξ)−ωj ξj

1≤j≤d

¡ ¢ E(ξ + hek )−ωj − E(ξ)−ωj = cj (ξ, h) E(ξ + hek ) − E(ξ) ,

where cj (ξ, h) :=

ωj −1 X −1 E(ξ + hek )−i E(ξ)−ωj +1+i . E(ξ + hek )E(ξ) i=0

From this, (3.2.4), (3.2.5), and step (2) we deduce ¯ ¡ ¢ ¡ ¢−1 a(ξ, h) ¯ E(ξ + hek ) q ξ − E(ξ) q ξ Rd E(ξ + hek ) − E(ξ) → −2E(ξ)−1

d X

¡ ¢2 ωj ξ j E(ξ)ωj

j=1

as h → 0. Similarly, ¯ ¡ ¢ a(ξ, h) ¯ E(ξ + hek ) q ek Rd → 2E−2ωk (ξ)ξk . By combining this with (3.2.3) and (3.2.5), we see that ³X ¡ ± ¢2 ´−1 E(ξ + hek ) − E(ξ) → ξ k E(ξ)1−2ωk ωj ξ j E(ξ)ωj h j=1 d

(3.2.6)

as h → 0. This shows that E is differentiable for ξ 6= 0, and that ∂k E(ξ) is given by the right side of (3.2.6). From this result and step (2) we deduce, by induction, ¡ q¢ that E belongs to C ∞ (Rd ) . (4) To prove the triangle inequality, we set a(ξ, η) :=

` X i=1

|ξi + ηi |2 ¡ ¢2νi , E(ξ) + E(η)

q ξ, η ∈ (Rd ) .

Then the claim follows, provided we show a(ξ, η) ≤ 1 =

` X |ξj + ηi |2 . E(ξ + η)2νi i=1

42

VI Auxiliary Material

Since a(t q ξ, t q η) = a(ξ, η), we can assume E(ξ) + E(η) = 1. Thus 0 < E(ξ) ≤ 1 implies ` X |ξi |2 |ξ|2 1= ≥ . E(ξ)2νi E( ξ)2 i=1 Hence |ξ| ≤ E(ξ). Similarly, |η| ≤ E(η). Thus ¡ ¢2 a(ξ, η) = |ξ + η|2 ≤ (|ξ| + |η|)2 ≤ E(ξ) + E(η) = 1 for ξ, η ∈ (Rd ).

¥

3.2.2 Remarks (a) Let Q be a ν-quasinorm on Rd and set dQ (ξ, ξ 0 ) := Q(ξ − ξ 0 ) ,

ξ, ξ 0 ∈ Rd .

Then dQ is a translation-invariant metric on Rd . (b) If [`, d, ν] is the trivial weight system [d, 1, 1], then N is the Euclidean norm on Rd . (c) Suppose [m, ν] is a parabolic weight system on Rm . Write (ξ, τ ) for the general point of Rm × R and set ¡ ¢1/2ν P(ξ, τ ) := |ξ|2ν + |τ |2 , (ξ, τ ) ∈ Rm × R . Then P is the natural (1, ν)-quasinorm on Rm+1 , the ν -parabolic quasinorm. Moreover, ¡ ¢ ¡ ¢1/2 δP (ξ, τ ), (ξ 0 , τ 0 ) := |ξ − ξ 0 |2 + |τ − τ 0 |2/ν defines the ν -parabolic metric on Rm+1 . It is equivalent to dP .

3.3

¥

Parametric Augmentations

For technical reasons it is often convenient to consider parameter-dependent settings. To introduce them, we recall that a cone in a vector space X is a nonempty subset C of X satisfying R+ C ⊂ C. Thus {0} ⊂ C, and C is trivial if it equals {0}. In analogy to (3.1.3), we define on C` an anisotropic dilation by t νq z := (tν1 z 1 , . . . , tν` z ` ) ,

t≥0,

z ∈ C` .

Then we fix for each i = 1, . . . , ` a closed cone Hi in C containing the positive half-line R+ + i0, if it is nontrivial. We set H := H1 × · · · × H` and Z := Rd × H. The general point of Z is denoted by ζ or (ξ, η) with η = (η 1 , . . . , η ` ) ∈ H. Note that Z is a closed cone in Rd × C` which is invariant under the action t νq ζ := (t νq ξ, t νq η) ,

t≥0,

ζ∈Z.

(3.3.1)

We embed R+ naturally in C` by identifying t ∈ R+ with (t + i 0, 0, . . . , 0) ∈ C` . Then R+ is the positive half-line in C` .

VI.3 Anisotropy

43

Throughout the rest of this section we assume H = H1 × · · · × H` is a closed cone in C` containing the positive half-line.

•

(3.3.2)

d

•

Z = R × H.

To simplify the notation, we put t q ζ := t νq ζ ,

σt := σtν ,

t≥0,

ζ∈Z,

so that σt is now defined on functions whose domain is Z. We say that Z is a parametric augmentation of Rd , associated with the weight system [`, d, ν]. It is trivial if H = R+ , that is, H1 = R+ and H2 = · · · = H` = {0}. Augmented Quasinorms For a : Z → E we put aη : Rd → E ,

ξ 7→ a(ξ, η) ,

We also set [η] = [η]ν :=

` ³X

|ηi |2ν/νi

η∈H. ´1/2ν

.

i=1

Identifying C` with R2` by identifying z = x + i y with (x, y) ∈ R2 , we see that [·]ν is the natural ν-quasinorm on R2` with respect to the reduced weight system [`, 2 · 1` , ν]. By a ν -quasinorm on Z we mean the restriction to Z of a ν-quasinorm on13 d R × C` = b Rd × R2` with respect to the reduced weight system £ ¤ d + `, (d, 2 · 1` ), (ν, ν) for Rd × R2d . Thus, if Q is a ν-quasinorm on Z, then Q ∈ C(Z) ,

Qη ∈ C ∞ (Rd ) ,

The natural ν -quasinorm on Z is defined by ¡ ¢1/2ν Λ(ζ) := N2ν (ξ) + [η]2ν ,

q η∈H.

ζ = (ξ, η) ∈ Z .

(3.3.3)

3.3.1 Remarks (a) Observe that the conventions for H mean that we can consider as many parameters as there are factors in the d-clustering (3.1.1) of Rd . This 13 a

= b means: a is identified with b.

44

VI Auxiliary Material

choice of H and the definition of [η]ν are made for notational simplicity, since they ` allow to stick throughout to q the weight vector ν. It is obvious that C could be replaced by Cn for any n ∈ N, and [η]ν by [η]µ for any weight vector µ = (µ1 , . . . , µn ) satisfying LCM(µ) = ν. This latter restriction can also be dropped if we work with the natural (ν, µ)-quasinorm, defined by ¡ ¢ 2ρ 1/2ρ Λ(ν,µ) (ζ) := N2ρ , ν (ξ) + [η]µ where µ := LCM(µ) and ρ := LCM(ν, µ). Such generalizations are left to the reader. (b) For most applications either the trivial parametric augmentation Z = Rd × R+ or the complex parametric augmentation Z = Rd × C suffices, where C is naturally identified with C × {0} × · · · × {0} ⊂ C` . Thus, in these cases, ¡ ¢1/2ν Λ(ζ) = N2ν (ξ) + |η|2ν/ν1 .

(3.3.4)

Clearly, instead of using the natural identifications we could use, for any i belonging to {1, . . . , `}, the embedding η 7→ (0, . . . , 0, η, 0, . . . , 0), where η is at position i. In other words, everything remains valid if ν1 in (3.3.4) is replaced by νi for any i ∈ {1, . . . , `}. This means that the parameter η ∈ R+ , resp. η ∈ C, can be given any of the weights ν1 , . . . , ν` . This fact will sometimes be used to simplify the presentation. ¥ Let Q be a ν-quasinorm on Z. Then © ª SQ := [Q = 1] = ζ ∈ Z ; Q(ζ) = 1 is the Q-quasisphere in Z, and q r Q : Z → SQ ,

ζ 7→ Q(ζ)−1 q ζ

(3.3.5)

is the Q-retraction. Since, by the 1-homogeneity of Q, ¡ ¢ Q rQ (ζ) = Q(ζ)−1 Q(ζ) = 1 ,

q ζ∈Z,

q it follows that rQ is indeed a continuous (that is, topological) retraction from Z onto SQ . 3.3.2 Lemma

¡q ¢ (i) Suppose M ∈ C Z, (0, ∞) is positively 1-homogeneous. Then [M = 1] is compact and M ∼ Λ.

VI.3 Anisotropy

45

q (ii) Assume z q∈ C and a ∈ C(Z, E) is positively z-homogeneous. If α ∈ Nd and ∂ξα a ∈ C(Z, E), then ∂ξα a is positively (z − α q ω)-homogeneous and, given any ν-quasinorm Q, ∂ξα a(ζ) = Qz−α

pω

(ζ)(∂ξα a) ◦ rQ (ζ)

(3.3.6)

and |∂ξα a(ζ)|E ≤ QRez−α

pω

(ζ) k(∂ξα a) ◦ rQ k∞ < ∞ ,

q ζ∈Z.

(3.3.7)

Proof (1) It is obvious that SΛ is bounded and closed in Z, hence in Rd × C` , since H is closed in C` . Thus SΛ is compact in Rd × C` , therefore in Z. Now it follows from the continuity of M that 1/c ≤ M(ζ) ≤ c for ζ ∈ SΛ . Consequently, q ¡ ¢ ¡ ¢ M(ζ) = M Λ(ζ) q rΛ (ζ) = Λ(ζ)M rΛ (ζ) , ζ∈Z, implies M ∼ Λ. Thus Λ ≤ cM which gives [M = 1] ⊂ [Λ ≤ c]. Hence [M = 1] is bounded. Now, similarly as above, we see that it is compact. This proves (i). (2) We differentiate σt a = tz a with respect to ξ and use Proposition 3.1.3(i) to obtain p ∂ξα (σt a) = tα ω σt (∂ξα a) = tz ∂ξα a . This shows that ∂ξα a is positively (z − α q ω)-homogeneous. By replacing in the above relation t by 1/t we find q p ∂ξα a(ζ) = tz−α ω σ1/t (∂ξα a)(ζ) , ζ∈Z. We substitute Q(ζ) for t and note that ¡ ¢ σ1/Q(ζ) (∂ξα a)(ζ) = ∂ξα a rQ (ζ) ,

q ζ∈Z.

Then we arrive at (3.3.6). The last estimate is obvious by the compactness of SQ and the continuity of ∂ξα a. ¥ 3.3.3 Remark Let Q be a ν-quasinorm on Rd . We define the Q-quasisphere in Rd and the Q-retraction by replacing Z by Rd in the above definitions. Then the assertions of the lemma apply in this situation as well, where now Λ is to be replaced by N. In particular, it follows that all ν-quasinorms on Rd are equivalent. ¥ Positive Homogeneity k For z ∈ C and k ∈ N we q denote by Hz (Z, E) the q vector space of all positively α z-homogeneous a ∈ C(Z, E) satisfying ∂ξ a ∈ C(Z, E) for α ∈ Nd with α q ω ≤ k.

46

VI Auxiliary Material

We endow it with the norm k·kHkz , defined by kakHkz := max k(∂ξα a) ◦ rΛ k∞ . α p ω≤k

(3.3.8)

Q

3.3.4 Lemma Hzk (Z, E) is a Banach space. If Q is a ν-quasinorm and k·kHkz is Q

defined by replacing Λ in (3.3.8) by Q, then k·kHkz ∼ k·kHkz . Proof

(1) It is obvious that k·kHkz is a norm. Let (aj ) be a Cauchy sequence. q It follows from (3.3.6) that (∂ξα aj ) is a Cauchy sequence in C(Z, E) with respect q to the topologyq of uniform convergence qon compact subsets of Z. By q the completeness of C(Z, E) there exist aα ∈ C(Z, E) with ∂ξα aj → aα in C(Z, E). Since ¡ ¢ C(Rd , E) ,→ D0 (Rd , E) and ∂ α belongs to L D0 (Rd , E) , we obtain ∂ α aj,η → ∂ α a0η q q in D0 (Rd , E) for η ∈ H. Consequently, ∂ξα a0 = aα ∈ C(Z, E) and ∂ξα aj → ∂ξα a0 in q C(Z, E) for α q ω ≤ k. In particular, (∂ξα aj ) converges, uniformly on the compact q set SΛ , towards ∂ξα a0 , and a0 is z-homogeneous. Since rΛ maps Z onto SΛ , it follows from (3.3.8) that Hzk (Z, E) is complete. q (2) Note rQ (ζ) = Q(ζ)−1 q ζ = (Λ/Q)(ζ) q rΛ (ζ) for ζ ∈ Z. Thus we deduce from the (z − α q ω)-homogeneity of ∂ξα a that (∂ξα a) ◦ rQ = (Λ/Q)z−α

pω

(∂ξα a) ◦ rΛ ,

a ∈ Hzk (Z, E) ,

α qω ≤ k .

q Observe that Λ/Q is positively 0-homogeneous on Z and continuous on the compact set SΛ . Hence there exists c = c(Rez) ≥ 1 such that 1/c ≤ |Λ/Q|Rez−α Q

This implies k·kHk ∼ k·kHk . z

z

pω

≤c,

α qω ≤ k .

¥

3.3.5 Remark Due to Remark 3.3.3, this lemma and its proof remain true if H = {0}, that is, Z = Rd . ¥ Clearly, Hzm (Z, E) ,→ Hzk (Z, E) for m > k. Hence we can set \ Hz∞ (Z, E) := Hzk (Z, E) , k

endowed with the obvious projective limit topology which makes it a Fr´echet space. 3.3.6 Lemma Let k ∈ N ∪ {∞} and z1 , z2 ∈ C. If E1 × E2 → E3 ,

(e1 , e2 ) 7→ e1 e2

VI.3 Anisotropy

47

is a multiplication14 , then its point-wise extension is a multiplication Hzk1 (Z, E1 ) × Hzk2 (Z, E2 ) → Hzk1 +z2 (Z, E3 ) . Proof

Note σt (ab) = σt (a)σt (b) ,

q q (a, b) ∈ C(Z, E1 ) × C(Z, E2 ) .

Using this, we deduce the assertion from Leibniz’ rule.

¥

Differentiating Inverses

¡ ¢ Next we consider elements of Hzk Z, Lis(E1 , E0 ) . For this we first establish a semi-explicit formula for derivatives of inverses which will be useful repeatedly. q ¡ ¢ 3.3.7 Lemma Suppose X is open in Rd , k ∈ N, and a ∈ C k X, Lis(E1 , E0 ) . Set ¡ ¢−1 a−1 (x) := a(x) for x ∈ X. Then ∂ α a−1 =

|α| X X

εβ a−1 (∂ β1 a)a−1 · · · · · (∂ βj a)a−1

(3.3.9)

j=1 β∈Bj,α

for 0 < α q ω ≤ k, where © ª Pj Bj,α := β := {β1 , . . . , βj } ; βi ∈ Nd , |βi | > 0, i=1 βi = α ¡ ¢ and εβ ∈ Z. In particular, ∂ α a−1 ∈ C X, L(E0 , E1 ) for α q ω ≤ k. Proof

Recall that Lis(E1 , E0 ) is open in L(E1 , E0 ) and inv : Lis(E1 , E0 ) → Lis(E0 , E1 ) ,

is smooth with ¡ ¢ ∂ inv(b) c = −b−1 c b−1 ,

b 7→ b−1

(3.3.10)

b ∈ Lis(E1 , E0 ) ,

c ∈ L(E1 , E0 ) , (3.3.11) ¡ ¢ (e.g., [AmE06, Proposition VII.7.2]). From this we infer a−1 ∈ C X, Lis(E1 , E0 ) and, by the chain rule, ∂j a−1 = −a−1 (∂j a)a−1 ,

1≤j≤d.

Now (3.3.9) follows by induction. The last assertion is then read off (3.3.9).

¥

14 A multiplication X × X → X of LCSs is a continuous bilinear map. Deviating from the 1 2 3 usage in Volume I, we do not require that it has norm at most one if the Xj are normed spaces. However, the present definition is not more general –– but more practical –– than the one of Volume I (cf. Example 1.3.1(g) in the Appendix). Furthermore, it is now convenient to write e1 e2 for e1 q e2 .

48

VI Auxiliary Material

As a first application of this formula we derive an estimate for the inverse of a positively homogeneous operator-valued map. ¡ ¢ k 3.3.8 Lemma ¡ Suppose z ∈ C and k ∈ N. If a ∈ H Z, Lis(E , E ) , then a−1 be1 0 z ¢ k longs to H−z Z, Lis(E0 , E1 ) and ¡ ¢ ka−1 kHk−z ≤ c kakHkz , ka−1 ◦ rΛ k∞ ka−1 ◦ rΛ k∞ . ¡q ¢ ¡q ¢ Proof (1) Since a ∈ C Z, Lis(E1 , E0 ) , (3.3.10) implies a−1 ∈ C Z, Lis(E0 , E1 ) . It is a consequence of a(ζ)a−1 (ζ) = 1E0 that 1E0 = σt (aa−1 ) = σt (a)σt (a−1 ) = tz aσt (a−1 ) and, similarly, 1E1 = σt (a−1 a) = tz σt (a−1 )a. Hence σt (a−1 ) = t−z a−1 ,

t>0.

(3.3.12)

Thus a−1 is positively (−z)-homogeneous. Moreover, it follows from Lemma 3.3.7 ¡q ¢ α −1 that ∂ξ a ∈ C Z, L(E0 , E1 ) for α q ω ≤ k. ¡q ¢ ¡q ¢ (2) If b ∈ C Z, L(E1 , E2 ) and c ∈ C Z, L(E0 , E1 ) , then (bc) ◦ rΛ = (b ◦ rΛ )(c ◦ rΛ ) .

(3.3.13)

Suppose 0 < α q ω ≤ k, 1 ≤ j ≤ |α|, and {β1 , . . . , βj } ∈ Bj,α . Then we get from (3.3.13) °¡ −1 β1 ° ¢ ° a (∂ a)a−1 · · · · · (∂ βj a)a−1 ◦ rΛ ° ξ ξ ∞ ≤ ka−1 ◦ rΛ kj+1 ∞

j Y

k(∂ξβi a) ◦ rΛ k∞ .

i=1

Now the assertion follows from (3.3.9).

¥

3.3.9 Example Suppose z ∈ C. Then Λz ∈ Hz∞ (Z) and kΛz kHkz ≤ c (1 + |z|)k kΛkHk1 ,

k∈N.

(3.3.14)

q Proof From Example 3.2.1(a) we infer that ∂ξα Λ ∈ C(Z) for α ∈ N. Hence Λ belongs to H1∞ (Z). We get by induction from ∂j Λz = zΛz−1 ∂j Λ ,

1≤j≤d,

(3.3.15)

q that ∂ξα Λz ∈ C(Z) for α ∈ Nd . Thus Λz ∈ Hz∞ (Z) since Λz is z-homogeneous. From (3.3.15) we also derive the second part of the claim. ¥

VI.3 Anisotropy

49

Slowly Increasing Functions Recall that OM (Rd , E) is the LCS of smooth slowly increasing E-valued functions on Rd (see Subsection III.4.1 or Subsection 1.1 of the Appendix) and that S(Rd , E) ,→ OM (Rd , E) ,→ S 0 (Rd , E) . 3.3.10 Lemma Let z ∈ C. Then ¡ ¢ (a 7→ aη ) ∈ L Hz∞ (Z, E), OM (Rd , E) ,

(3.3.16)

η 6= 0 ,

uniformly with respect to z in bounded sets. Proof

Using |ξi | ≤ |ξ| ≤ max{1, |ξ|}, we find ¡ ¢ [η] ≤ Λ(ζ) ≤ c [η] (1 + |ξ|2ν )1/2ν ,

ζ = (ξ, η) ∈ Z .

(3.3.17)

Define Qi : Z → R by ¡ ¢1/2ν Q1 (ζ) := [η]2ν + |ξ|2ν ,

Q2 (ξ, η) := (|η|2 + |ξ|2 )1/2 .

Then Q1 and Q2 are quasinorms with respect to the trivial parametric augmentation of the reduced trivial weight system [1, 1, 1]. Thus Q1 ∼ Q2 by Lemma 3.3.2(i). In particular, 2ν (1 + |·| )1/2ν = Q1 (·, 1) ∼ Q2 (·, 1) = h·i . From this and (3.3.17) we deduce [η] ≤ Λη ≤ c(|η|)h·i .

(3.3.18)

Let B be a bounded subset of C, α ∈ Nd , and ϕ ∈ S(Rd ). We fix k ∈ N with Rez − α q ω ≤ k for z ∈ B. Then we infer from (3.3.7) and (3.3.18) that aη is slowly increasing and, recalling (1.1.1), ¯ ¢ ¯ pω ¡ α |ϕ∂ξα aη |E = ¯Λz−α ϕ (∂ξ a) ◦ rΛ η ¯E η p ω −k ≤ ΛRez−α h·i qk,0 (ϕ) kakHαz p ω ≤ c(|η|) qk,0 (ϕ) kakHαz p ω η for η 6= 0 and z ∈ B. This proves the assertion.

¥

Now we can improve Example 3.3.9. 3.3.11 Lemma Suppose η 6= 0. Then 15 z 7→ Λzη 1OM is a continuous representation ¡ ¢ of (C, +) in Laut OM (Rd ) . 15 Recall

that, given an LCS X , we write 1X for idX ∈ L(X ).

50

VI Auxiliary Material

Proof (1) Example 3.3.9 and Lemma 3.3.10 imply Λzη 1OM ∈ OM (Rd ). Conse¡ ¢ quently, see Theorem 1.6.4 of the Appendix, Λzη 1OM = (u 7→ Λzη u) ∈ L OM (Rd ) . Thus, since Λz1 +z2 = Λz1 Λz2 and Λ0 = 1Rd , it suffices to prove the asserted continuity. It follows from Z 1 Λh − 1 = h Λth log Λ dt , h∈C, 0

that Z |∂ξα (Λh − 1)| ≤ |h|

1 0

|∂ξα (Λth log Λ)| dt ,

h∈C.

(3.3.19)

(2) We evaluate ∂ξα (Λz log Λ) for α ∈ Nd and z ∈ C. Note that ∂jα (Λz log Λ) = (Λz−1 ∂j Λ)(1 + z log Λ) ,

1≤j≤d.

(3.3.20)

Suppose |α| ≥ 2. Fix j ∈ {1, . . . , d} with α − ej ≥ 1. Then, by (3.3.20) and Leibniz’ rule, α−e

∂ξα (Λz log Λ) = ∂ξ j ∂j (Λz log Λ) X ³ α − ej ´ β α−e −β = ∂ξ (Λz−1 ∂j Λ)∂ξ j (1 + z log Λ) β β≤α−ej α−ej

(Λz−1 ∂j Λ)(1 + z log Λ) X ³ α − ej ´ β α−e −β +z ∂ξ (Λz−1 ∂j Λ)∂ξ j log Λ . β

= ∂ξ

(3.3.21)

β 0. Then we can fix ` ∈ {1, . . . , d} with α − ej − β − e` ≥ 0. Thus α−ej −β

∂ξ

α−ej −β−e`

log Λ = ∂ξ

(Λ−1 ∂` Λ) .

(3.3.22)

(3) From Example 3.3.9 we know that Λz ∈ Hz∞ (Z). Hence it follows from Lemmas 3.3.2(ii) and 3.3.6 that ∞ Λz−1 ∂j Λ ∈ Hz−ω (Z) , j

1≤j≤d.

Using Lemma 3.3.2(ii) once more, we thus obtain ¢ p ¡ ∂ξγ (Λz−1 ∂j Λ) = Λz−ωj −γ ω ∂ξγ (Λz−1 ∂j Λ) ◦ rΛ

(3.3.23)

for γ ∈ Nd and 1 ≤ j ≤ d. By also employing Lemma 3.3.6, j −γ p ω |∂ξγ (Λz−1 ∂j Λη )| ≤ ΛRez−ω kΛz−1 ∂j ΛkHkz−ω η η j Rez−ωj −γ p ω z−1 ≤ c(k)Λη kΛ kHkz−1 kΛkHk+1

1−ωj

for γ ∈ Nd with γ q ω ≤ k.

VI.3 Anisotropy

51

(4) It follows from Example 3.3.9 that j −γ p ω |∂ξγ (Λz−1 ∂j Λη )| ≤ c(k) |z|k ΛRez−ω , η η

1≤j≤d,

γ q ω ≤ k . (3.3.24)

From this we get α−ej

|∂ξ

(Λz−1 ∂j Λη )| ≤ c(k) |z|k ΛRez−α η η

pω

and, due to (3.3.22), α−ej −β

|∂ξβ (Λz−1 ∂j Λη )∂ξ η

log Λ| ≤ c(k) |z|k ΛRez−α η

pω

for α q ω ≤ k and β < α − ej . Using these estimates, we obtain from (3.3.20) and (3.3.21) that pω |∂ξα (Λz−1 log Λη )| ≤ c(k) |z|k ΛRez−α (1 + |z| | log Λη |) (3.3.25) η η for 0 < α q ω ≤ k and z ∈ C. (5) Suppose h ∈ C with |h| ≤ 1, and 0 ≤ t ≤ 1. Then −1 − k ≤ −1 − α q ω ≤ Re(th − α q ω) ≤ 1 − α q ω ≤ 1 ,

α qω ≤ k ,

and (3.3.25) imply, due to Λη ≥ [η] and log Λη ≤ c(η)Λη , 2 |∂ξα (Λth η log Λη )| ≤ c(k, η)Λη ,

α qω ≤ k .

Thus it follows from (3.3.19) that |∂ α (Λhη − 1)| ≤ c(k, η) |h| Λ2η ,

α qω ≤ k ,

|h| ≤ 1 .

(3.3.26)

We fix z ∈ C. By Leibniz’ rule, ¡ ¢ X ³ α ´ α−β z β h ∂ξα (Λz+h − Λz ) = ∂ξα Λz (Λh − 1) = ∂ Λ ∂ξ (Λ − 1) β ξ β≤α

for h ∈ C. From this, (3.3.26), Lemma 3.3.2(ii), and Example 3.3.9 we get |∂ α (Λz+h − Λzη )| ≤ c(k, η) |h| Λ|z|+2 , η η

α qω ≤ k ,

|h| ≤ 1 .

We choose k ∈ N with k ≥ |z| + 2. Then this estimate and (3.3.18) imply |∂ξα (Λz+h − Λzη )| ≤ c(k, η) |h| h·ik , η

α qω ≤ k ,

|h| ≤ 1 .

Consequently, given z ∈ C, α ∈ Nd , and η 6= 0, there exist c ≥ 1 and k ∈ N such that kϕ∂ α (Λz+h − Λzη )k∞ ≤ c |h| qk,0 (ϕ) , η

α ∈ Nd ,

|h| ≤ 1 ,

ϕ ∈ S(Rd ) .

This proves Λz+h → Λzη in OM (Rd ) as h → 0. Now the assertion follows by apη pealing once more to Theorem 1.6.4 of the Appendix. ¥

52

VI Auxiliary Material

3.4

Fourier Multipliers and Multiplier Spaces

Large parts of what is discussed in this volume depend on Fourier analytic techniques. For this reason we now embark on investigations of classes of functions whose members can serve as symbols of Fourier multiplier operators. Elementary Fourier Multiplier Theorems Let E1 × E2 → E3 ,

(e1 , e2 ) 7→ e1 e2

(3.4.1)

be a multiplication. For m ∈ S 0 (Rd , E1 ) we define a linear map m(D) with domain ¡ ¢ © ª dom m(D) := u ∈ S 0 (Rd , E2 ) ; mb u ∈ S 0 (Rd , E3 ) by16

¡ ¢ u ∈ dom m(D) ,

m(D)u := F −1 mF = F −1 (mb u) ,

provided the point-wise b) → mb u, is well-defined. Then ¡ ¢extension of (3.4.1), (m, u m(D) maps dom m(D) ⊂ S 0 (Rd , E2 ) into S 0 (Rd , E3 ). It is the Fourier multiplier with symbol m. Our first ‘Fourier multiplier theorem’ concerns smooth symbols. 3.4.1 Theorem Let X ∈ {S, S 0 }. Then ¡ ¢ ¡ ¡ ¢¢ m 7→ m(D) ∈ L OM (Rd , E1 ), L X (Rd , E2 ), X (Rd , E3 ) . Proof Theorem 1.6.4 of the Appendix guarantees that there exists a unique hypocontinuous bilinear map OM (Rd , E1 ) × X (Rd , E2 ) → X (Rd , E3 ) ,

(3.4.2)

the point-wise multiplication induced by (3.4.1), ¡which coincides on S with the ¢ natural point-wise multiplication. Since F ∈ Laut X (Rd , E) , we see that OM (Rd , E1 ) × X (Rd , E2 ) → X (Rd , E3 ) ,

(m, u) 7→ mb u

is a hypocontinuous bilinear map as well. Consequently, the bilinear map OM (Rd , E1 ) × X (Rd , E2 ) → X (Rd , E3 ) , is also hypocontinuous. This implies the claim. 16 We

(m, u) 7→ F −1 (mb u)

¥

do not indicate its dependence on (3.4.1) which will always be clear from the context.

VI.3 Anisotropy

53

Recalling (3.3.3), we set Jηz := Λzη (D) , and

z∈C,

J z := J1z ,

η∈H,

(3.4.3)

J := J 1 .

Since, in the classical isotropic case (where the reduced weight system is the trivial one), J −s is a Bessel kernel for s > 0, in the present general setting J −s might be called ν -anisotropic Bessel kernel. 3.4.2 Proposition Let X ∈ {S, S¡0 }. Then z¢ 7→ J z is a strongly continuous linear representation of (C, +) in Laut X (Rd , E) . Proof

This is an easy consequence of Lemma 3.3.11 and Theorem 3.4.1.

¥

3.4.3 Example As usual, ∆ = ∆d = −|D|2 = ∂12 + · · · + ∂d2 denotes the Laplace operator on Rd . We write ∆xi if only the variables xi := (x1i , . . . , xdi i ) are considered. The following differential, respectively pseudodifferential, operators A1 := 1 − ∆ , A3 := 1 − ∂t2 + ∆2m

A2 := 1 − ∆x1 + ∆2x2 , q , A4 := 1 − ∂t2 + ∆3m

are toplinear automorphisms of S(Rd , E) and S 0 (Rd , E). Proof Observe that Ak = J 2ν for 1 ≤ k ≤ 3, where the weight system [`, d,¤ ν] £ equals the trivial one [1, d, 1] if k = 1 (the isotropic case), 2, (d1 , d2 ), (2, 1) if k = 2, and the parabolic weight system [m, 2] if k = 3. Furthermore, A4 = J ν with respect to the parabolic weight system [m, 3]. Thus the assertions follow from the preceding proposition. ¥ Our principal interest, in what follows, are Fourier multiplier theorems for operator-valued symbols. Here we content ourselves with a very simple though useful result. ¡ ¢ By17 L1 (Rd , E) ,→ S 0 (Rd , E) and F ∈ L S 0 (Rd , E) , the image F L1 (Rd , E) of L1 (Rd , E) under F is a well-defined Banach space. The Riemann-Lebesgue lemma, the density of S(Rd , E) in L1 (Rd , E) and in C0 (Rd , E), and the invariance of S(Rd , E) under F imply d

d

S(Rd , E) ,→ F L1 (Rd , E) ,→ C0 (Rd , E) .

(3.4.4)

3.4.4 Theorem Suppose E1 × E2 → E3 , (e1 , e2 ) 7→ e1 e2 is a multiplication. If X ∈ { BUC, C0 , Lq , 1 ≤ q < ∞ } , 17 See

(VII.1.2.1) below.

54

then

VI Auxiliary Material

¡

¢ ¡ ¡ ¢¢ a 7→ a(D) ∈ L F L1 (Rd , E1 ), L X (Rd , E2 ), X (Rd , E3 ) .

Moreover, ¡ ¢ ¡ ¡ ¢¢ a 7→ a(D) ∈ L F L1 (Rd , E1 ), L L∞ (Rd , E2 ), BUC(Rd , E3 ) .

(3.4.5)

(3.4.6)

The norms of these linear maps are bounded by 1. Proof (1) It follows from Theorem 1.9.9 of the Appendix that convolution is a multiplication of Banach spaces L1 (Rd , E1 ) × X2 (Rd , E2 ) → X3 (Rd , E3 ) , of norm at most 1 if

(a, u) 7→ a ∗ u

ª (BUC, BUC), (C0 , C0 ), (Lq , Lq ), (L∞ , BUC) , ¡ ¢ where 1 ≤ q < ∞. From this we infer that the map a 7→ (u 7→ F −1 a ∗ u) possesses properties (3.4.5) and (3.4.6). (2) Suppose a ∈ S(Rd , E1 ) and u ∈ S 0 (Rd , E2 ). Then we deduce from (3.3.16) and (3.4.2) that ab u ∈ S 0 (Rd , E3 ) and, by the convolution theorem (Theorem 1.9.10 of the Appendix), a(D)u = F −1 (ab u) = F −1 a ∗ u. Thus, by step (1) and since17 d 0 d X2 (R , E2 ) ,→ S (R , E2 ), (X2 , X3 ) ∈

©

ka(D)ukX3 (Rd ,E3 ) ≤ kakF L1 (Rd ,E1 ) kukX2 (Rd ,E2 ) ,

u ∈ X2 (Rd , E2 ) ,

provided a ∈ S(Rd , E1 ). Now the assertion follows from the density of the latter space in FL1 (Rd , E1 ). ¥ Fourier Multiplier Spaces Now we introduce a space of E-valued functions on Rd , a Fourier multiplier space, as follows: We set k(ν) := |ω| (2 + max ν). Then M(Rd , E) = Mν (Rd , E) is the linear subspace of C(Rd , E) consisting of all u satisfying ∂ α u ∈ C(Rd , E) for α q ω ≤ k(ν) and pω α kukM := max kΛα ∂ uk∞ < ∞ , 1 α p ω≤k(ν)

(3.4.7)

endowed with the norm k·kM . In Subsections 3.6, VII.2.4, and VII.5.3 it will be shown that the elements of M(Rd , E) are Fourier multipliers for large classes of function spaces. We take k(ν) as regularity index to allow for easy proofs. Lesser regularity requirements would suffice, but this is not important for what follows and we do not give details (see, however, the notes to Section 2).

VI.3 Anisotropy

55

3.4.5 Lemma (i) M(Rd , E) is a Banach space. (ii) If E1 × E2 → E3 is a multiplication, then its point-wise extension is a multiplication M(Rd , E1 ) × M(Rd , E2 ) → M(Rd , E3 ) . Proof (1) We write M for M(Rd , E) etc. Let (aj ) be a Cauchy sequence in M. pω α Then (Λα ∂ aj ) is a Cauchy sequence in BC, for α q ω ≤ k(ν). Hence there exist 1 pω α pω α α b ∈ BC with Λα ∂ aj → bα in BC for α q ω ≤ k(ν). Set aα := Λ−α b ∈ C. 1 1 α α 0 Then ∂ aj → a in C, hence in D . By the continuity of the distributional derivative on D0 , it follows ∂ α aj → ∂ α a0 in D0 . Hence ∂ α a0 = aα ∈ C for α q ω ≤ k(ν). pω α pω α 0 pω α Consequently, Λα ∂ aj → Λα ∂ a in C. Since kΛα ∂ aj k∞ ≤ c for j ∈ N, 1 1 1 α pω α 0 0 it follows kΛ1 ∂ a k∞ < ∞ for α q ω ≤ k(ν). Thus a ∈ M and, letting k → ∞ in kaj − ak kM , we deduce from the definition of Cauchy sequences that aj → a0 in M. Thus M is complete. (2) Claim (ii) follows from Leibniz’ rule. ¥ Resolvent Estimates For later use, we present here and in the next subsection a series of results which will help to determine Fourier multipliers in concrete situations. The reader may want to skip the rest of this section at the first reading and return when these results are actually employed. We recall that Sϑ := [ | arg z| ≤ ϑ] ∪ {0} ⊂ C ,

0≤ϑ≤π ,

where arg z ∈ (−π, π] is the principal value of the argument of z. In the following, we assume • E1 ,→ E0 ,

κ≥1,

0≤ϑ 0 we denote by τ Σ the contour obtained from Σ by applying the dilation z 7→ τ z. Then we infer from a = Λs a ◦ rΛ that Λs (ζ)Σ is a positively oriented contour containing ¡ ¢ ¡ ¢ σ a(ζ) = Λs (ζ)σ a ◦ rΛ (ζ) in its interior. ˚π−ϑ → C be holomorphic. Since Λs (ζ)Σ ⊂ S ˚π−ϑ , the Cauchy inteLet h : S gral Z ¡ ¢−1 1 h(a)(ζ) := h(λ) λ − a(ζ) dλ (3.5.10) 2πi Λs (ζ)Σ q is well-defined in L(E) for ζ ∈ Z. It follows from Cauchy’s theorem that h(a)(ζ) is independent of the particular contour Σ. In fact, the well-known Dunford calculus¡ (cf. ¢[DuS57, Section VII.1]) shows that h(a)(ζ) depends on the values of h on σ a(ζ) only. It is the purpose of the following considerations to show that Σ in (3.5.10) can be replaced by Γ := ∂W , the positively oriented boundary of W , and that q ¡ ¢ then the integral converges in Mη Rd , L(E) for each η ∈ H. For this we prepare some technical results. 3.5.3 Lemma Suppose U is open in C and q h : U → C is holomorphic, V is open in Rd and g ∈ C m (V, U ) for some m ∈ N, and α ∈ Nd satisfies |α| = m. Then ∂ α (h ◦ g) =

m X j=1

where

n

B(j, α) :=

h(j) ◦ g

X

ω(k, β)(∂ β1 g)k1 · · · (∂ βm g)km ,

(k,β)∈B(j,α)

¡ ¢m (k, β) ; k = (k1 , . . . , km ) ∈ Nm , β = (β1 , . . . , βm ) ∈ Nd \{0} , o Pm k1 + · · · + km = j, k β = α i i i=1

and ω(k, β) ∈ N. Proof

Let m = 1 and α = ei . Then, by the chain rule, ∂ α (h ◦ g) = ∂i (h ◦ g) = (h0 ◦ g)∂i g = (h0 ◦ g)∂ α g .

Hence the assertion holds in the present case. From this, ∂ α+ei = ∂i ∂ α , and the product rule we obtain the claim by induction. ¥ ˚ψ → C is holomorphic for some ψ ∈ (ϕ, π − ϑ). Let 3.5.4 Lemma Suppose h : S ˚ψ satisfying |z| ≥ R0 . there exist δ > 0 and R0 ≥ 1 such that |z|δ |h(z)| ≤ 1 for z ∈ S

VI.3 Anisotropy

61

q Then, given η ∈ H, there is an Rη ≥ R0 such that ¢¯ p ω ¯¯ α ¡ Λα ∂ξ h(Λsη z) ¯ ≤ c(|α|)[η]−δs |z|−δ η q for α ∈ (Nd ) and z ∈ W with |z| ≥ Rη . Proof

We infer from Λs ∈ Hs (Z) and Lemma 3.3.2(ii) that ¯¡ β s ¢k ¯ p ω)k k ¯ ∂ (Λη z) ¯ ≤ c(k, β)Λ(s−β |z| η ξ

q for k ∈ N and β ∈ Nd \{0}. Thus Lemma 3.5.3 implies |∂ξα h(Λsη z)| ≤ c(α)

|α| X

|h(j) (Λsη z)| Λjs−α η

pω

|z|j

(3.5.11)

j=1

˚ψ . for z ∈ S We set r := sin(ψ − ϕ)/2. Then the disc [ |λ £− z| ≤ r |z| ] is for z ∈ W ¤con˚ψ . Hence this is also true for the disc |λ − Λs (ζ)z| ≤ rΛs (ζ) |z| for tained in S η η q ζ ∈ Z and z ∈ W . Consequently, by Cauchy’s formula, Z ¡ ¢ j! h(λ) dλ h(j) Λs (ζ)z = , j∈N. 2πi |λ−Λs (ζ)z|=rΛs (ζ) |z| (λ − Λs (ζ)z)j+1 © ª We put Rη := max 1, 1/(1 − r)[η]s R0 . If z ∈ W satisfies |z| ≥ Rη

and

|λ − Λsη (ζ)z| = rΛsη (ζ) |z| ,

then |λ| ≥ (1 − r)Λsη (ζ) |z| ≥ (1 − r)[η]s |z| ≥ R0 . Thus, invoking the assumption, ¯ (j) ¡ s ¢¯ ¡ ¢−j −δ ¯h Λη (ξ)z ¯ ≤ j! r−j (1 − r)−δ [η]−δs Λsη (ξ) |z| |z| for ξ ∈ Rd , z ∈ W , and j ∈ N. By inserting this estimate into the right side of (3.5.11), the assertion follows. ¥ Dunford Integral Representations Now we are ready for the proof of the representation theorem for holomorphic functions of multipliers. ˚ψ → C is holomorphic for some ψ ∈ (ϕ, π − ϑ). 3.5.5 Proposition Suppose h : S Let there exist δ > 0 such that |z|δ h(z) → 0 as |z| → ∞. Denote by Γ the positively

62

VI Auxiliary Material

¡ ¢ oriented boundary of W . Then h(aη ) ∈ Mη Rd , L(E) and ¯ ¡ ¢¯ kh(aη )kMη ≤ c(κ, ϑ) sup max ¯h Λsη (ξ)z ¯

(3.5.12)

ξ∈Rd z∈Γ

q ¡ ¢ for a ∈ Ps Z, L(E); κ, ϑ and η ∈ H. Furthermore, Z 1 h(aη ) = h(λ)(λ − aη )−1 dλ (3.5.13) 2πi Λsη Γ ¡ ¢ in Mη Rd , L(E) . q d q q Proof ¡ (1) Throughout ¢ this proof ζ ∈ Z, α, β ∈ N with α ω, β ω ≤ k(ν), and a ∈ Ps Z, L(E); κ, ϑ . Since |a ◦ rΛ (ζ)| ≤ kakHs ≤ κ and the norm of a linear operator is an upper bound for its spectral radius, it follows that ¡ ¢ ˚ ∩ [ |z| ≤ κ] . σ a ◦ rΛ (ζ) ⊂ W (3.5.14) (2) We denote by ΣR the positively oriented boundary of W ∩ [ |z| ≤ R] for R ≥ 2κ. Then we claim ¯¡ ¢ ¯ ¯ z − a ◦ rΛ (ζ) −1 ¯ ≤ 3κ/|z| , z ∈ ΣR . (3.5.15) Indeed, if |z| ≥ 2κ ≥ 2 |a ◦ rΛ (ζ)|, then ¯¡ ¢ ¯ ¯ z − a ◦ rΛ (ζ) −1 ¯ ¯¡ ¢−1 ¯ ¯ ≤ 2/|z| = |z|−1 ¯ 1 − z −1 a ◦ rΛ (ζ)

ΓR

by Lemma 3.5.2. If |z| = 1/2κ, then

Σ0R

¯¡ ¢−1 ¯ ¯ ≤ 1/2 , |z| ¯ a ◦ rΛ (ζ) ¯¡ ¢−1 ¯ ¯ ≤ κ. since (3.5.3) guarantees that ¯ a ◦ rΛ (ζ) Hence, invoking Lemma 3.5.2 once more, we arΣR = Σ0R + ΓR rive at ¯¡ ¢ ¯ ¯¡ ¢ ¯ ¯¡ ¢ ¯ ¯ z − a ◦ rΛ (ζ) −1 ¯ ≤ ¯ a ◦ rΛ (ζ) −1 ¯ ¯ z(a ◦ rΛ (ζ))−1 − 1 −1 ¯ ≤ 2κ = 1/|z| . Lastly, if | arg z| = ϕ, then (3.5.9) implies ¯¡ ¢ ¯ ¯ z − a ◦ rΛ (ζ) −1 ¯ ≤ 3κ/(1 + |z|) ≤ 3κ/|z| . Thus (3.5.15) follows from κ ≥ 1. Consequently, ¯¡ s ¯¡ ¢ ¯ ¢ ¯ ¯ Λ (ζ)z − a(ζ) −1 ¯ = Λ−s (ζ) ¯ z − a ◦ rΛ (ζ) −1 ¯ ≤ 3κ/Λs (ζ) |z| for z ∈ ΣR .

(3.5.16)

VI.3 Anisotropy

63

q (3) Fix ζ0 ∈ Z. By the upper semicontinuity and its comq of the spectrum ¡ ¢ pactness, there exists a neighborhood U of ζ0 in Z such that σ a(U ) is contained in the interior of Λs (ζ0 )ΣR . Thus, by Cauchy’s theorem, Z ¡ ¢−1 1 h(a)(ζ) = h(λ) λ − a(ζ) dλ , ζ∈U . 2πi Λs (ζ0 )ΣR Consequently, ∂ξα h(a)(ζ) =

Z

1 2πi

Λs (ζ

0 )ΣR

h(λ)∂ξα (λ − a)−1 (ζ) dλ ,

ζ∈U .

q This holds, in particular, for ζ = ζ0 . Hence, ζ0 being arbitrary in Z, we find Z 1 α ∂ξ h(a)(ζ) = h(λ)∂ξα (λ − a)−1 (ζ) dλ (3.5.17) 2πi Λs (ζ)ΣR q for ζ ∈ Z. (4) Suppose β 6= 0. From (3.3.7) and (3.5.16) we derive ¯ ¢ p ¯¡ p Λβ ω ¯ ∂ξβ (λ − a)(λ − a)−1 | s ¯ = Λβ ω |∂ξβ a| |(Λs z − a)−1 | λ=Λ z ± ≤ 3κΛs kakHs Λs |z| ≤ c(κ) for z ∈ ΣR . Using this, Lemma 3.3.7 and (3.5.16) once more, it follows ¯ ¢ p ¯¡ Λα ω ¯ ∂ξα (λ − a)−1 | s ¯ ≤ c(κ) |(Λs z − a)−1 | ≤ c(κ)Λ−s |z|−1 λ=Λ z

(3.5.18)

for z ∈ ΣR . Now we deduce from (3.5.17) Z ¯ ¡ α ¢ 1 ¯¯ pω α ¯ s α pω −1 s Λα |∂ h(a )| ≤ h(Λ z)Λ ∂ (λ − a ) Λ dz ¯ ¯ η η η η η |λ=Λsη z η 2π ΣR Z ≤ c(κ) |h(Λsη z) dz/z| ΣR

≤ c(κ, ϑ) sup max |h(Λsη z)| . ξ∈Rd z∈ΣR

¡ ¢ This shows that h(aη ) belongs to Mη Rd , L(E) and kh(aη )kMη ≤ c(κ, ϑ) sup max |h(Λsη z)| ξ∈Rd z∈ΣR

(3.5.19)

¡ ¢ for any R ≥ 2κ. Since |Λsη (ξ)z| ≥ [η]s |z|, the assumption implies h Λsη (ξ)z → 0 as |z| → ∞, uniformly for ξ ∈ Rd . Hence, by choosing R sufficiently large, we infer from the maximum principle for holomorphic functions that ¯ ¡ ¯ ¡ ¢¯ ¢¯ max ¯h Λsη (ξ)z ¯ = max ¯h Λsη (ξ)z ¯ . z∈ΣR

z∈Γ

From this and (3.5.19) it follows that (3.5.12) applies.

64

VI Auxiliary Material

(5) We fix Rη ≥ 2κ such that ¡ s ¢δ ¯ ¡ s ¢¯ [η] |z| ¯h [η] z ¯ ≤ 1 , |z| ≥ Rη , z ∈ W . ¯ ¯ ¯ ¡ ± ¢ ¡ ± ¢ ¯ Then Λsη (ξ) |z| = [η]s ¯ Λsη (ξ) [η]s z ¯ and ¯ Λsη (ξ) [η]s z ¯ ≥ |z| give ¡ s ¢δ ¯ ¡ ¢¯ Λη (ξ) |z| ¯h Λsη (ξ)z ¯ ≤ 1 , ξ ∈ Rd , z ∈ W , |z| ≥ Rη . We set

(3.5.20)

Σ0R := ΣR ∩ [ |z| = R] = { Rei t ; −ϕ ≤ t ≤ ϕ } .

Then we get from (3.5.20) and (3.5.16) ¯Z ¡ ¢−1 ¯¯ ¯ h(λ) λ − aη (ξ) dλ¯ ¯ Λsη (ξ)Σ0R

Z

≤ Σ0R

¯ ¡ s ¢¯ ¯¡ ¢ ¯ ¯h Λη (ξ)z ¯ ¯ Λsη (ξ)z − aη (ξ) −1 ¯ Λsη (ξ) d|z|

(3.5.21)

≤ c(κ, ϑ)[η]−δs R−δ for ξ ∈ Rd and R ≥ Rη . Put ΓR := ΣR ∩ Γ and 1 IR (aη ) := 2πi

Z Λsη ΓR

h(λ)(λ − aη )−1 dλ

for R ≥ 2κ. By Cauchy’s theorem, Z Z ´ 1 ³ h(aη ) = + h(λ)(λ − aη )−1 dλ . 2πi Λsη ΓR Λsη Σ0R From this and (3.5.21) we deduce h(aη ) = lim IR (aη ) R→∞ Z 1 = h(λ)(λ − aη )−1 dλ 2πi Λsη Γ

ΓR0 \ΓR ΓR

¡ ¢ in BC Rd , L(E) .

Σ0R Σ0R0

(6) Let ∆R0 ,R be the positively oriented boundary of W ∩ [R ≤ |z| ≤ R0 ] for 2κ ≤ R < R0 < ∞. Since λ 7→ h(λ)(λ − a)−1 is holomorphic in a neighborhood of W ∩ [ |z| ≥ 2κ], Cauchy’s the∆R 0 R orem implies Z 1 0= h(λ)(λ − a)−1 dλ 2πi Λs ∆R0 ,R Z Z ´ 1 ³ 0 = IR (a) − IR (a) + − h(λ)(λ − a)−1 dλ . 2πi Λs Σ0 0 Λs Σ0R R

VI.3 Anisotropy

65

Thus pω α Λα ∂ η

pω ³ ¡ ¢ Λα η 0 IR (aη ) − IR (aη ) = ∂α 2πi

Z

Z

´

− Σ0R

Σ0R0

h(Λsη z)(Λsη z − aη )−1 Λsη dz .

q From this, Lemma 3.5.4, (3.5.18), and Leibniz’ rule we deduce, for η ∈ H, Z ´ ³Z ¯ α p ω α¡ ¢¯ ¯Λη ∂ IR0 (aη ) − IR (aη ) ¯ ≤ c[η]−δs + |z|−1−δ d|z| Σ0R

Σ0R0

≤ c[η]−δs R−δ → 0 ¡ ¢ ¡ ¢ as R → ∞. Thus IR (aη ) R≥2κ is a Cauchy net in Mη Rd , L(E) for R → ∞. Since ¡ d ¢ Mη is a Banach such ¡ d space, ¢ there exists I(η) ∈ Mη R ¡, L(E) ¢ that IR (aη ) converges in M R , L(E) towards I(η) , hence in BC Rd , L(E) by M ,→ BC. Now step (5) implies I(η) = h(aη ). This proves the last part of the claim. ¥ Powers and Exponentials Now we consider two most important choices for h, namely, power functions and exponentials. Recall that λ 7→ log λ = log |λ| + i arg λ and λ 7→ hz (λ) := λz = ez log λ ,

λ∈C,

(3.5.22)

are the principal value of the logarithm and the power function, respectively. By the Dunford calculus, Z q ¡ ¢−1 1 z a (ζ) = hz (a)(ζ) = λz λ − a(ζ) dλ ∈ L(E) , ζ∈Z, 2πi Λs (ζ)Γ ¡ ¢ for a ∈ Ps Z, L(E); κ, ϑ and z ∈ C. 3.5.6 Proposition Suppose Rez < 0. Then Z 1 z aη = λz (λ − aη )−1 dλ 2πi Λsη Γ

(3.5.23)

¡ ¢ in Mη Rd , L(E) and kazη kMη ≤ c(κ, ϑ)[η]sRez e|Imz| ϕ

(3.5.24)

q ¡ ¢ for η ∈ H and a ∈ Ps Z, L(E); κ, ϑ . Proof lows

Since Re(z log λ) = Rez log |λ| − (Imz) arg λ and |λz | = eRe(z log λ) , it fol|λz | ≤ |λ|Rez e|Imz| (π−ϑ) ,

˚π−ϑ . λ∈S

66

VI Auxiliary Material

˚π−ϑ . FurtherHence, given 0 < δ < −Rez, we see that |λδ | λz → 0 as λ → ∞ in S more, ¡ ¢Rez |Imz| ϕ |λz | ≤ [η]s /2κ e , λ ∈ Λsη Γ . Thus the assertion follows from Proposition 3.5.5.

¥

Of outstanding importance is the symbol class ¡ ¢ ¡ ¢ Ps Z, L(E); κ := Ps Z, L(E); κ, π/2 .

(3.5.25)

In this case ϕ = ϕ(κ) ∈ (0, π/2). Consequently, £ ¤ ¡ ¢± W ⊂ Rez ≥ ω(κ) , ω(κ) := cos ϕ(κ) 2κ .

(3.5.26)

Now we consider the entire analytic function λ 7→ et (λ) := e−tλ ,

t>0.

Then e−ta(ζ) := et (a)(ζ) Z ¡ ¢−1 1 = e−tλ λ − a(ζ) dλ 2πi Λs (ζ)Σ

W

0 ω

q ¡ ¢ with ζ ∈ Z is, for a ∈ Ps Z, L(E); κ , the Dunford representation of the analytic semigroup { e−ta(ζ) ; t ≥ 0 } on E, generated by a(ζ) ∈ L(E).

[Rez ≥ ω]

W ⊂ [Rez ≥ ω]

3.5.7 Proposition If t > 0, then −taη

e

1 = 2πi

Z

¡ ¢−1 e−tλ λ − a(ζ) dλ Λsη Γ

¡ ¢ in Mη Rd , L(E) and s

ke−taη kMη ≤ c(κ)e−t[η]

ω(κ)

q ¡ ¢ for η ∈ H and a ∈ Ps Z, L(E); κ . ˚ψ for any ψ ∈ (ϕ, π/2). By Proof It is obvious that |z|δ et (z) → 0 as |z| → ∞ in S (3.5.26), s |e−tλ | = e−tReλ ≤ e−t[η] ω(κ) , λ ∈ Λsη W . Hence Proposition 3.5.5 implies the claim.

¥

VI.3 Anisotropy

67

¡ ¢ Lastly, we exhibit a simple example of a symbol in Ps Z, L(E); κ . ¡ ¢ 3.5.8 Example Put a := Λs 1E . Then a ∈ Ps Z, L(E); κ , where © ª κ := sup (1 + |λ|)/|1 + λ| ; λ ∈ Sπ/2 . Proof

Since a ◦ rΛ = 1E , it holds σ(a ◦ rΛ ) = {1}. Thus Sπ/2 is contained in ¯¡ ¢−1 ¯ ¯ = |1 + λ|−1 ≤ κ(1 + |λ|)−1 for the resolvent set of −a ◦ rΛ and ¯ λ + a ◦ rΛ (ζ) λ ∈ Sπ/2 . Now the assertion follows from Example 3.3.9. ¥

3.6

Dyadic Partitions of Unity

In this subsection we introduce partitions of unity of Rd which are compatible with the dilation (3.1.3). They are fundamental for the Fourier-analytic theory of Besov spaces, discussed in detail in Section 2. A nonempty subset M of Rd is symmetric (with respect to the origin), if x ∈ M implies −x ∈ M . It is ν -starshaped, if t q M ⊂ M for 0 ≤ t ≤ 1. Suppose Ω is a bounded open 0-neighborhood in Rd which is symmetric and ν-starshaped, a ν -admissible 0-neighborhood of Rd . We set Ω0 := 2 q Ω ,

Σ := 2 q Ω0 \2−1 q Ω0 ,

Ωj := 2j−1 q Σ ,

j≥1.

(3.6.1)

Then Ωk is open and symmetric for k ∈ N, Ωj ∩ Ωk = ∅ for |j − k| ≥ 2, and S d d Ω k≥0 k = R . We say that (Ωk ) is the ν -dyadic open covering of R induced by Ω. Assume that ψ is Ω-adapted, that is, b

ψ ∈ D(Ω0 ) , We put

ψ | (3/2) q Ω = 1 = 1Ω .

ψ=ψ,

ψe := ψ − σ2 ψ ,

ψ0 = ψ ,

ψj := σ2−j ψe ,

(3.6.2)

j≥1.

Then b

ψk ∈ D(Ωk ) ,

ψk = ψk ,

n X

ψj = σ2−n ψ ,

k, n ∈ N .

(3.6.3)

ξ ∈ Rd ,

(3.6.4)

j=0

Since lim σ2−n ψ(ξ) = lim ψ(2−n q ξ) = ψ(0) = 1 ,

n→∞

n→∞

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VI Auxiliary Material

we see that ∞ X

ψj = 1 on Rd .

(3.6.5)

j=0

Thus (ψk ) is a smooth partition of unity on Rd , subordinate to the open cover¡ ¢ ing (Ωk ). We call (Ωk ), (ψk ) ν -dyadic partition of unity on Rd induced by (Ω, ψ). 3.6.1 Examples (a) Let Q be a ν-quasinorm on Rd . Then Ω := [Q < 1] is a ν-admissible 0-neighborhood. Note that Ω0 = [Q < 2] ,

Ωj = [2j−1 < Q < 2j+1 ] ,

j≥1.

q (b) Let d1 , d2 ∈ N satisfy d1 + d2 = d. Assume [`i , di , νi ] is a weight system for Rdi such that £ ¤ [`, d, ν] = `1 + `2 , (d1 , d2 ), (ν1 , ν2 ) . Let Ωi be a νi -admissible 0-neighborhood in Rdi and suppose ψ i is Ωi -adapted. Then Ω := Ω1 × Ω2 is a νi -admissible 0-neighborhood of Rd and ψ := ψ 1 ⊗ ψ 2 is Ω-adapted. ¥ Preliminary Fourier Multiplier Theorems d ¡Now we fix¢a ν-admissible 0-neighborhood Ω of R dand an Ω-adapted ψ, and let (Ωk ), (ψk ) be the ν-dyadic partition of unity on R induced by (Ω, ψ).

3.6.2 Lemma Suppose X ∈ {S, S 0 }. Then Proof

P∞ k=0

ψk (D)u = u in X (Rd , E) if u ∈ X .

It is an easy consequence of Proposition 3.1.3(i) and (3.6.4) that ° ¡ ¢° lim °ϕ ∂ α (σ2−n ψ − 1) °∞ = 0 , ϕ ∈ S(Rd ) . n→∞

Hence we see from (3.6.3) that (3.6.5) converges in OM (Rd ). Now the assertion follows from Theorem 3.4.1. ¥ In the next lemma we collect some Fourier multiplier results for the sequence (ψk ) which will lead, among other things, to a Fourier multiplier theorem for Besov spaces. 3.6.3 Lemma (i) Let E0 × E1 → E2 be a multiplication of Banach spaces and X ∈ { BUC, C0 , Lq , 1 ≤ q < ∞ } .

VI.3 Anisotropy

Then ¡

69

³ ¢ ¡ ¢´ a 7→ (ψk a)(D) ∈ L M(Rd , E0 ), `∞ L(X (Rd , E1 ), X (Rd , E2 ))

and ³ ¡ ¢ ¡ ¢´ a 7→ (ψk a)(D) ∈ L M(Rd , E0 ), `∞ L(L∞ (Rd , E1 ), BUC(Rd , E2 )) . ¡ ¢ ¡ ¢ (ii) ψk (D) ∈ L S(Rd , E) ∩ L S 0 (Rd , E) , k∈N. (iii) Let 1 ≤ q ≤ ∞ and s ∈ R. Then there exists ` ∈ N such that sup k2ks ψk (D)ukq ≤ c q`,` (u) ,

u∈S ,

(3.6.6)

k∈N

and

Proof

¡

¢ ¡ ¢ ψk (D) ∈ `s∞ L(S(Rd , E), C0 (Rd , E)) .

(1) Let a ∈ M(Rd , E0 ). If we show kψk akF L1 ≤ c kakM ,

k∈N,

(3.6.7)

then assertion (i) follows from Theorem 3.4.4. q e 2k a) and Proposition 3.1.3(ii) and (iii) imply (2) If k ∈ N, then ψk a = σ2−k (ψσ e 2k a) F −1 (ψk a) = 2|ω| k σ2k F −1 (ψσ and, consequently, e 2k a)k1 = kψσ e 2k akF L , kψk akF L1 = kF −1 (ψσ 1

q k∈N.

Hence (3.6.7) is satisfied if e 2k a)k1 ≤ c kakM , kF −1 (ψa)k1 + kF −1 (ψσ

q k∈N.

(3.6.8)

(3) By Leibniz’ rule and Proposition 3.1.3(i), X³α ´ p e 2k a) = e 2k (Dβ a) . Dα (ψσ 2kβ ω Dα−β ψσ β β≤α

We denote by χY the characteristic function of a subset Y of Rd . Then, since e ⊂ Σ for γ ∈ Nd , we get supp(Dγ ψ) X p e 2k a)(ξ)|E ≤ c |Dα (ψσ 2kβ ω |∂ β a(2k q ξ)|E0 χΣ (ξ) 0 β≤α

q for ξ ∈ Rd and k ∈ N.

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VI Auxiliary Material

We fix r > 1 with 1/r ≤ N(ξ) ≤ r for ξ ∈ Σ. Then ¡ ¢β p ω ¡ ¢β p ω p 2kβ ω ≤ 2k rN(ξ) = rN(2k q ξ) , ξ∈Σ, Since N ≤ Λ1 , it follows e 2k a)(ξ)|E ≤ c |Dα (ψσ 0

X

Λβ1

pω

k≥1.

(3.6.9)

(2k q ξ) |∂ β a(2k q ξ)|E0 χΣ (ξ) (3.6.10)

β≤α

≤ c kakM χΣ (ξ) q for ξ ∈ Rd , α q ω ≤ k(ν), and k ∈ N. p Similarly, |∂ β a|E0 ≤ Λβ1 ω |∂ β a|E0 ≤ kakM implies |Dα (ψa)|E0 ≤ c kakM χΩ0 ,

α q ω ≤ k(ν) .

(3.6.11)

From (3.6.10) and (3.6.11) we see that these functions are integrable and e 2k a)k1 ≤ c kakM , kDα (ψa)k1 + sup kDα (ψσ

α q ω ≤ k(ν) .

k≥1

Hence, by the Riemann-Lebesgue lemma (cf. Section III.4.2), ¡ ¢ (−1)|α| xα F −1 (ψa) = F −1 Dα (ψa) and

¡ ¢ e 2k a) = F −1 Dα (ψσ e 2k a) (−1)|α| xα F −1 (ψσ

belong to C0 (Rd , E0 ) and ° e 2k a)k∞ ≤ c kakM kxα F −1 (ψa)k∞ + sup °xα F −1 (ψσ

(3.6.12)

k≥1

for α q ω ≤ k(ν). Thus, choosing α = 0 and α := 2 · 1d , e 2k a)(x)|E |F −1 (ψa)(x)|E0 + sup |F −1 (ψσ 0 k≥1

© Qd ª ≤ c kakM min 1, j=1 |xj |−2

(3.6.13)

for x ∈ Rd . Consequently, we obtain (3.6.8) by integrating over Rd . This proves assertion (i). (4) Since ψk ∈ D(Rd ) ,→ OM (Rd ), claim (ii) follows from Theorem 3.4.1. ¡ ¢ (5) Let u ∈ S = S(Rd , E). Then, using F −1 (1 − ∆)|ω| v = h·i2 |ω| F −1 v for v ∈ S 0 , we estimate kψk (D)ukq = kh·i−2 |ω| h·i2 |ω| ψk (D)ukq ° ¡ ¢° ≤ c kh·i2 |ω| ψk (D)uk∞ = c °F −1 F h·i2 |ω| F −1 (ψk u b) °∞ ° ¡ ¢° = c °F −1 (1 − ∆)|ω| (ψk u b) °∞ ≤ c k(1 − ∆)|ω| (ψk u b)k1 ,

(3.6.14)

the last inequality being once more a consequence of the Riemann-Lebesgue lemma.

VI.3 Anisotropy

71

From ∂ α ψk = ∂ α σ2−k ψe = 2−kα we get

e ∞, k∂ α ψk k∞ ≤ k∂ α ψk

pω

σ2−k ∂ α ψe

α ∈ Nd ,

(3.6.15)

k∈N.

Recall that ψk is supported in Ωk and write ξ = 2k q ξe with ξe ∈ Σ for k ≥ 1. Then e ≥ 1/r that 2k ≤ r2k N(ξ) e = rN(ξ). Hence we deduce from N(ξ) ¡ ¢|s| α 2ks |∂ α ψk (ξ)| ≤ rN(ξ) |∂ ψk (ξ)| ≤ c(α, s)hξi|s| ,

ξ ∈ Rd ,

(3.6.16)

for α ∈ Nd , where the last inequality follows from (3.3.18) and (3.6.15). We fix m ∈ N with m ≥ |s|. Then (3.6.16), (1 − ∆)|ω| = (1 + D12 + · · · + Dd2 )|ω| , and Leibniz’ rule imply 2ks kh·i2 |ω| (1 − ∆)|ω| (ψk u b)k∞ ≤ c

X

kh·im+2 |ω| ∂ α u bk∞ .

|α|≤2 |ω|

Hence it follows from (3.6.14) that 2ks kψk (D)ukq ≤ c 2ks k(1 − ∆)|ω| (ψk u b)k1 ≤ c 2ks kh·i2 |ω| (1 − ∆)2 |ω| (ψk u b)k∞ ≤ c qm+2 |ω|,2 |ω| (b u)

(3.6.17)

for u ∈ S and k ∈ N. Due to F ∈ L(S), there exists ` ∈ N such that qm+2 |ω|,2 |ω| (b u) ≤ c q`,` (u) ,

u∈S .

Thus, by (3.6.17), we get (3.6.6). Since ψk (D)u ∈ S ⊂ C0 by (ii), the second part of (iii) is clear in this case as well. ¥ For later use we record the following observations on the preceding proof. 3.6.4 Remarks (a) It holds ° 2 |ω| ¡ ¢° sup °Λ1 F −1 σ2k (ψk+j a) °1 ≤ c(j) kakM k≥0

for a ∈ M(Rd , E0 ) and j ∈ N. Proof (1) Assume j = 0. Let ` := |ω|. From (3.3.18) and the multinomial theorem we deduce X³`´ ¡ ¢ 2` 1 2 d 2 ` Λ2` (x) ≤ c hxi = c 1 + (x ) + · · · + (x ) = c x2β . (3.6.18) 1 β |β|≤`

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VI Auxiliary Material

Moreover, β q ω + 2 |ω| ≤ max ω |β| + 2 |ω|. Thus, setting α := β + 21, it follows that α q ω ≤ k(ν) if |β| ≤ `. Hence (3.6.12) and (3.6.18) imply, letting α = 0, resp. α = β + 21 with |β| ≤ `, ¯¡ 2` −1 ¢ ¯ ¯ Λ1 F (ψa) (x)¯

E0

¯¡ ¢ ¯ −1 e + sup ¯ Λ2` (ψσ2k a) (x)¯E 1 F k≥1

©

0

≤ c kakM min 1,

Qd

j=1 |x

j −2

|

ª

e 2k a for k ≥ 1 implies the claim. for x ∈ Rd . Now σ2k (ψk a) = ψσ (2) Suppose j > 0. Then supp(ψj ) ⊂ Ωj = 2j q Σ. Now the assertion follows by replacing ψe and Σ by ψj and Ωj , respectively, in step (2) of the proof of Lemma 3.6.3. ¥ (b) In the proof of part (i) of this lemma we need only derivatives of order at most 2 |ω|. Hence this part remains valid if M is replaced by M2 |ω| . ¥

3.7

Notes

This section consists of revisions and augmentations of Chapter 2 of [Ama09]. The anisotropic dilations which we consider are tailored to fit our purposes, the investigation of parabolic differential operators in particular. Reduced weight systems are introduced so that we can work with the minimal number of parameters. This will become clear when we consider anisotropic function spaces and differential operators in later sections (also see Example 3.4.3). Weight systems introduced here are a convenient tool for collecting relevant data. The anisotropic dilations associated with a weight system form a subclass of the general one-parameter group of dilations on the Euclidean d-space. A systematic study of the latter is given in [SteW78]. In step (4) of the proof of Example 3.2.1(b) we follow J. Johnsen and W. Sickel [JoS07] (also see [CaT75]). All papers in which anisotropic dilations play a role, and which we know of (e.g., [CaT75], [Yam86]), employ the natural quasinorm E, which is already introduced in [FaR66]. In contrast, we base our investigations on the equivalent natural quasinorm N which is easier to work with and particularly well-adapted for our purposes. It should be noted that our use of ‘quasinorm’ is somewhat nonstandard and has a different meaning than the same notion employed by other authors (e.g., [RS96], [Tri83]). The multiplier estimates in Subsection 3.5 are of a preparatory nature. Particular care is taken to exhibit their dependence on the constants characterizing the symbol classes. In order to reduce the number of those constants, we do not give optimal estimates but ¡ ¢ content ourselves with qualitative bounds. In particular, if A ∈ Ps Z; L(E); κ , then the estimates depend on κ only, (and on [`, d, ν]

VI.3 Anisotropy

73

and E, which are fixed throughout). This will considerably facilitate the proof of a priori estimates for concrete parabolic differential equations in later chapters. The dyadic partition of unity introduced in Subsection 3.6 is an anisotropic variant of the well-known Littlewood–Paley decomposition (e.g., [Ste93]). Starting with J. Peetre’s paper [Pee67], it has been the basis for the Fourier-analytic approach to general isotropic Besov, Bessel potential, and Triebel–Lizorkin spaces (see one of the many books by H. Triebel, e.g; [Tri83], for detailed expositions and historic comments). The anisotropic case is usually modeled straightforwardly after the isotropic one by means of Example 3.6.1(a) with Q = E (e.g., [Yam86] or [Joh95]). In contrast to those works, we use the slightly more general and flexible definition (3.6.1). This does pay off in the proof of the trace theorem in Subsection VIII.1.2. The preliminary Fourier multiplier results of Lemma 3.6.3 constitute an anisotropic extension of the results of Section 4 in [Ama97]. q For simplicity, we do not consider dyadic decompositions of (Rd ) , although simple modifications would suffice (cf. [Ama97]). Such extensions would be needed if we wanted to study homogeneous function spaces on Rd , which is not our intention.

Chapter VII

Function Spaces This chapter forms the core of this volume. Besides of the classical function spaces, the basic scales of Besov and Bessel potential spaces are introduced and studied in great detail. Of course, Banach-space-valued anisotropic spaces are considered throughout. The first section is devoted to the spaces of bounded continuous functions and the Sobolev spaces. Since we do not make any assumption on the target Banach spaces, the vector-valued Sobolev spaces do not fit into the framework of the general Besov and Bessel potential spaces. For this reason a separate study is appropriate. The following two sections explore Besov spaces. Particular attention is paid to the subclass of little H¨older spaces. The latter play a seminal role in the study of nonlinear parabolic boundary value problems, realized in the third volume of this treatise. Up to this point, no restrictions on the target Banach spaces are necessary, except for reflexivity assumptions in duality theorems. This is different in the case of Bessel potential spaces whose investigation is executed in Section 4. For most of their profound properties, restrictions on the geometry of the target Banach spaces have to be imposed. In Section 5, Triebel–Lizorkin spaces are introduced and their main properties are explored. In our setup they play only a technical role. The principal reason for their inclusion is the fact that they provide sharpenings of some embedding theorems without restrictions on the target Banach spaces. This is used to prove anisotropic vector-valued replications of the classical Gagliardo—Nirenberg inequalities. There are three more sections in this chapter. In the first one we establish optimal and nearly optimal point-wise multiplier theorems. They are essential for the study of boundary value problems under minimal regularity assumptions. © Springer Nature Switzerland AG 2019 H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4_7

75

76

Function Spaces

Section 7 explores compact embedding theorems. After having related Besov and Bessel potential spaces on compact intervals to our general setup, we provide far-reaching extensions of the well-known Aubin–Lions theorem. In the last section we introduce the technical tool of parameter-dependent function spaces and establish a number of seminorm estimates. This material is useful for proving resolvent estimates for elliptic and parabolic differential operators in the next volume.

VII.1 Classical Spaces

1

77

Classical Spaces

Classical Sobolev spaces and spaces of bounded continuous functions are, of course, well-known. However, in this section we are interested in vector-valued anisotropic versions of those spaces, defined on corners. Unfortunately, in the vector-valued case some of the most important function spaces, namely spaces of bounded continuous functions and Sobolev spaces, do not fit into any of the general scales of Banach spaces discussed in the subsequent sections. For this reason they have to be treated separately. After having introduced appropriate definitions in the first two subsections, we prove in the third subsection the basic restriction-extension theorems. This allows to carry over results proved on the full Euclidean space to spaces defined on corners. As a first application we show in Subsection 1.4 that, even on corners, anisotropic Sobolev spaces, which are defined by a completion procedure, can be characterized by integrability conditions of distributional derivatives. This is then used, in the following subsection, to establish their reflexivity, given appropriate restrictions, and to give an explicit representation of spaces of negative order. The last subsection contains some elementary embedding properties. It should be observed that, in this whole section, there are no restrictions on the Banach space, except for occasional reflexivity assumptions. Nevertheless, we can establish most of the properties, known in the scalar-valued setting, for vector-valued anisotropic Sobolev spaces. It is assumed throughout that • K is a corner in Rd . • X ∈ {Rd , K}. Moreover,1 ∂ := (∂1 , . . . , ∂d ) = (∂x1 , . . . , ∂xd ) ,

∂xi := (∂x1i , . . . , ∂xdi ) ,

1≤i≤`.

i

We indiscriminately use ∇ := ∂ ,

∇xj := ∂xj ,

∇xi := ∂xi ,

1≤j≤d,

1≤i≤`.

If convenient, we identify ∇ = ∂ with the Fr´echet derivative on X. Then ∇m := ∇ ◦ ∇m−1 ,

∇xmi := ∇xi ◦ ∇xm−1 , i

∇0 := ∇x0i := id

q for m ∈ N. Finally, we recall that assumption (VI.3.1.20) holds throughout. 1 The reader is cautioned to distinguish carefully between xi ∈ R, a coordinate of x ∈ Rm , and xi ∈ Rdi .

78

1.1

VII Function Spaces

Bounded Continuous Functions

First we show that uniform continuity ‘from the right’ already implies uniform continuity. 1.1.1 Lemma Suppose K is closed and u : X → E. Then u is uniformly continuous iff lim u(x + y) = u(x), uniformly for x ∈ X .

y→0 y∈X

(1.1.1)

Proof It is clear that the condition is necessary. Let x, z ∈ X and denote by (e1 , . . . , ed ) the standard basis of Rd . Define Pj i h := z i − xi ∈ R and hj := hj ej for 1 ≤ j ≤ d. Set x0 := x and xj := x + i=1 hi for 1 ≤ j ≤ d. Then xd = xd−1 + hd = z and |u(z) − u(x)|E ≤

d X

|u(xj−1 + hj ) − u(xj )|E .

j=1

Put yj := |hj | ej . Then yj ∈ X and ( |u(xj−1 + yj−1 ) − u(xj−1 )|E , |u(xj−1 + hj ) − u(xj−1 )|E = |u(xj−1 − yj + yj ) − u(xj−1 − yj )|E ,

if hj ≥ 0 , if hj < 0 .

It follows from (1.1.1) that |u(xj−1 + hj ) − u(xj−1 )|E ≤ sup |u(x + yj ) − u(x)|E → 0 x∈X

as yj → 0. This proves that (1.1.1) is sufficient.

¥

For the reader’s convenience, we recall the definition of C0 (X, E). Let X be a σ-compact metric space. Then u ∈ C(X, E) vanishes at infinity, if for each ε > 0 there exists a compact set K such that |u(x)|E < ε for x ∈ / K. Suppose X is a σ-compact metric space which is a dense subset of some topological space Y . If u ∈ C0 (X, E) and y ∈ Y \X, then u(x) → 0 as x → y in Y . Hence u has a unique extension u ∈ C(Y, E), and u | Y \X = 0. Thus we can (and do) identify u with u. This applies, in particular, to X := K and Y := K if K is not closed. Banach Spaces of Bounded Continuous Functions We simply write k·kr for k·kLr (X,E) , 1 ≤ r ≤ ∞, if (X, E) is clear from the context. Furthermore, k·k∞ is also used for the supremum norm of B(X, E), the Banach space of bounded E-valued functions on X.

VII.1 Classical Spaces

79

Recall that j ∈ J ∗ iff ∂j K ∩ K = ∅. We define © ª BC(K, E) := u ∈ BC(K, E) ; u | ∂j K = 0, j ∈ J ∗ and BUC(K, E) :=

©

u ∈ BC(K, E) ; u is uniformly continuous

(1.1.2) ª

.

Then C0 (X, E) ,→ BUC(X, E) ,→ BC(X, E) ,→ B(X, E) and each space on the left of one of the symbols ,→ is a closed linear subspace of the one on the right. Furthermore, d

S(X, E) ,→ C0 (X, E) ,

d

S(X, E) ,→ Lq (X, E) ,

as is well-known and easily seen. Let m ∈ νN. We put X |||·|||m/ν,r := k∂ α · kr , α p ω≤m

1≤r≤∞.

(1.1.3)

(1.1.4)

We denote by BC m/ν (X, E) the linear subspace of BC(X, E) such that ∂ α u exists for α ∈ Nd with α q ω ≤ m (in the classical sense) and belongs to BC(X, E).

(1.1.5)

It is endowed with the norm |||·|||m/ν,∞ . If X = K, then u ∈ BC m/ν (K, E) implies, due to (1.1.2), ∂jk u|∂j K = 0

for

0 ≤ k ≤ m/νj and j ∈ J ∗ .

(1.1.6)

1.1.2 Lemma BC m/ν (X, E) is a Banach space. Proof (1) Let (uj ) be a Cauchy sequence. Then (∂ α uj ) is a Cauchy sequence in BC(X, E) for α q ω ≤ m. Hence there exist uα ∈ BC(X, E) with k∂ α uj − uα k∞ converging to 0 as j → ∞, and u0 = u. Assume α = ek . Then Z 1 uj (x + hek ) − u(x) = h ∂k uj (x + thek ) dt , x, x + hek ∈ X . 0

Letting j → ∞, we get ¡ ¢ h−1 u(x + hek ) − u(x) =

Z

1

uek (x + thek ) dt .

0

From this we infer that ∂k u exists and equals uek . Now we find inductively that ∂ α u exists and equals uα ∈ BC(X, E) for α q ω ≤ m. This proves the completeness of BC m (X, E). ¥

80

VII Function Spaces

1.1.3 Remark BC 0/ν (K, E) = BC(K, E).

¥

Spaces of bounded and uniformly continuously differentiable functions are introduced by BUC m/ν (X, E) := { u ∈ BC m/ν (X, E) ;

(1.1.7)

∂ α u is uniformly continuous for α q ω ≤ m } . This is a closed linear subspace of BC m/ν (X, E), hence a Banach space. The mean-value theorem implies BC m/ν (X, E) ,→ BUC k/ν (X, E) ,

k 0 and u ∈ X m (X, E). Then there exists ρ > 0 such that |∂ α u(x)|E < ε/3 , |x|∞ ≥ ρ , α q ω ≤ m . (1.1.11) ¡ ¢ We fix χ ∈ D Rd , [0, 1] satisfying χ(x) = 1 for |x|∞ ≤ 1 and χ(x) = 0 for |x|∞ ≥ 2, and set χr (x) := χ(x/r) for r > 0. Leibniz’ rule implies (cf. step (2) of the proof

VII.1 Classical Spaces

81

of Lemma VI.1.1.3) k∂ α (χr u − u)k∞ ≤ k(χr − 1)∂ α uk∞ + c(α)r−1 |||u|||m/ν,∞

(1.1.12)

for α q ω ≤ m, where c(0) := 0. Note k(χr − 1)∂ α uk∞ ≤ sup |∂ α u(x)|E . |x|∞ ≥r

From this, (1.1.11), and (1.1.12) we see that we can fix r0 ≥ ρ such that u0 := χr0 u ∈ BUC m/ν (X, E) satisfies |||u0 − u|||m/ν,∞ ≤ 2ε/3 .

(1.1.13)

Suppose X =¡K. Theorem 1.3.1, which we prove ¢ below, implies that (RK , EK ) is an r-e pair for BUC n/ν (Rd , E), BUC n/ν (K, E) , n ∈ νN. If X = Rd , then we set (RX , EX ) := (id, id). We put v0 := EX u0 . Note that v0 (x) = 0 ,

x ∈ X ∩ [ |·|∞ ≥ 2r0 ] .

(1.1.14)

Let { ϕξ ; ξ > 0 } be a mollifier.2 Recall that ϕξ ∗ v0 ∈ BUC ∞ (Rd , E) ,

supp(ϕξ ∗ v0 ) ⊂ supp(v0 ) + [ |·|∞ ≤ ξ] ,

(1.1.15)

and ϕξ ∗ v0 → v0 in BUC m/ν (Rd , E) as ξ → 0. Hence we can fix η > 0 such that |||ϕη ∗ v0 − v0 |||m/ν,∞ < ε/3 .

(1.1.16)

Assume K is closed. We set v := RX (ϕη ∗ v0 ). Then v ∈ BUC ∞ (X, E). Moreover, (1.1.14) and (1.1.15) imply supp(v) ⊂ [ |·|∞ ≤ 2r0 + 2η]. Thus v belongs to D(X, E) ⊂ S(X, E). Since RX is the operator of point-wise restriction and RX v0 equals u0 , we get from (1.1.13) and (1.1.16) |||v − u|||m/ν,∞ ≤ |||v − RX v0 |||m/ν,∞ + |||RX v0 − u|||m/ν,∞ ≤ |||ϕη ∗ v0 − v0 |||m/ν,∞ + |||u0 − u|||m/ν,∞ < ε . This proves that S(X, E) is dense in X m (X, E) if K is closed. (3) Suppose K is not closed. We can assume K = L × M with L := (0, ∞)` for some ` ∈ {1, . . . , d} where M is a closed corner in Rd−` . We denote by { Tt ; t ∈ R } the group of right translations on Rd defined by Tt w(x) := w(x − th) ,

x, h := (1` , 0) ∈ Rd ,

for w : Rd → E. It is easy to see (cf. Theorem X.7.6 in [AmE08]) that it is strongly continuous on BUC m/ν (Rd , E). 2 Subsection

III.4.

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VII Function Spaces

It follows from (VI.1.1.18) that EK = EL ◦ EM , where EL is the extension by zero. We set vt := Tt EK u0 for t > 0. Then vt ∈ BUC m/ν (Rd , E), vt → EK u0 in BUC m/ν , and vt (y) = 0 , y ∈ { x ∈ Rd ; xi ≤ t for 1 ≤ i ≤ ` } ∪ { x ∈ K ; |x|∞ ≥ 2r0 } .

(1.1.17)

We fix τ > 0 such that |||vτ − EK u0 |||m/ν,∞ ≤ ε/6 kRK k ,

(1.1.18)

¡ ¢ where kRK k is the norm of RK ∈ L BUC m/ν (Rd , E), BUC m/ν (K, E) . Next we choose ζ ∈ (0, τ /2) such that w0 := ϕζ ∗ vτ satisfies kw0 − vτ km/ν,∞ ≤ ε/6 kRK k .

(1.1.19)

Then w0 ∈ BC ∞ (Rd , E) and supp(w0 ) b K, as follows from (1.1.15) and (1.1.17). Hence w := RK w0 ∈ D(K, E) ⊂ S(K, E) and RK w0 (x) = w0 (x) for x ∈ K by Remark VI.1.2.4(b). Thus we find kw − u0 kBC m/ν (K,E) = kRK (w0 − EK u0 )kBC m/ν (K,E) ≤ kRK k kw0 − EK u0 kBC m/ν (Rd ,E) ¡ ¢ ≤ kRK k |||w0 − vτ |||m/ν,∞ + |||vτ − EK u0 |||m/ν,∞ ≤ ε/3 , due to (1.1.18) and (1.1.19). Using (1.1.13), we find |||w − u|||m/ν,∞ < ε. Hence S(K, E) is also dense in X m (K, E) if K is not closed. The theorem is proved. ¥ Vector Measures Let X and Y be LCSs. We refer to Proposition V.1.4.8 for a proof and precise explanation of d

X ,→ Y

=⇒

Y 0 ,→ X 0

(1.1.20)

with respect to h·, ·iY . We denote by MBV (X∗ , E) the Banach space of E-valued vector measures ∗ on X of bounded variation. Then (the generalized Riesz representation) Theorem 2.0.4 of the Appendix guarantees C0 (X∗ , E)0 = MBV (X∗ , E 0 ) with respect to the duality pairing Z hµ, uiC0 := u dµ , µ ∈ MBV (X∗ , E 0 ) , X∗

u ∈ C0 (X∗ , E) .

(1.1.21)

(1.1.22)

VII.1 Classical Spaces

Hence

83

¡ ¢ Tµ := u 7→ hµ, uiC0 ∈ S(X∗ , E)0 = S 0 (X, E 0 )

by Theorem VI.1.3.1. Furthermore, the map µ 7→ Tµ is linear, continuous, and injective from MBV (X∗ , E 0 ) into S 0 (X, E 0 ). Thus we identify MBV (X∗ , E 0 ) with a linear subspace of S 0 (X, E 0 ) by identifying µ with Tµ . Then MBV (X∗ , E 0 ) ,→ S 0 (X, E 0 )

(1.1.23)

This shows that (1.1.21) holds with respect to h·, ·iC0 . Suppose E is reflexive. Set −m/ν

C0

¡ m/ν ¢0 (X, E) := C0 (X∗ , E 0 )

(1.1.24)

with respect to h·, ·iC0 . Then d

m/ν

S(X∗ , E 0 ) ,→ C0

d

(X∗ , E 0 ) ,→ C0 (X∗ , E 0 ) ,

(1.1.25)

(1.1.20), and (1.1.21) imply ¡ ¢0 −m/ν MBV (X∗ , E) = C0 (X∗ , E 0 ) ,→ C0 (X, E) ,→ S 0 (X, E) .

(1.1.26)

Using Remark 2.0.1 of the Appendix, we identify L1 (X∗ , E 0 ) = L1 (X, E 0 ) (Theorem 1.2.1 below) with a closed linear subspace of MBV (X∗ , E 0 ) by identifying f ∈ L1 (X∗ , E 0 ) with f dx ∈ MBV (X∗ , E 0 ). Then

Z hf dx, ϕiC0 (X∗ ,E) =

X

­ ® f (x), ϕ(x) E dx = hf, ϕiS(X∗ ,E) d

for f ∈ L1 (X, E 0 ) and ϕ ∈ S(X∗ , E). Since S(X∗ , E) ,→ C0 (X∗ , E), it follows that the duality pairing h·, ·iC0 : MBV (X∗ , E 0 ) × C0 (X∗ , E) → C is a (natural) restriction-extension (restriction in the first variable and extension in the second one) of the duality pairing h·, ·iS(X∗ ,E) : S 0 (X, E 0 ) × S(X∗ , E) → C . Thus it it feasible to simply write h·, ·i for either of them if no confusion seems likely.

(1.1.27)

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VII Function Spaces

1.2

Sobolev Spaces

In this subsection we introduce anisotropic E-valued Sobolev spaces. For this we need the concept of regular E-valued distributions. Regular Distributions Suppose X ∈ {BC, BUC, C0 , Lr } ,

1≤r≤∞,

and set r := ∞ if X ∈ {BC, BUC, C0 }. Then, given u ∈ X (X, E), H¨older’s inequality implies ¯Z ¯ ¯ ¯ |hu, ϕiX |E = ¯ uϕ dx¯ ≤ kukr kϕkr0 ≤ c kukr q2d,0 (ϕ) X

E

for ϕ ∈ S(X∗ ), where c := kh·i−2d kr0 . Thus ¡ ¢ ¡ ¢ Tu := ϕ 7→ hu, ϕiX ∈ L S(X∗ ), E = S 0 (X, E) and

¡ ¢ (u 7→ Tu ) ∈ L X (X, E), S 0 (X, E) .

It is clear that u 7→ Tu is injective. Hence we can (and do) identify X (X, E) with a linear subspace of S 0 (X, E) by identifying u with the regular tempered X-distribution Tu . Thus d

X (X, E) ,→ S 0 (X, E) ,

(1.2.1) d

the density being a consequence of S(X, E) ⊂ X (X, E) and S(X, E) ,→ S 0 (X, E), guaranteed by Theorem VI.1.2.3(i) (cf. (1.3.16) of the Appendix). It follows that, given α ∈ Nd and u ∈ X (X, E), the X-distributional derivative ∂ α u is defined in S 0 (X, E). Basic Definitions Let 1 ≤ q < ∞. We define (E-valued anisotropic Lq ) Sobolev spaces of order m/ν on X by3 Wqm/ν (X, E) is the completion of S(X, E) in Lq (X, E) with respect to the norm |||·|||m/ν,q . 3 In

(1.2.2)

the standard isotropic case ` = 1 = ν, that is, ω = 1, we write s for s/1, whenever s ∈ R.

VII.1 Classical Spaces

85

Since |||·|||m/ν,q ≥ |||·|||0/ν,q = k·kq , this definition is meaningful and the second 0/ν

embedding in (1.1.3) implies Wq

(X, E) = Lq (X, E). Hence, by (1.2.1),

d

d

d

S(X, E) ,→ Wqm/ν (X, E) ,→ Lq (X, E) ,→ S 0 (X, E) .

(1.2.3)

By H¨older’s inequality, Z 0

hu , uiLr :=

X

­ 0 ® u (x), u(x) E dx

is well-defined for (u0 , u) ∈ Lr0 (X, E 0 ) × Lr (X, E) and is called Lr duality pairing. Since X and X∗ differ by a d-dimensional Lebesgue null set only, Lr (X, E) = Lr (X∗ , E) ,

(1.2.4)

using obvious identifications.4 Thus (VI.1.3.2) and (1.2.1) imply hu0 , uiS(X∗ ,E) = hu0 , uiLr if

either (u0 , u) ∈ S(X, E 0 ) × Lr (X, E) and r < ∞ , or

(u0 , u) ∈ Lr0 (X, E 0 ) × S(X∗ , E) and r > 1 .

From this and the density of S in S 0 and in Lq it follows that h·, ·iS(X∗ ,E) is uniquely determined by h·, ·iLr , and vice versa. For this reason we simply write h·, ·i for either of them and call it duality pairing, respectively Lr or X distributional duality pairing, according to the context, if clarity requires it. This is consistent with (1.1.27). For the reader’s convenience we recall the following fundamental duality result. 1.2.1 Theorem Suppose E is reflexive or E 0 is separable. Then Lq (X, E)0 = Lq0 (X, E 0 ) ,

1≤q 1, then Lq (X, E) is reflexive iff E is reflexive. Proof (1) The first assertion follows from [DU77, Theorem 1 on p. 98, Theorem 1 on p. 79, Corollary 4 on p. 82] and by the usual procedure for extending results from finite measure spaces to the σ-finite case (e.g., [HeS65, Proof of Thorem 20.19]. Also see [KuJF77, Section 2.2] and [GaGZ74, Satz IV.1.14] for special cases. 4 Recall that the elements of L are equivalence classes of measurable functions differing by r null sets only.

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VII Function Spaces

(2) Let q > 1. If E is reflexive, then the reflexivity of Lq (X, E) is a consequence of (1). The converse assertion is implied by the isometry of E to the closed linear subspace { ϕe ; e ∈ E } of Lq (X, E), where ϕ ∈ Lq (X) satisfies kϕkq = 1, and the fact that a closed linear subspace of a reflexive space is reflexive. ¥ 1.2.2 Remark It should be noted that this theorem remains true if X is replaced by an arbitrary σ-compact metrizable space and the Lebesgue measure by a positive Radon measure. ¥ Sobolev spaces of negative order are defined ‘by duality’:

Suppose m ∈ νN and E is reflexive. Then ¡ m/ν ¢0 Wp−m/ν (X, E) := Wp0 (X∗ , E 0 ) , 1 0 be given. We fix T ≥ 1 such that Z ∞ 2 kuk∞ |h` (t)| dt < ε/2 . T

Then we choose δ > 0 such that Z T ¯ ¡ ck ¢ − u¡−ty k , yck ¢¯¯ dt < ε/2 |h` (t)| ¯u −txk , x E 0

for (x, y) ∈ (−∞, 0) × ∂k K k

with

|x − y| < δ .

(1.3.10)

Consequently, |ε`k u(x) − ε`k u(y)|E < ε if (1.3.10) is satisfied. From this and the arguments used in steps (2) and (3) it follows that EK u and RK u are uniformly continuous if u is so. Now we easily derive that the assertion holds if X = BUC m/ν .

90

VII Function Spaces

¡ ¢ (7) By Theorem VI.1.2.3, (RK , EK ) is an r-e pair for S(Rd , E), S(K, E) . m/ν From this and step (6) we get the claim for X = C0 . (8) Suppose E is reflexive and q > 1. By the above, we know that (RK∗ , EK∗ ) ¡ m/ν ¢ m/ν is an r-e pair for Wq0 (Rd , E 0 ), Wq0 (K∗ , E 0 ) . Thus Remark VI.1.2.1(c) guar¡ ¢ ¡ −m/ν d ¢ −m/ν antees that (EK∗ )0 , (RR∗ )0 is an r-e pair for Wq (R , E), Wq (K, E) , −m/ν due to (1.2.5) and E 00 = E. Now the claim for X = Wq follows from Corollary VI.1.3.2. m/ν

(9) Lastly, by replacing Wq0 applies to X =

m/ν

in step (8) by C0

, we see that the assertion

−m/ν C0 .¥

1.3.2 Corollary (i) RK u = RK u for u ∈ EK X (K, E). ¡ ¢ (ii) (RK , EK ) is an r-e pair for BC ∞ (Rd , E), BC ∞ (K, E) . Proof The first assertion is immediate from Remark VI.1.2.4(b) and the second one is obvious. ¥

1.4

Distributional Derivatives

q Let m ∈ ν N. For clarity, we temporarily denote the X-distributional derivative of α order α of u ∈ S 0 (X, E) by ∂w u. m/ν Suppose u ∈ BC (X, E) and α q ω ≤ m. It follows from (VI.1.1.2) and (1.1.6) that h∂ α u, ϕiX = (−1)|α| hu, ∂ α ϕiX ,

ϕ ∈ S(X∗ , E) .

Hence α ∂ α u = ∂w u,

u ∈ BC m/ν (X, E) ,

α qω ≤ m .

m/ν

(1.4.1)

Assume u ∈ Wq (X, E). Definition (1.2.2) guarantees the existence of a sequence (uj ) in S(X, E) such that (∂ α uj ) converges for α q ω ≤ m in Lq (X, E) towards some uα , where u0 = u. We set ∂sα u := uα and call it strong (X-)derivative of u of order α. The construction of the completion implies that ∂sα u is uniquely determined by u, independently of the particular sequence (uj ). α Since ∂w is an extension of the classical derivative on S(X, E), it holds h∂ α uj , ϕiX = (−1)|α| huj , ∂ α ϕiX ,

ϕ ∈ S(X∗ , E) .

Thus, letting j → ∞, we get h∂sα u, ϕiX = (−1)|α| hu, ∂ α ϕiX ,

ϕ ∈ S(X∗ , E) .

VII.1 Classical Spaces

91

Hence α ∂sα u = ∂w u,

u ∈ Wqm/ν (X, E) ,

α qω ≤ m .

(1.4.2)

The following theorem shows that, conversely, the elements of BUC m/ν (X, E), m/ν respectively Wq (X, E), can be characterized by their distributional derivatives. 1.4.1 Theorem Let K be closed in Rd , X ∈ {Rd , K }, and m ∈ νN. (i) Suppose X ∈ {BC, BUC}. Then u ∈ X m/ν (X, E)

α ∂w u ∈ X (X, E) for α q ω ≤ m .

iff

α Moreover, ∂ α u = ∂w u. m/ν

(ii) Let 1 ≤ q < ∞. Then u ∈ Wq α Furthermore, ∂sα u = ∂w u.

α (X, E) iff ∂w u ∈ Lq (X, E) for α q ω ≤ m.

α Proof (1) It follows from (1.4.1) that u ∈ X m/ν (X, E) implies ∂w u ∈ X (X, E) α α q and ∂w u = ∂ u for α ω ≤ m. m/ν

Similarly, (1.4.2) implies that it is a consequence of u ∈ Wq (X, E) that α α ∂w u ∈ Lq (X, E) and ∂w u = ∂sα u for α q ω ≤ m. (2) Assume X = Rd . Let u ∈ X (Rd , E) be such that v := ∂d,w u, the distributional derivative of u with respect to the last coordinate, also belongs to X (Rd , E). Fix t0 ∈ R and set Z t w(y, t) := v(y, τ ) dτ , (y, t) ∈ Rd−1 × R = Rd . t0

Then

w ∈ C(Rd , E) ,→ L1,loc (Rd , E) ,→ D0 (Rd , E)

and

Z hw, ∂d ϕiRd =

Z Z Rd−1

Z

t

v(y, τ ) dτ ∂d ϕ(y, t) dt dy R

Z

t0

=− Rd−1

R

v(y, t)ϕ(y, t) dt dy = −hv, ϕiRd

for ϕ ∈ D(Rd ). This shows that the distributional derivative of w with respect to the last coordinate equals v. Hence ∂d (u − w) = v − v = 0. Thus it follows from Example 1.8.6(b) of the Appendix that u − w = u1 ⊗ 1 for some u1 ∈ D0 (Rd−1 , E). Since u and w are continuous, u1 ∈ C(Rd−1 , E). Observe that u = w + u1 ⊗ 1 is point-wise differentiable with respect to xd and ∂d u = ∂d w = v ∈ X (Rd , E). From this we obtain, by permutation of coordinates and induction, that α ∂w u ∈ X (Rd , E) for α ∈ Nd with α q ω ≤ m implies u ∈ X m/ν (Rd , E). α (3) Suppose X = Rd . Let u ∈ Lq¡(Rd , E) satisfy ∂w u ∈ Lq (Rd , E) for α ∈ Nd ¢ ∞ d q with α ω ≤ m. We choose ψ ∈ C R , [0, 1] such that ψ(x) = 1 for |x| ≤ 1,

92

VII Function Spaces

and ψ(x) = 0 for |x| ≥ 2. For r ∈ (0, 1] and x ∈ Rd we put ψr (x) := ψ(rx). Then ψr (x) = 1 for |x| ≤ 1/r, and ψr (x) = 0 for |x| ≥ 2/r. Moreover, ∂ γ ψr = r|γ| (∂ γ ψ)r implies the estimates k∂ γ ψr k∞ ≤ k∂ γ ψk∞ for 0 < r ≤ 1 and γ ∈ Nd . By Leibniz’ rule X³α ´ α β ∂w (ψr u) = ∂ α−β ψr ∂w u. (1.4.3) β β≤α

Consequently, α qω ≤ m ,

α β k∂w (ψr u)kq ≤ c max k∂w ukq , β≤α

0 0 be given and fix r ∈ (0, 1) with w

|||ψr u − u|||m/ν,q < ε/2 .

(1.4.6)

Assume { ϕξ ; ξ > 0 } is a mollifier. Since ψr u has compact support, ϕξ ∗ (ψr u) too has compact support. Hence ϕξ ∗ (ψr u) ∈ D(Rd , E) ,→ S(Rd , E) ,

ξ>0,

by (III.4.2.13). Furthermore, by (III.4.2.10) and (III.4.2.23), (III.4.2.24), ¡ ¢ α α ∂ α ϕξ ∗ (ψr u) = ϕξ ∗ ∂w (ψr u) −−→ ∂w (ψr u) , α qω ≤ m , ξ→0

in Lq (Rd , E). This shows that there exists η > 0 such that w

|||ϕη ∗ (ψr u) − ψr u|||m/ν,q < ε/2 . From this and (1.4.6) it follows that there is a sequence (uj ) in S(Rd , E) such w that |||uj − u|||m/ν,q → 0 as j → ∞. Thus (uj ) is a Cauchy sequence in the norm m/ν α |||·|||m/ν,q , that is, in Wq (Rd , E), and ∂ α uj → ∂w u in Lq (Rd , E) for α q ω ≤ m. α α q This proves ∂w u = ∂s u for α ω ≤ m.

VII.1 Classical Spaces

93

(4) Suppose X = K and X ∈ {BC, BUC, Lq }. Set © ª α X m (K, E) := u ∈ X (K, E) ; ∂w u ∈ X (K, E), α q ω ≤ m . ˚ it is clear that EK∗ ∂ α ϕ = ∂ α ϕ for ϕ ∈ S(K∗ ) and α ∈ Nd , provided Since K∗ = K, we identify ϕ with its trivial extension over Rd . Let u ∈ X m (K, E) and ϕ ∈ S(K∗ ). Then, if α q ω ≤ m, α h∂w u, ϕiK = (−1)|α| hu, ∂ α ϕiK = (−1)|α| hRK EK u, ∂ α ϕiK .

Due to Lemma VI.1.2.2 and the above observation, the last term equals α (−1)|α| hEK u, EK∗ ∂ α ϕiK = (−1)|α| hEK u, ∂ α ϕiRd = h∂w EK u, ϕiRd . α By steps (2) and (3), we know that ∂w EK u equals ∂ α EK u if X ∈ {BC, BUC}, α α and ∂w EK u = ∂s EK u for X = Lq . Writing ∂ = ∂ if X ∈ {BC, BUC}, and ∂ = ∂s if X = Lq , we see that ­ ® ­ ® ­ ® α h∂w u, ϕiK = ∂ α EK u, ϕ Rd = ∂ α EK u, EK∗ ϕ Rd = RK ∂ α EK u, ϕ K .

Since K is closed, (1.4.1), (1.4.2), and Theorem VI.1.2.3(iii) guarantee that RK ∂ α E K u = ∂ α RK E K u = ∂ α u . Consequently, ­ ® α h∂w u, ϕi = ∂ α u, ϕ ,

ϕ ∈ S(K∗ ) ,

α qω ≤ m .

α Thus ∂w u = ∂ α u for α q ω ≤ m and u ∈ X (K, E). This proves the theorem.

¥

Henceforth, we write again ∂ α for the X-distributional derivatives.

1.5

Reflexivity

In this subsection we extend some well-known facts from the theory of classical Sobolev spaces to the present setting. m/ν

1.5.1 Theorem Let E be reflexive, 1 < p < ∞, and m ∈ νZ. Then Wp reflexive. Proof

(X, E) is

Since the dual of a reflexive space is reflexive, we can assume m ∈ νN.

We endow { α ∈ Nd ; α q ω ≤ m } with the lexicographical ordering, denote ¡ ¢N its cardinality by N , and equip Z := Lp (X, E) with the `1 norm. It follows from Theorem 1.2.1 and the fact that products of reflexive spaces are reflexive that Z is reflexive. The map u 7→ (∂ α u)α p ω≤m is an isometry from the dense

94

VII Function Spaces m/ν

linear subspace S(X, E) of Wp (X, E) into Z. Hence it has a unique isometric m/ν m/ν extension S over Wp (X, E). Since Wp (X, E) is complete, im(S) is a closed m/ν linear subspace of Z. Consequently, Wp (X, E) is reflexive, being isomorphic to a closed linear subspace of a reflexive Banach space. ¥ For the reader’s convenience we recall some basic facts from linear functional analysis. Let Y be a subset of some Banach space X. Then © ª Y ⊥ := x0 ∈ X 0 ; hx0 , yiX = 0, y ∈ Y , the annihilator, or polar set, of Y (in X 0 ), is a closed linear subspace of X 0 . 1.5.2 Lemma Suppose Y is a linear subspace of a Banach space X. Then f : X 0 /Y ⊥ → Y 0 ,

x0 + Y ⊥ 7→ x0 | Y

is an isometric isomorphism. Proof Let z 0 ∈ x0 + Y ⊥ . Then z 0 − x0 ∈ Y ⊥ . Hence (z 0 − x0 ) | Y = 0. Thus f is a well-defined linear map. Given y 0 ∈ Y 0 , by the Hahn-Banach theorem there is an x0 ∈ X 0 with x0 ⊃ y 0 . Consequently, f (x0 + Y ⊥ ) = x0 | Y = y 0 . This shows that f is surjective. For any z 0 ∈ x0 + Y ⊥ it holds kf (x0 + Y ⊥ )kY 0 = kz 0 | Y kY 0 ≤ kz 0 kX 0 . Thus

kf (x0 + Y ⊥ )kY 0 ≤

inf

z 0 ∈x0 +Y ⊥

kz 0 kX 0 = kx0 + Y ⊥ kX 0 /Y ⊥ .

On the other hand, the Hahn-Banach theorem guarantees the existence of v 0 ∈ X 0 with v 0 ⊃ x0 | Y and kv 0 kX 0 = kx0 | Y kY 0 . Hence kf (x0 + Y ⊥ )kY 0 = kx0 | Y kY 0 = kv 0 kX 0 ≥ kx0 + Y ⊥ kX 0 /Y ⊥ . Therefore f is an isometry. In particular, it is injective.

¥

Now we can give an explicit description of Sobolev spaces of negative order. q 1.5.3 Theorem Let E be reflexive, m ∈ ν N, and 1 < p < ∞. Then u belongs to −m/ν Wp (X, E) iff there exist uα ∈ Lp (X, E) such that X u= (−1)|α| ∂ α uα . (1.5.1) p α ω≤m Then kukW −m/ν (X,E) = inf max kuα kp ∼ inf p α p ω≤m

X α p ω≤m

kuα kp ,

(1.5.2)

VII.1 Classical Spaces

95

the infimum being taken over all representations of u of the form (1.5.1). Furthermore, X m/ν hv, ui = h∂ α v, uα i , v ∈ Wp0 (X∗ , E 0 ) , (1.5.3) α p ω≤m for any such representation. Proof We use the notation of the proof of Theorem 1.5.1 and write X for the ¡ ¢N Banach space Lp0 (X∗ , E 0 ) . By Y we mean the closed linear subspace thereof which is the image of the isometry m/ν

T : Wp0

(X∗ , E 0 ) → X ,

u 7→ (∂ α u)α p ω≤m .

It follows that ¡ m/ν ¢0 T 0 : Y 0 → Wp0 (X∗ , E 0 ) = Wp−m/ν (X, E) is an isometric isomorphism. By Lemma 1.5.2, f : X 0 /Y ⊥ → Y 0 ,

v + Y ⊥ 7→ v |Y

is an isometric isomorphism, where ³¡ ´ ¡ ¢N ¢N . v = (vα ) ∈ X 0 = Lp (X, E) , k·k`∞ (Lp ) = Lp (X, E) , ¡ ¢ with `∞ (Lp ) := `∞ {1, . . . , N }, Lp (X, E) (cf. Theorem VI.2.1.1). Hence u belongs −m/ν to Wp (X, E) iff there exists u + Y ⊥ ∈ X 0 /Y ⊥ such that u = T 0 f (u + Y ⊥ ) = T 0 (u |Y) .

(1.5.4)

If v ∈ S(X∗ , E 0 ), then ­ ® hu, vi = hv, uiW −m/ν (X,E) = v, T 0 (u | Y) W −m/ν (X,E) p p X = hT v, uiX 0 = hu, T viX = huα , ∂ α vi . α p ω≤m Assume v = ϕ ⊗ e0 ∈ S(X∗ ) ⊗ E 0 . Then ­ ® huα , ∂ α vi = huα , ∂ α ϕiX∗ , e0 E 0 ­ ® = (−1)|α| h∂ α uα , ϕiX∗ , e0 E 0 = (−1)|α| h∂ α uα , vi

(1.5.5)

d by (VI.1.2.3). Theorem 1.3.6 of the Appendix guarantees that ) ⊗ E 0 is dense ¡ S(R ¢ d 0 d in S(R , E ). This, the fact that (RK , EK ) is an r-e pair for S(R , E 0 ), S(K∗ , E 0 ) ,

96

VII Function Spaces

and Lemma VI.1.1.2 imply that S(K∗ ) ⊗ E 0 is dense in S(K∗ , E 0 ). Hence we get from (1.5.5), Lp (X, E) ,→ S 0 (X, E), and (VI.1.2.6) that D X E X hu, vi = huα , ∂ α vi = (−1)|α| ∂ α uα , v , v ∈ S(X∗ , E 0 ) . (1.5.6) p p α ω≤m α ω≤m m/ν

Now the density of S(X∗ , E 0 ) in Wp0 (X∗ , E 0 ) implies that (1.5.1) and (1.5.3) ¡ ¢N hold for this particular choice of u ∈ Lp (X, E) . We deduce from (1.5.4) and the isometry of T 0 and f that kukW −m/ν (X,E) = kf (u + Y ⊥ )kY 0 = ku + Y ⊥ kX 0 /Y ⊥ . p

¡ ¢N Note that v ∈ Y ⊥ iff v ∈ X 0 = Lp (X, E) and D E X X 0 = hv, T viX 0 = h∂ α v, vα i = v, (−1)|α| ∂ α vα α p ω≤m α p ω≤m for v ∈ S(X∗ , E 0 ). Now the assertions follow from (1.5.5)–(1.5.8).

1.6

(1.5.7)

(1.5.8)

¥

Embeddings

In this short subsection we present some easy embedding theorems for Sobolev spaces. More sophisticated ones will be given in later sections (cf. Theorems 2.6.5, 2.6.6, 4.1.4, 4.3.2, and 5.6.5). Let E0 and E1 be Banach spaces with E1 ,→ E0 . Suppose X ∈ {BC, BUC, C0 , Wq } .

(1.6.1)

Then X m1 /ν (X, E1 ) ,→ X m0 /ν (X, E0 ) ,

m0 , m1 ∈ νN ,

m1 ≥ m0 .

(1.6.2)

Indeed, this is clear if X = Rd . Otherwise, it follows from this fact and Theorem 1.3.1. Next we suppose that (E0 , E1 ) is a densely injected Banach couple, that is, d

E1 ,→ E0 . Then we prove the density of embedding (1.6.2) if X 6= BC. For this we need the following observation. d

d

1.6.1 Lemma If E1 ,→ E0 , then S(X, E1 ) ,→ S(X, E0 ). Proof (1) It is clear that S(X, E1 ) ,→ S(X, E0 ). By Corollary V.2.4.2 and embeddings (III.4.1.5) we know d

d

D(Rd , E1 ) ,→ D(Rd , E0 ) ,→ S(Rd , E0 ) . Now the assertion is clear if X = Rd (cf. (1.3.15) and (1.3.16) of the Appendix).

VII.1 Classical Spaces

97

(2) Let X = K. Then the claim follows from step (1), Theorem VI.1.2.3, and Lemma VI.1.1.2. ¥ The following density statements are now easily obtained from this lemma. 1.6.2 Theorem Let (E0 , E1 ) be a densely injected Banach couple and m0 , m1 ∈ νZ with m1 ≥ m0 , and 1 ≤ q < ∞. Then 5 d

Wqm1 /ν (X, E1 ) ,→ Wqm0 /ν (X, E0 )

(1.6.3)

and m1 /ν

C0

d

m0 /ν

(X, E1 ) ,→ C0

(X, E0 ) ,

provided m0 ≥ 0. If E0 and E1 are reflexive, then (1.6.3) holds for m1 ≤ 0 also. Proof (1) If m0 ≥ 0, then the assertions follow from (1.6.2), Lemma 1.6.1, and definitions (1.1.10) and (1.2.2). (2) Suppose E0 and E1 are reflexive and m1 ≤ 0. Recall d

E1 ,→ E0

d

E00 ,→ E10

=⇒

(1.6.4)

(cf. Proposition V.1.4.8). From this and step (1) we get −m0 /ν

Wq0

d

−m1 /ν

(X∗ , E00 ) ,→ Wq0

(X∗ , E10 ) .

Since these spaces are reflexive, we obtain the second statement, due to definition (1.2.5), by invoking (1.6.4) once more. ¥ d

1.6.3 Remark If E1 ,→ E0 and E0 and E1 are reflexive, then −m1 /ν

C0

−m0 /ν

(X, E1 ) ,→ C0

(X, E0 )

for m0 , m1 ∈ νN with m0 ≥ m1 . Proof This follows from definition (1.1.10) and the arguments of step (2) of the preceding proof. However, since the C0 spaces are not reflexive, we get the desired embedding from (1.1.20). ¥ Our next theorem guarantees that the elements of BUC m/ν (X, E) can be approximated by smooth functions. 5 See

Theorem 5.6.5 below for an important improvement of this embedding.

98

VII Function Spaces d

1.6.4 Theorem If m ∈ νN, then BC ∞ (X, E) ,→ BUC m/ν (X, E). Proof

The continuity of this embedding follows from (1.1.9).

(1) If X = Rd , then the assertion is obtained by a standard mollification argument (e.g., [AmE08, Theorem X.7.11] and Subsections 1.2 and 1.3 of the Appendix). (2) Let X = K. Then we get the claim by invoking step (1), Theorem 1.3.1 and its corollary, and Lemma VI.1.1.2. ¥

1.7

Notes

Theorem 1.3.1 is an amplification and sharpening of Theorem 4.4.3(i) of [Ama09]. Part (2) of the proof of Theorem 1.4.1 follows L. Schwartz [Schw66, Th´eor`eme VII] (also see L. H¨ormander [H¨or83, Theorem 3.1.7]). In the isotropic scalar-valued setting, the fact that strong and weak derivatives of Sobolev functions coincide, has first been proved by N.G. Meyers and J. Serrin [MeS64]. The remaining results of this section are straightforward extensions of classical results for scalar-valued isotropic spaces.

VII.2 Besov Spaces

2

99

Besov Spaces

This and the next section contain a comprehensive exposition of the theory of anisotropic Banach-space-valued Besov spaces on corners. It extends the Fourier analytic approach from the isotropic scalar-valued case on Euclidean spaces to the present setting. After having introduced the somewhat technical definition of Besov spaces, we present, in the second subsection, the basic embedding theorems for these scales of Banach spaces. In addition, we put some emphasis on little Besov spaces, which will play an important role in connection with differential equations. In Subsection 2.3 we prove a duality theorem for anisotropic Besov spaces, which, under the assumption that the target space E is reflexive or has a separable dual, is the perfect analogue of the duality theorem for isotropic scalar-valued Besov spaces. The next subsection contains the fundamental lifting and Fourier multiplier theorems. On the basis of these results we derive, in Subsection 2.6, useful renorming theorems and clarify the relations between Besov spaces and the classical spaces studied in Section 1. Subsection 2.7 contains the interpolation properties enjoyed by Besov spaces. Everything mentioned above is derived for Besov spaces on the ‘full’ Euclidean space. In the last subsection we provide extensions to ‘corner spaces’ by building on the restriction-extension results of Section VI.1 and Subsection 1.3. Throughout this section •

s∈R,

1 ≤ q, r ≤ ∞ .

(2.0.1)

We simply write X for X (Rd , E) if the latter is a vector subspace of S 0 = S 0 (Rd , E), provided no confusion seems likely. We also remind the reader that assumption (VI.3.1.20) holds throughout.

2.1

The Definition

We suppose X ∈ {BUC, C0 , Lq } . We fix¡ a ν-admissible 0-neighborhood Ω of Rd and an Ω-adapted ψ ∈ D(Rd ). ¢ Then (Ωk ), (ψk ) is the ν-dyadic partition of unity on Rd induced by (Ω, ψ). By applying Lemma VI.3.6.3 with E1 = E2 = E3 := C and a = 1 = 1Rd , we get ψk (D) ∈ L(X ) ,

kψk (D)kL(X ) ≤ c ,

(2.1.1)

kψk (D)kL(L∞ ,BUC) ≤ c

(2.1.2)

and ψk (D)(L∞ ) ⊂ BUC , for k ∈ N.

100

VII Function Spaces

Preliminary Estimates We introduce a vector subspace of S 0 by © ª 0 s `s/ν r X := u ∈ S ; (ψk (D)u) ∈ `r (X )

(2.1.3)

and endow it with the norm °¡ ¢° u 7→ kuk`s/ν X := ° ψk (D)u °`s (X ) . r

r

Moreover, s/ν

c0 X :=

¡©

ª ¢ u ∈ S 0 ; (ψk (D)u) ∈ cs0 (X ) , k·k`s/ν X . ∞

(2.1.4)

Of course, these definitions depend on the choice of (Ω, ψ). The next lemma shows, however, that another choice of these quantities leads to the same spaces, except for equivalent norms. 2.1.1 Lemma Let Ωi be a ν-admissible of Rd and assume ψ i is ¡ i 0-neighborhood ¢ i i Ω -adapted for i = 1, 2. Denote by (Ωk ), (ψk ) the ν-dyadic partition of unity s/ν,i s/ν,i on Rd induced by (Ωi , ψ i ). Write `r X , resp. c0 X , if (ψk ) in (2.1.3), resp. in (2.1.4), is replaced by (ψki ). Then . `s/ν,1 X = `s/ν,2 X r r Proof

and

s/ν,1

c0

. s/ν,2 X = c0 X .

We can fix m ∈ N with 2−m q Ω1 ⊂ Ω2 ⊂ 2m q Ω1 . Then (recall (VI.3.6.1)) ² ¡ ¢ Σ2 = 2 q Ω2 \2−1 q Ω2 ⊂ 2 q (2m q Ω1 ) 2−1 q 2−m q Ω1 ² = 2m q (2 q Ω1 ) 2−m q (2−1 q Ω1 ) .

We put, for k ∈ N, m+1 X

χ1k,m :=

1 ψk+i ,

ψj1 := 0 for j < 0 .

i=−m−1

From (VI.3.6.3) we infer that χ1k,m =

k+m+1 X j=0

ψj1 −

k−m−2 X

ψj1 = σ2−(k+m+1) ψ 1 − σ2−(k−m−2) ψ 1 .

j=0

Note that ψ 1 (2−(k+m+1) q ξ) = 1

if

2−(k+m+1) q ξ ∈ (3/2) q Ω1 ,

(2.1.5)

VII.2 Besov Spaces

101

that is, if ξ ∈ 2k+m q (3 q Ω1 ). Thus, in particular, σ2−(k+m+1) ψ 1 (ξ) = 1

¡ ¢ ξ ∈ 2k q 2m q (2 q Ω1 ) .

if

Furthermore, ψ 1 (2−(k−m−2) q ξ) = 0 if 2−(k−m−2) q ξ ∈ / 2 q Ω1 , that is, if ¡ ¢ ξ∈ / 2k−m−1 q Ω1 = 2k q 2−m q (2−1 q Ω1 ) . This shows that χ1k,m (ξ) = 1

if

¡ ² ¢ ξ ∈ 2k q (2m q (2 q Ω1 )) 2−m q (2−1 q Ω1 )

for k ≥ 1, and χ10,m (ξ) = 1 if ξ ∈ 2m q Ω1 . Thus, by Ω2 ⊂ 2m q Ω1 , and since (2.1.5) implies ¡ ² ¢ Ω2k = 2k q Σ2 ⊂ 2k q (2m q (2 q Ω1 )) 2−m q (2−1 q Ω1 ) , we obtain χ1k,m | Ω2k = 1 for k ∈ N. From this and supp(ψk2 ) ⊂ Ω2k we get ψk2 = ψk2 χ1k,m ,

k∈N.

(2.1.6)

Now it follows from (2.1.1) that m+1 X

kψk2 (D)ukX ≤ c kχ1k,m (D)ukX ≤ c

1 kψk+i (D)ukX

i=−m−1

for u ∈ X and k ∈ N. This implies m+1 X

2ks kψk2 (D)ukX ≤ c 2(m+1) |s|

1 2(k+i)s kψk+i (D)ukX .

i=−m−1

Consequently, °¡ 2 ¢° ° ψk (D)u ° s/ν,1

`sr (X )

°¡ ¢° ≤ c(m, s) ° ψk1 (D)u °`s (X ) , r

s/ν,2

that X¢ ,→ `r¡ X . Now¢ we obtain the assertion by interchanging the roles ¡ is, `r of (Ω1k ), (ψk1 ) and (Ω2k ), (ψk2 ) . ¥ 2.1.2 Corollary Set χk := ψk−1 + ψk + ψk+1 ,

ψ−1 := 0 ,

k∈N.

Then ψk χk = χk ψk = ψk for k ∈ N. Proof This follows from (2.1.6) if Ωi := Ω and ψ i := ψ for i = 1, 2, since then m = 0. ¥

102

VII Function Spaces

A Retraction-Coretraction Pair We introduce linear maps R : `sr (X ) → S 0 , and

(vk ) 7→

P

k χk (D)vk

¡ ¢ u 7→ ψk (D)u .

s Rc : `s/ν r X → `r (X ) ,

The following lemma shows, in particular, that they are well-defined. 2.1.3 Lemma s/ν (i) `r X is a Banach space and (R, Rc ) is a retraction-coretraction pair for ¡ s ¡ s/ν ¢ s/ν ¢ `r (X ), `r X and for cs0 (X ), c0 X . s/ν

s/ν

(ii) `r L∞ = `r BUC. s/ν

s/ν

(iii) `r C0 is a closed linear subspace of `r BUC. s/ν

s/ν

(iv) c0 X is a closed linear subspace of `∞ X . Proof (1) Suppose v = (vk ) ∈ `sr (X ) and 0 ≤ m < n. By the evenness of ψk (cf. (III.4.2.2)), n DX

E χk (D)vk , ϕ

k=m

Rd

=

n X

hvk , χk (D)ϕiRd ,

ϕ ∈ S(Rd ) .

(2.1.7)

k=m

Fix t < s. Then, by Corollary 2.1.2 and (ψj ψk )(D) = ψj (D)ψk (D), n n+1 ¯D X E ¯ X °¡ ¢° ¯ ¯ χk (D)vk , ϕ ≤3 k2tk vk kX ° ψk (D)ϕ °`−t . ¯ ¯ d ∞ R

k=m

E

(2.1.8)

k=m−1

By Lemma VI.3.6.3(iii), there exists ` ∈ N such that °¡ ¢° ° ψk (D)ϕ ° −t ≤ c q`,` (ϕ) , ϕ ∈ S(Rd ) . ` ∞

Using this, we deduce from (2.1.8) n n+1 ¯D X E ¯ X ¯ ¯ χk (D)vk , ϕ d ¯ ≤ c q`,` (ϕ) k2tk vk kX , ¯ k=m

R

E

ϕ ∈ S(Rd ) .

(2.1.9)

k=m−1

Since `sr (X ) ,→ `t∞ (X )

(2.1.10) ¡­Pn ® ¢ by (VI.2.2.5), we infer from v ∈ `sr (X ) and (2.1.9) that k=0 χk (D)vk , ϕ Rd is a Cauchy sequence in E, uniformly with respect to ϕ in bounded subsets of S(Rd ).

VII.2 Besov Spaces

103

¡ Pn ¢ ¡ ¢ d Thus ), E = S 0 . By Proposik=0 χk (D)vk is a Cauchy sequence in L S(RP tion VI.1.2.5 we know that S 0 is complete. Hence Rv = k χk (D)vk exists in S 0 . From (2.1.9) and (2.1.10) we get ϕ ∈ S(Rd ) .

|hRv, ϕiRd |E ≤ c q`,` (ϕ) kvk`t∞ (X ) ≤ c q`,` (ϕ) kvk`sr (X ) , This implies ¡ ¢ R ∈ L `sr (X ), S 0 .

(2.1.11)

¡ ¢ s Rc ∈ L `s/ν r X , `r (X ) .

(2.1.12)

It is obvious that

If u ∈ S 0 , then ψk (D)u ∈ S 0 and χk (D)ψk (D)u ∈ S 0 by Lemma VI.3.6.3(ii). Hence we get from Corollary 2.1.2 χk (D)ψk (D)u = F −1 χk FF −1 ψk F u = F −1 χk ψk F u = ψk (D)u . s/ν

Thus, given u ∈ `r X , (2.1.11) and (2.1.12) imply RRc u = lim

n→∞

n X

χk (D)ψk (D)u = lim

n→∞

k=0

n X

ψk (D)u = u

in S 0 ,

(2.1.13)

k=0

due to Lemma VI.3.6.2. (2) Note supp(ψj ) ∩ supp(χk ) = ∅ if |j − k| ≥ 3. Hence, using (2.1.11) and Lemma VI.3.6.3(ii), ψj (D)Rv = lim ψj (D) n→∞

n X

χk (D)vk =

k=0

j+2 X

ψj (D)χk (D)vk

k=j−2

for j ∈ N and v ∈ `sr (X ). From this and (2.1.1) we get kψj (D)RvkX ≤ c

j+2 X

kvk kX ,

j∈N,

v ∈ `sr (X ) .

(2.1.14)

k=j−2

This implies

°¡ ¢° kRvk`s/ν X = ° ψk (D)Rv °`s (X ) ≤ c kvk`sr (X ) , r

r

that is, ¡ ¢ R ∈ L `sr (X ), `s/ν r X .

(2.1.15)

Together with (2.1.11)–(2.1.13) this shows that (R, Rc ) is a retraction-coretraction ¡ s/ν ¢ pair for `sr (X ), `r X .

104

VII Function Spaces

(3) Suppose v ∈ `s∞ (X ) and 2sk kvk kX → 0 as k → ∞. Then we see from ¡ s/ν ¢ (2.1.14) that 2sj kψj (D)RvkX → 0 as j → ∞. Thus R ∈ L cs0 (X ), c0 X , due to (2.1.15) with r = ∞. By (2.1.12) and the definition of Rc , it is obvious that Rc be¡ s/ν ¢ longs to L c0 X , cs0 (X ) . Hence, taking (2.1.13) into consideration, we see that ¡ s/ν ¢ (R, Rc ) is a retraction-coretraction pair for cs0 (X ), c0 X . ¡ s/ν ¢ (4) Since (R, Rc ) is a retraction-coretraction pair for `sr (X ), `r X , we know s/ν from Lemma I.2.3.1 that `r X is isomorphic to a closed linear subspace of the s/ν Banach space `sr (X ). Hence it is complete. Analogously, we see that c0 X is complete. s/ν

s/ν

s/ν

(5) Clearly, `r BUC is a linear subspace of `r L∞ . Given u ∈ `r L∞ , we know ψk (D)u ∈ L∞ for k ∈ N. Hence χk (D)u ∈ L∞ for k ∈ N. Thus, by (2.1.2), ψk (D)u = ψk (D)χk (D)u ∈ BUC ,

k∈N.

(2.1.16)

s/ν

This implies u ∈ `r BUC and proves (ii). s/ν

s/ν

(6) The fact that `r C0 is a closed linear subspace of `r BUC follows, since it is complete. The last assertion is obvious. ¥ It should be noted that (R, Rc ) is a universal retraction-coretraction pair, that is, these operators are independent of s and r. The Final Definition We define (anisotropic) Besov spaces (of E-valued distributions) on Rd by s/ν s/ν Bq,r = Bq,r (Rd , E) := `s/ν r Lq .

Thus s/ν u ∈ Bq,r

iff

u ∈ S0

and

¡

(2.1.17)

¢ 2ks ψk (D)u ∈ `r (Lq ) .

Furthermore, °¡ ¢° kukB s/ν = ° 2ks kψk (D)ukq °` . r

q,r

(2.1.18)

s/ν

Observe that, by Lemma 2.1.1, Bq,r (Rd , E) is independent of the particular choice of (Ω, ψ), except for equivalent norms.

2.2

Embedding Theorems

The following elementary observation is the basis for the proof of the subsequent and many more embedding theorems.

VII.2 Besov Spaces

105

2.2.1 Lemma Let Xi and Yi be LCSs and suppose (r, rc ) is an r-c pair for (X1 , Y1 ) d

and (X2 , Y2 ). If X1 ,→ X2 , then Y1 ,→ Y2 , with dense embedding if X1 ,→ X2 . Proof

Let iX : X1 ,→ X2 . Then iY := r ◦ iX ◦ rc ∈ L(Y1 , Y2 ) ,

¡ ¢ iY (y) = r rc (y) = y ,

y ∈ Y1 .

Hence iY : Y1 ,→ Y2 . If X1 is dense in X2 , then Y1 = r(X1 ) is dense in Y2 = r(X2 ) by Lemma VI.1.1.2. ¥ By means of this lemma we can now easily derive most assertions of the adjacent basic embedding theorem for Besov spaces. 2.2.2 Theorem Suppose 1 ≤ r0 , r1 ≤ ∞ and −∞ < s0 < s1 < ∞. Then d

s/ν s/ν S ,→ Bq,r ,→ Bq,r ,→ S 0 , 1 0

r 1 < r0 ,

(2.2.1)

and s1 /ν s0 /ν Bq,r ,→ Bq,r . 1 0

(2.2.2)

If q0 , q1 ∈ [1, ∞] satisfy s1 − |ω|/q1 = s0 − |ω|/q0 , then 6 /ν /ν Bqs11,r ,→ Bqs00,r .

(2.2.3)

All embeddings (2.2.1) and (2.2.2) are dense if max{q, r0 } < ∞. The one in (2.2.3) is dense if max{q0 , r} < ∞ . Proof (1) The middle part of (2.2.1) and (2.2.2) follow from Theorem VI.2.2.2(i) and Lemmas 2.1.3 and 2.2.1. (2) Suppose t > s. We get from (2.2.2) and Lemma VI.3.6.3(iii) that there exists m ∈ N such that °¡ ¢° kukB s/ν ≤ c kukB t/ν = c ° ψk (D)u °`t (L ) ≤ c qm,m (u) q,r

q,∞

∞

q

s/ν

for u ∈ S. This proves S ,→ Bq,r . s/ν

Assume max{q, r} < ∞ and u ∈ Bq,r . Let ε > 0. Since Rc u ∈ `sr (Lq ), by Theorem VI.2.2.2(i) there is v = (vk ) ∈ cc (Lq ) such that kv − Rc uk`sr (Lq ) < ε/2. Using the density of S in Lq , we see that we can find w = (wk ) ∈ cc (Lq ) with wk ∈ S and kw − vk P`sr (Lq ) < ε/2. Lemma VI.3.6.3(ii) guarantees that ψk (D)wk ∈ S. Hence Rw = k χk (D)wk ∈ S. From this and Lemma 2.1.3(i) we get ku − RwkB s/ν ≤ kRk kRc u − wk`sr (Lq ) q,r ¡ ¢ ≤ kRk kRc u − vk`sr (Lq ) + kv − wk`sr (Lq ) ≤ ε kRk . 6 Corollary

5.6.4 below contains an important complement to this embedding.

106

VII Function Spaces s/ν

This shows that S is dense in Bq,r , if max{q, r} < ∞. Thus the first embedding of (2.2.1) as well as its density if max{q, r1 } < ∞ have been shown. s/ν

(3) From (2.1.11)–(2.1.13) we get Bq,r0 ,→ S 0 . The density of this embedding d

s/ν

follows from S ,→ S 0 and S ,→ Bq,r0 . (4) Suppose q1 < q0 . Imagine we have shown that the following anisotropic version of Nikol0 ski˘ı’s inequality applies: kϕ(D)ukq0 ≤ cρ|ω| (1/q1 −1/q0 ) kϕ(D)ukq1

(2.2.4)

for ϕ ∈ D(Rd ) with supp(ϕ) ⊂ ρ q Ω0 and ρ > 0, and u ∈ S 0 with ϕ(D)u ∈ Lq1 . Then, letting ρ := 2k and ϕ := ψk , we obtain from s1 − s0 = |ω| (1/q1 − 1/q0 ) 2ks0 kψk (D)ukq0 = 2ks1 2k(s0 −s1 ) kψk (D)ukq0 ≤ c 2ks1 kψk (D)ukq1 for k ∈ N. This implies °¡ ¢° kukB s0 /ν = ° ψk (D)u °`s0 (Lq q0 ,r

r

0)

°¡ ¢° ≤ c ° ψk (D)u °`s1 (Lq r

1)

= c kukB s1 /ν q1 ,r

s /ν

for u ∈ Bq11,r , that is, (2.2.3). The corresponding density assertion follows then from the density of S in these spaces if max{q, r} < ∞. Hence it remains to prove (2.2.4). (5) We first assume ρ = 1. We fix λ ∈ D(Rd ) with λ | Ω0 = 1. Then ϕ = λϕ. Since ϕ, λ ∈ D(Rd ) ⊂ OM (Rd ), it follows from (VI.3.4.2), and since F belongs to Laut(S) ∩ Laut(S 0 ), that λ(D)ϕ(D)u = (F −1 λF)(F −1 ϕF)u = F −1 λϕF u = ϕ(D)u ,

u ∈ S0 .

By the convolution theorem, λ(D)ϕ(D) = F −1 λ ∗ ϕ(D)u (cf. Remark 1.9.11(b) in the Appendix). Thus, since F −1 λ ∈ S ,→ Lq10 , we get from Young’s inequality (III.4.2.22) kϕ(D)uk∞ ≤ kF −1 λkq10 kϕ(D)ukq1 = c kϕ(D)ukq1 . Consequently, 1 /q0 kϕ(D)ukq0 ≤ kϕ(D)uk1−q kϕ(D)ukqq11 /q0 ≤ c kϕ(D)ukq1 . ∞

This proves (2.2.4) in the present case. (6) Now suppose ρ 6= 1 and supp(ϕ) ⊂ ρ q Ω0 . Then supp(σρ ϕ) ⊂ Ω0 . By Proposition VI.3.1.3(iii), ¡ ¢ ¡ ¢ (σρ ϕ)(D)u = F −1 (σρ ϕ)b u = F −1 σρ (ϕσ1/ρ u b) = σ1/ρ F −1 ϕF (σρ u) . Hence, by (ii) of Proposition VI.3.1.3, k(σρ ϕ)(D)ukq = ρ|ω|/q kϕ(D)(σρ u)kq .

VII.2 Besov Spaces

107

Thus, by step (5), ρ|ω|/q0 kϕ(D)(σρ u)kq0 = k(σρ ϕ)(D)ukq0 ≤ c k(σρ ϕ)(D)ukq1 = cρ|ω|/q1 kϕ(D)(σρ u)kq1

(2.2.5)

for u ∈ S 0 with ϕ(D)(σρ u) ∈ Lq1 . Proposition VI.3.1.3 also implies σt ∈ Laut(Lq1 ). Thus we infer from (2.2.5) that (2.2.4) applies. ¥ Little Besov Spaces We define very little Besov spaces by s/ν s/ν s/ν ˚q,r ˚q,r B =B (Rd , E) is the closure of S in Bq,r .

(2.2.6)

We also introduce little Besov spaces by s/ν d (s+ν)/ν s/ν bs/ν in Bq,r . q,r = bq,r (R , E) is the closure of Bq,r

(2.2.7)

The following lemma gives useful characterizations of these Banach spaces. 2.2.3 Lemma It holds:

s/ν ˚q,r B

and bs/ν q,r

 s/ν  Bq,r ,      cs/ν L , q 0 = s/ν  `r C0 ,      s/ν c0 C0 ,

if max{q, r} < ∞ , if q < ∞ , r = ∞ , if q = ∞ , r < ∞ ,

(2.2.8)

if q = r = ∞ ,

 s/ν  B ,   q,r s/ν ˚ = Bq,∞ ,    s/ν c0 BUC ,

if r < ∞ , if q < ∞ , r = ∞ ,

(2.2.9)

if q = r = ∞ .

Moreover, d

t/ν Bq,r ,→ bs/ν q,r ,

t>s.

(2.2.10)

Proof (1) If max{q, r} < ∞, then we get (2.2.8) from Theorem 2.2.2. (2) Lemma 2.1.3 (ii)–(iv) imply that each space on the right side of (2.2.8) s/ν is a closed linear subspace of Bq,r . Hence it suffices to show that S is contained and dense in these spaces. Set Y := `sr (X ) if r < ∞, Y := cs0 (X ) if r = ∞, where X := Lq if q < ∞ and X := C0 for q = ∞. By Theorem VI.2.2.2 we know that cc (X ) is dense in Y.

108

VII Function Spaces s/ν

s/ν

Let u ∈ `r X if r < ∞, resp. u ∈ c0 X otherwise. Since S is dense in X , we can approximate Rc u arbitrarily closely in Y by compactly supported sequences whose elements belong to S. Thus we can find, for any given ε > 0, a finitely supported sequence w = (wk ) in Y ∩ S N satisfying kw − Rc ukY < ε. From this it follows, as in step (2) of the preceding proof, that Rw ∈ S and ku − RwkB s/ν < ε. This q,r proves (2.2.8). (3) If max{q, r} < ∞, then we know from Theorem 2.2.2 that d d s/ν ˚t/ν = B t/ν ,→ S ,→ B Bq,r , q,r q,r s/ν

t>s.

s/ν

This implies bq,r = Bq,r and (2.2.10) in the present case. d

(4) By Theorem VI.2.2.2(i), `tr (L∞ ) ,→ `sr (L∞ ) if r < ∞ and t > s. From this and Lemmas 2.1.3 and 2.2.1 we get d

t/ν s/ν B∞,r ,→ B∞,r , s/ν

rs.

s/ν

This proves b∞,r = B∞,r as well as (2.2.10) if r < ∞. (5) Set X := Lq if q < ∞, and X := BUC if q = ∞. Then Theorem VI.2.2.2(i) and Lemmas 2.1.3 and 2.2.1 imply d

s/ν

`t/ν ∞ X ,→ c0 X ,

t>s.

Now (2.2.9) and (2.2.10) follow for q < ∞ and r = ∞ from (2.2.8), and from Lemma 2.1.3(ii) if q = r = ∞ . ¥ The following results complement Theorem 2.2.2. 2.2.4 Theorem (i) Let q < ∞ and 1 ≤ r0 , r1 < ∞. If either s0 = s1 and r1 < r0 , or s1 > s0 , then d d s1 /ν d s0 /ν d ˚s0 /ν 0 /ν S ,→ Bq,r ,→ Bq,r ,→ Bq,∞ = bsq,∞ ,→ S 0 . 1 0 d

s/ν

s/ν

s/ν

d

˚∞,∞ ,→ b∞,∞ ,→ B∞,∞ ,→ S 0 . (ii) S ,→ B s/ν d ˚t/ν s/ν d t/ν ˚q,r (iii) B 1 ,→ Bq,r0 , bq,r1 ,→ bq,r0 for s > t and 1 ≤ r0 , r1 ≤ ∞.

(iv) Suppose s1 > s0 and 1 ≤ q0 , q1 ≤ ∞ satisfy s1 − |ω|/q1 = s0 − |ω|/q0 . Then d ˚s1 /ν ,→ ˚s0 /ν , B B q1 ,r q0 ,r

d

/ν /ν bsq11 ,r ,→ bsq00 ,r .

VII.2 Besov Spaces

109

Proof (1) All assertions (i)–(iii), except the second embedding in (ii), follow directly from Theorem 2.2.2 and the characterization of the little Besov spaces given in (2.2.9). s/ν s/ν s/ν s/ν s/ν s/ν ˚∞,∞ ˚∞,∞ (2) Since B = c0 C0 and b∞,∞ = c0 BUC, we get B ,→ b∞,∞ from Theorem VI.2.2.2(ii) and Lemmas 2.1.3(i) and 2.2.1. ˚ and b. ¥ (3) Claim (iv) is a consequence of (2.2.3) and the definition of B Embeddings With Varying Target Spaces 2.2.5 Theorem Let E1 ,→ E0 . Then s/ν s/ν Bq,r (Rd , E1 ) ,→ Bq,r (Rd , E0 ) .

(2.2.11)

If E1 is dense in E0 , then d ˚s/ν (Rd , E1 ) ,→ ˚s/ν (Rd , E0 ) . B B q,r q,r

(2.2.12)

Proof Since, obviously, Lq (Rd , E1 ) ,→ Lq (Rd , E0 ), we obtain (2.2.11) from Theorem VI.2.2.2(ii), and, once more, from Lemmas 2.1.3(i) and 2.2.1. The second assertion follows now from Lemma 1.6.1. ¥

2.3

Duality

Assuming E to be reflexive or having a separable dual, we prove in this subsection a duality theorem for vector-valued anisotropic Besov spaces. It is a vast generalization of the isotropic scalar result. Recall the definition of h·, ·i in Subsection 1.2 s/ν s/ν ˚q,r and B = Bq,r if max{q, r} < ∞. 2.3.1 Theorem Suppose E is either reflexive or has a separable dual . Then ¡ s/ν d ¢ d 0 ˚ (R , E) 0 = B −s/ν B q,r q 0 ,r 0 (R , E )

(2.3.1) s/ν

with respect to h·, ·i. If 1 < q, r < ∞ and E is reflexive, then Bq,r (Rd , E) is also reflexive. Proof

As before, we omit (Rd , E). Moreover, (Rd , E 0 ) is replaced by E 0 .

(1) We set Xq := Lq for q < ∞, and X∞ := C0 . It follows from Lemma 2.2.3 that ( s/ν ˚q,r B =

`s/ν r Xq ,

if r < ∞ ,

s/ν c0 Xq

if r = ∞ .

,

(2.3.2)

110

VII Function Spaces

We put Xq00 := Lq0 (E 0 ) if q < ∞, and X10 := MBV (E 0 ). Then, with respect to h·, ·i, Xq00 = (Xq )0 , due to Theorem 1.2.1 and (1.1.21) (recall (1.1.27)). Thus we get from Theorem VI.2.2.2(iii) ¡ s ¢0 ¡ s ¢0 0 0 `r (Xq ) = `−s c0 (Xq ) = `−s (2.3.3) 1 (Xq 0 ) , r 0 (Xq 0 ) if r < ∞ , with respect to h ·, ·ii. d s/ν ˚q,r (2) Since S ,→ B , we get from (1.1.20) and Theorem VI.1.3.1

˚s/ν )0 ,→ S 0 (E 0 ) (B q,r with respect to h·, ·i. Thus ˚s/ν )0 , u 0 ∈ (B q,r

0 hu0 , uiB ˚s/ν = hu , ui , q,r

u∈S .

(3) It follows from (2.3.2) and Lemma 2.1.3(i) that ( ¡ s ¢ ˚s/ν , if r < ∞ , L `r (Xq ), B q,r R∈ ¡ ¢ s/ν ˚q,∞ L cs0 (Xq ), B , if r = ∞ .

(2.3.4)

(2.3.5)

Hence, by (2.3.3), ¡ s/ν 0 −s 0 ¢ ˚ ) , ` 0 (X 0 ) . R 0 ∈ L (B q,r q r ¡ ¢ We define Su0 := χk (D)u0 . It is obvious that ¡ −s/ν ¢ 0 S ∈ L Bq0 ,r0 (E 0 ), `−s r 0 (Lq 0 (E )) .

(2.3.6)

(2.3.7)

−s/ν

Let u0 ∈ Bq0 ,r0 (E 0 ). Then, by H¨older’s inequality, |hhSu0 , vii| ≤ kSu0 k`−s kvk`sr (Lq ) 0 0 (Lq 0 (E ))

(2.3.8)

r

for v ∈ `sr (Lq ). Since C0 is a closed linear subspace of L∞ and cs0 is one of `s∞ , it follows from (2.3.7) and (2.3.8) that |hhSu0 , vi i| ≤ c ku0 kB −s/ν (E 0 ) kvk , q 0 ,r 0

`sr (Xq )

where v ∈ if r < ∞, and v ∈ Rc is a coretraction for R,

cs0 (Xq )

otherwise. Thus, by (2.3.5) and since

|hhSu0 , Rc ui i| ≤ c ku0 kB −s/ν (E 0 ) kukB ˚s/ν q,r

q 0 ,r 0

s/ν ˚q,r for u ∈ B . By the evenness of ψk , X­ ® X­ 0 ® hSu0 , Rc uii = χk (D)u0 , ψk (D)u = u , χk (D)ψk (D)u k

D

0

= u,

X k

E

k

χk (D)ψk (D)u = hu0 , RRc ui = hu0 , ui

VII.2 Besov Spaces

111

−s/ν

for u0 ∈ Bq0 ,r0 (E 0 ) ,→ S 0 (E 0 ) and u ∈ S. From this and step (2) it follows −s/ν s/ν 0 ˚q,r Bq0 ,r0 (E 0 ) ,→ (B ) .

P 0 0 0 (4) For v 0 ∈ `−s r 0 (Xq 0 ) we set T v := k ψk (D)vk . Since T has a similar strucc ture as R , we infer from the proof of Lemma 2.1.3 that ¡ −s/ν 0 ¢ 0 T ∈ L `−s Xq 0 . (2.3.9) r 0 (Xq 0 ), `r 0 Suppose µk ∈ MBV (E 0 ). Then ψk (D)µk = F −1 ψk ∗ µk . Hence ° ° ° ° kF −1 ψk k1 = 2|ω| k °σ2k F −1 ψe°1 = °F −1 ψe°1 , k≥1, implies ψk (D)µk ∈ L1 (E 0 ) and kψk (D)µk kL1 (E 0 ) ≤ c kµk kMBV (E 0 ) ,

k∈N,

by Proposition 2.0.2 of the Appendix. Consequently, ¡ ¢ −s/ν 0 T ∈ L `−s L1 (E 0 ) . r 0 (MBV (E )), `r 0

(2.3.10)

Since Xq00 = Lq0 (E 0 ) if 1 < q 0 ≤ ∞, it follows from (2.3.9), (2.3.10), and (2.1.17) ¡ ¢ −s/ν 0 0 that T ∈ L `−s r 0 (Xq 0 ), Bq 0 ,r 0 (E ) for all q and r. Thus, by (2.3.6), ¡ s/ν 0 −s/ν 0 ¢ ˚ ) , B 0 0 (E ) . T R 0 ∈ L (B q,r q ,r s/ν 0 ˚q,r Let u0 ∈ (B ) and u ∈ S. Then, setting v 0 := R0 u0 , we deduce from step (2) DX E X­ ® X­ 0 ® hT R0 u0 , ui = ψk (D)vk0 , u = ψk (D)vk0 , u = vk , ψk (D)u k

k

k

= hv 0 , Rc ui i = hR0 u0 , Rc uii = hu0 , RRc ui = hu0 , ui . s/ν

−s/ν

˚q,r )0 ,→ B 0 0 (E 0 ). Together with step (3) we thus obtain (2.3.1). This proves (B q ,r (5) Suppose 1 < q, r < ∞. Then Xq = Lq is reflexive by Theorem 1.2.1. Hence Theorem VI.2.2.2(iii) guarantees that `sr (Xq ) is reflexive. By Lemmas I.2.3.1 and s/ν s/ν 2.1.3(i), Bq,r = `r Xq is isomorphic to a closed linear subspace of `sr (Xq ). Hence it is reflexive. ¥ 2.3.2 Remark Since there are no restrictions on E in (1.1.21) and (VI.2.2.6), we see that the equality ¡ s/ν d ¢ d 0 ˚ (R , E) 0 = B −s/ν B ∞,r 1,r 0 (R , E ) is valid for any Banach space E.

¥

112

2.4

VII Function Spaces

Fourier Multiplier Theorems

First we prove an important ‘lifting theorem’. Recall definition (VI.3.4.3) of J. ˚ b} and t ∈ R. Then J t belongs to Lis(B s+t , B s ) 2.4.1 Theorem Let B ∈ {B, B, q,r q,r t −1 −t and (J ) = J . Proof find

By means of Leibniz’ rule, Example VI.3.3.9, and Lemma VI.3.3.2(ii) we pω

|∂ α (Λt1 ψk )| ≤ c max(Λt−β 1 β≤α

|∂ α−β ψk |) .

(2.4.1)

¡ ¢1/2ν Writing ξ = 2k q η ∈ Ωk with η ∈ Ω0 , it follows from Λ(ξ) = 1 + (2k N(η)2ν ) and 1/c ≤ N(η) ≤ c for η ∈ Σ, that 2k /c ≤ Λ(ξ) ≤ c 2k ,

ξ ∈ Ωk ,

k∈N.

Hence we get from (2.4.1) Λα 1

pω

(α−β) p ω

|∂ α (Λt1 ψk )| ≤ cΛt1 max Λ1 β≤α

|∂ α−β ψk |

(α−β) p ω −k(α−β) p ω

= cΛt1 max Λ1 β≤α

2

¯ α−β ¯ ¯∂ ψe¯ ≤ c 2kt

for k ≥ 1. Since |∂ γ ψ0 | ≤ c for γ ∈ Nd , it follows 2−kt Λt1 ψk ∈ M(Rd ) ,

k2−kt Λt1 ψk kM ≤ c ,

k∈N.

(2.4.2)

Let X ∈ {BUC, C0 , Lq } .

(2.4.3)

We infer from (2.4.2) and Lemma VI.3.6.3 k(ψj 2−kt Λt1 ψk )(D)kL(X ) ≤ c ,

k, j ∈ N .

Note that ψk = ψk χk = ψk χ2k implies ψk Λt1 = 2kt (2−kt Λt1 ψk χk ) = 2kt

1 ³X

´ (2−kt Λt1 ψk )ψk+i χk .

i=−1

Hence we obtain from (2.4.4) kψk (D)J t ukX ≤ c 2kt kχk (D)ukX ≤ c

1 X i=−1

2(k+i)t kψk+i (D)ukX

(2.4.4)

VII.2 Besov Spaces

113

for k ∈ N, provided u ∈ S 0 is such that ψk (D)u ∈ X. From this we deduce

and

kRc J t uk`sr (X ) ≤ c kRc uk`s+t (X ) r

if

Rc u ∈ `s+t r (X ) ,

kRc J t ukcs0 (X ) ≤ c kRc ukcs+t (X )

if

Rc u ∈ cs+t 0 (X ) .

0

Thus we get from Lemma 2.1.3(i) kJ t uk`s/ν X = kRRc J t uk`s/ν X ≤ c kRc J t uk`sr (X ) ≤ c kRc uk`s+t (X ) r r

r

≤ c kuk`(s+t)/ν X , r

(s+t)/ν

if u ∈ `r

X . Similarly, kJ t ukcs/ν X ≤ c kukc(s+t)/ν X , 0

0

(s+t)/ν

u ∈ c0

X .

Now the assertion follows from (2.1.17) and Lemmas 2.1.3(ii) and 2.2.3.

¥

The following general Fourier multiplier theorem for Besov spaces is obtained by a suitable modification of the arguments used in the preceding proof. ˚ b}. Then: 2.4.2 Theorem Let B ∈ {B, B, ¡ ¢ ¡ ¡ d ¢ ¡ s/ν ¢¢ s/ν (i) m 7→ m(D) ∈ L M R , L(E0 , E1 ) , L Bq,r (Rd , E0 ), Bq,r (Rd , E1 ) . ¡ ¢ ¡ ¢ (ii) Assume m1 ∈ M Rd , L(E0 , E1 ) and m2 ∈ M Rd , L(E1 , E2 ) . Then ¡ ¢ m2 m1 ∈ M Rd , L(E0 , E2 ) (2.4.5) and m2 m1 (D) = m2 (D)m1 (D). Proof (1) Let (2.4.3) be satisfied and write Xi := X (Rd , Ei ). Using ψk χk = ψk , Lemma VI.3.6.3(i) gives kψk (D)m(D)ukX1 = k(ψk χk )(D)m(D)ukX1 = k(ψk m)(D)χk (D)ukX1 ≤ c kmkM

1 X

kψk+i (D)ukX0

i=−1

for u ∈ S 0 with ψj (D)u ∈ X0 for j ∈ N. This implies for u ∈ S 0

and

kRc m(D)uk`sr (X1 ) ≤ c kmkM kRc uk`sr (X0 )

if

Rc u ∈ `sr (X0 ) ,

kRc m(D)ukcs0 (X1 ) ≤ c kmkM kRc ukcs0 (X0 )

if

Rc u ∈ cs0 (X0 ) .

Now we apply Lemma 2.1.3(i), similarly as in the last part of the preceding proof, to arrive at assertion (i).

114

VII Function Spaces

(2) Since the composition L(E1 , E2 ) × L(E0 , E1 ) → L(E0 , E2 ) ,

(m2 , m1 ) 7→ m2 m1

is a multiplication, (2.4.5) follows from Lemma VI.3.4.5(ii). Claim (ii) is then implied by F −1 m2 m1 F = F −1 m2 FF −1 m1 F . ¥ Occasionally, the following simpler Fourier multiplier theorem is also useful. ˚ b}. Then 2.4.3 Theorem Assume B ∈ {B, B, ¡ ¢ ¡ ¡ ¢ ¡ s/ν d ¢¢ s/ν m 7→ m(D) ∈ L F L1 Rd , L(E0 , E1 ) , L Bq,r (R , E0 ), Bq,r (Rd , E1 ) . Proof Let (2.4.3) be satisfied. Since ψk (D) and m(D) commute, it follows from Theorem VI.3.4.4 that kψk (D)m(D)ukX (Rd ,E1 ) ≤ kmkF L1 kψk (D)ukX (Rd ,E0 ) ,

k∈N,

provided ψk (D)u ∈ X (Rd , E0 ). Now the assertion follows by invoking the arguments of the preceding proof. ¥ By combining Theorem 2.4.2 with Theorem 2.4.1 we obtain Fourier multiplier theorems involving Besov spaces of different order. −s ˚ 2.4.4 ¡ dProposition ¢ Let B ∈ {B, B, b} and s, t ∈ R. Suppose Λ1 a is an element of M R , L(E1 , E0 ) . Then ¡ (s+t)/ν d ¢ t/ν (i) a(D) ∈ L Bq,r (R , E1 ), Bq,r (Rd , E0 ) and ka(D)k ≤ c kΛ−s 1 akM , t ∈ R. ¡ d ¢ s −1 (ii) Assume,¡ in addition, that ¢ a ∈ C R , Lis(E1 , E0 ) and Λ1 a belongs to the space B Rd , L(E0 , E1 ) . Then ¡ (s+t)/ν d ¢ t/ν a(D) ∈ Lis Bq,r (R , E1 ), Bq,r (Rd , E0 ) , a(D)−1 = a−1 (D) , ¡ ¢ s −1 and ka(D)−1 k ≤ c kΛ−s k∞ . 1 akM , kΛ1 a

Proof

(1) It follows from Theorem 2.4.2 that ¡ (s+t)/ν d ¢ (s+t)/ν (Λ−s (R , E1 ), Bq,r (Rd , E0 ) 1 a)(D) ∈ L Bq,r

(2.4.6) ¢ −s s 0 d and k(Λ−s 1 a)(D)k ≤ c kΛ1 akM . Proposition VI.3.4.2 gives J ∈ Laut S (R , E0 ) , (s+t)/ν and (2.2.1) guarantees Bq,r (Rd , E0 ) ,→ S 0 (Rd , E0 ). Hence we infer from (2.4.6) ¡

−1 s a(D)u = F −1 aF u = F −1 Λs1 (Λ−s Λ1 FF −1 Λ−s 1 a)F = F 1 aF u

= J s (Λ−s 1 a)(D)u (s+t)/ν

for u ∈ Bq,r

(Rd , E1 ). Now (i) follows from (2.4.6) and Theorem 2.4.1.

VII.2 Besov Spaces

115

(2) Let the ¡additional hypotheses be satisfied. Then Lemma VI.3.4.6(i) im¢ plies Λs1 a−1 ∈ M Rd , L(E0 , E1 ) . From this and Theorem 2.4.2 we obtain, similarly as above, that ¡ t/ν d ¢ (s+t)/ν a−1 (D) = J −s (Λs1 a−1 )(D) ∈ L Bq,r (R , E0 ), Bq,r (Rd , E1 ) ¡ ¢ s −1 and ka−1 (D)k ≤ c kΛ−s k∞ . Furthermore, part (ii) of Theorem 2.4.2 1 akM , kΛ1 a implies a−1 (D)a(D) = (a−1 a)(D) = 1B(s+t)/ν (Rd ,E1 ) . q,r

Similarly, a(D)a

−1

(D) = 1Bt/ν (Rd ,E0 ) . This proves (ii). q,r

¥

d 2.4.5 Remarks (a) Suppose s ∈ R and a ∈ Hs (Z, E). Then Λ−s 1 a1 ∈ M(R , E) −s and kΛ1 a1 kM ≤ c kakHs .

Proof Example VI.3.3.9 and Lemma VI.3.3.6 imply Λ−s a ∈ H0 (Z, E) and the −s estimate kΛ−s akH0 ≤ c kakHs . Hence Λ−s a)1 ∈ M(Rd , E) by Lemma 1 a1 = (Λ −s −s VI.3.4.5, and kΛ1 a1 kM ≤ kΛ akH0 . This implies the claim. ¥ (b) Proposition 2.4.4 is also valid if M is replaced by FL1 . Proof

2.5

Invoke Theorem 2.4.3.

¥

Operators of Positive Type

As a first application of the multiplier results in the preceding subsection we prove resolvent estimates for Fourier multiplier operators. They are of importance for the study of elliptic operators. The second main theorem of this subsection is a representation theorem for holomorphic functions of such operators. It is the basis for kernel representations of semigroups on Besov spaces, for example. Such applications are postponed to Volume III. We use the notation introduced in Subsections VI.3.4 and VI.3.5, the classes (VI.3.4.8) and (VI.3.5.2) in particular. Resolvent Estimates ˚ b}, κ ≥ 1, 0 ≤ ϑ < π, and s ≥ 0. 2.5.1 Theorem Suppose E1 ,→ E0 , B ∈ {B, B, If ¡ ¢ a ∈ Ps Z, L(E1 , E0 ); κ, ϑ , (2.5.1) then ¡ ¡ (s+t)/ν d ¢ ¢ t/ν a1 (D) ∈ P L Bq,r (R , E1 ), Bq,r (Rd , E0 ) ; c(κ), ϑ for t ∈ R.

(2.5.2)

116

VII Function Spaces

¡ ¢ Proof Let (2.5.1) be satisfied. Then a ∈ Hs Z, L(E1 , E0 ) and kakHs ≤ c(κ). ¡ d ¢ −s Hence Λ−s 1 a ∈ M R , L(E1 , E0 ) and kΛ1 akM ≤ c(κ) by Remark 2.4.5(a). From Example VI.3.3.9 and Remark VI.3.4.7(a) we infer ¡ d ¢ Λ−s kΛ−s 1 λ ∈ M R , L(E1 , E0 ) , 1 λkM ≤ c |λ| for7 λ ∈ C. Now we deduce from Proposition 2.4.4(i) that ¡ (s+t)/ν d ¢ t/ν λ + a1 (D) ∈ L Bq,r (R , E1 ), Bq,r (Rd , E0 ) , kλ + a1 (D)k ≤ c(κ)(1 + |λ|) for λ ∈ C and t ∈ R. (2) We get from Lemma VI.3.5.1 and Proposition 2.4.4(i) that ¡ t/ν d ¢ (js+t)/ν (λ + a1 )−1 (D) ∈ L Bq,r (R , E0 ), Bq,r (Rd , Ej )

(2.5.3)

and (1 + |λ|)1−j k(λ + a1 )−1 (D)k(j) ≤ c(κ) ,

λ ∈ Sϑ ,

j = 0, 1 ,

t∈R,

where k·k(j) is the norm in the space occurring in (2.5.3). ¡ ¢ (3) It follows from Lemma VI.3.5.1 that λ + a1 ∈ C Rd , Lis(E1 , E0 ) and ¡ ¢ Λs1 (λ + a1 )−1 ∈ B Rd , L(E0 , E1 ) with kΛs1 (λ + a1 )−1 k∞ ≤ c(κ) for λ ∈ Sϑ . Thus, since ¡ ¢−1 Λs1 (λ + a1 ) = Λ−s , 1 (λ + a1 )

λ ∈ Sϑ ,

we infer from part (ii) of Proposition 2.4.4 and step (1) that ¡ ¢−1 (λ + a1 )−1 (D) = λ + a1 (D) ,

λ ∈ Sϑ .

From this, step (1) (with λ = 0), and step (2) we get the assertion.

(2.5.4) ¥

2.5.2 Remark Let κ ≥ 1 and ϑ = π/2. Then definition (VI.3.5.7) shows that ϕ equals π/2 − arcsin(1/2κ). Recall that Γ stands ¡ for the¢positively oriented boundary of W = Sϕ ∩ [ |z| ≥ 1/2κ]. Suppose A ∈ P L(E); κ . It follows that Z 1 e−tA := e−tλ (λ − A)−1 dλ , t>0, (2.5.5) 2πi Γ is well-defined in L(E). We put e−0A := 1E . Then the Dunford–Taylor functional calculus shows that { e−tA ; t ≥ 0 } is an analytic semigroup on E, that is, in L(E), which is strongly continuous iff A is densely defined. Hereby, analytic means that 7 As

usual, we identify λ ∈ C with λiE1 ,E0 , where iE1 ,E0 is the injection E1 ,→ E0 .

VII.2 Besov Spaces

117

˚arcsin(1/2κ) . t 7→ e−tA : (0, ∞) → L(E) has an analytic extension over the sector S Furthermore, A is uniquely determined by this semigroup and vice versa. −A is said to be the infinitesimal generator of { e−tA ; t ≥ 0 }. This is justified, since im(e−tA ) ⊂ dom(A) and ∂(e−tA ) = −Ae−tA for t > 0. Proof See Section 2.1 in Chapter 2 of A. Lunardi [Lun95]. Based on the Dunford– Taylor representation (2.5.5), there is carried out a detailed study of analytic semigroups, which are not necessarily strongly continuous. ¥ ¡ ¢ 2.5.3 Corollary Suppose a ∈ Ps L(E1 , E0 ); κ . Then −a1 (D) generates an analytic t/ν semigroup on Bq,r (Rd , E0 ). A Representation Theorem Now we restrict our considerations to the case where E0 = E1 , that is, we assume ¡ ¢ a ∈ Ps Z, L(E); κ, ϑ and use the notations of Subsection VI.3.5. It follows from Theorem 2.5.1 that ¡ ¡ (s+t)/ν t/ν ¢ ¢ ¡ ¢ t/ν a1 (D) ∈ P L Bq,r , Bq,r ; c(κ), ϑ ⊂ P L(Bq,r ); c(κ), ϑ .

(2.5.6)

Hence ¡ ¢ 1 h a1 (D) := 2πi

Z

¡ ¢−1 h(λ) λ − a1 (D) dλ

(2.5.7)

Γ

s/ν

is a well-defined element ¡ of L(B ¢ q,r ). The following theorem gives a natural representation formula for h a1 (D) in terms of h(a1 ). ˚ψ → C be holomorphic for some ψ ∈ (ϕ, π − ϑ) and let 2.5.4 Theorem Let h : S there exist δ > 0 such that |z|δ h(z) → 0 as |z| → ∞. Then ¡ ¢ h a1 (D) = h(a1 )(D) .

(2.5.8)

¡ ¢ t/ν Proof (1) We set X := M Rd , L(E) and A := a1 , respectively X := L(Bq,r ) and A := a1 (D). Furthermore, κ e := κ ∨ c(κ), where c(κ) is the constant occurring in e is the positively oriented boundary of W f := Wκ,ϑ (2.5.6), and ϕ e := ϕ(e κ, ϑ). Then Γ e e (cf. (VI.3.5.7)) and ΓR := Γ ∩ [ |z| ≤ R] for R > 2e κ. In (VI.3.5.13) we can replace Γ e Similarly, in (2.5.7) we deform Γc(κ),ϑ into Γ. e Then by Γ. 1 IR (A) := 2πi

Z h(λ)(λ − A)−1 dλ ∈ L(X ) , ΓR

2e κ 0, and a ∈ Ps Z, L(E) . Then 2.5.5 Theorem Let B ∈ {B, (t+s)/ν t/ν t/ν a1 (D) ∈ Lis(Bq,r , Bq,r ) ∩ BIP(Bq,r ),

(s+t)/ν

t∈R. t/ν

Proof First we note that, by Theorem 2.2.4, (Bq,r , Bq,r ) is a densely embedded Banach couple. ¡ ¢ Suppose a ∈ Ps Z, L(E); κ, ϑ . Theorem 2.5.1 implies that a1 (D) is a toplin(t+s)/ν t/ν ear isomorphism from Bq,r onto Bq,r and ¡ ¢ t/ν A := a1 (D) ∈ P L(Bq,r ); c(κ), ϑ . Thus8 Theorem III.4.6.5 implies that Az is well-defined for z ∈ C and Az = hz (A) if Rez < 0. ¡ ¢ Proposition VI.3.5.6 guarantees that az1 ∈ M Rd , L(E) if Rez < 0. Hence we can apply Theorem 2.5.4 to obtain Az = hz (A) = hz (a1 )(D) ,

Rez < 0 .

Hence we deduce from estimate (VI.3.5.24) and Theorem 2.4.2 that kAz kL(Bt/ν ) ≤ c(κ, ϑ)e|Imz| ϕ , q,r

Rez < 0 , t/ν

where ϕ = ϕ(κ, ϑ). Now the fact that A belongs to BIP(Bq,r ) is a consequence of Lemma III.4.7.4. ¥ Interpolation-Extrapolation Scales Lastly, we give a simple application of the preceding theorem which sheds a new s/ν s/ν ˚q,r light on the Banach space scales B and bq,r for s ∈ R. 0/ν ˚ b}. Then J ∈ BIP(Bq,r 2.5.6 Theorem Suppose B ∈ {B, ) and the interpolation£ ¤ 0/ν extrapolation scale (Eα , Aα ) ; α ≥ −m , generated by (E0 , A) := (Bq,r , J) and . α/ν [·, ·]θ , 0 < θ < 1, satisfies Eα = Bq,r for m ∈ N. 8 Observe

the change of notation.

120

VII Function Spaces

¡ ¢ Proof Example VI.3.5.8 guarantees Λ = Λ1E ∈ P1 Z, L(E) . Hence we infer from 0/ν Theorem 2.5.5 that J = Λ1 (D) ∈ BIP(Bq,r ). Now the assertion follows from Theorems V.1.5.4 and 2.4.1. ¥ 2.5.7 Remark It is a consequence of this theorem, Theorem V.1.5.4, and the fact s/ν that the spaces Bq,r are defined for all s ∈ R, that the two-sided fractional power 0/ν scale generated by (Bq,r , J) is well-defined. ¥ ˚ b} and 0 < θ < 1. Then 2.5.8 Corollary Suppose B ∈ {B, . sθ /ν s0 /ν s1 /ν [Bq,r , Bq,r ]θ = Bq,r , −∞ < s0 < s1 < ∞ , where sθ = (1 − θ)s0 + θs1 . Proof

This follows from Theorems 2.5.6 and V.1.5.4.

¥

In Subsection 2.7 we shall give another proof for these interpolation theorems.

2.6

Renorming by Derivatives

First we show that Besov spaces behave naturally with respect to differentiation. ˚ b}. Then 2.6.1 Theorem Suppose B ∈ {B, B, (s+α p ω)/ν s/ν ∂ α ∈ L(Bq,r , Bq,r ),

α ∈ Nd .

Proof Set a(ξ) := ξ α . Then a is smooth, positively α q ω-homogeneous, and ∂ β a vanishes for β ∈ Nd unless β ≤ α. Hence ∂ β a is positively¡(α − β) q ω-homogeneous ¢ by Lemma VI.3.3.2. Thus we deduce from ∂ β a(ξ) = ∂ β a Λ1 q rΛ (ξ) and the compactness of [Λ = 1] ∩ Rd that (α−β) p ω |∂ β a| ≤ cΛ1 , β≤α. (2.6.1) By Leibniz’ rule and ∂ β a = 0 if β  α, X³ γ ´ pω pω β ∂ γ (Λ−α a) = ∂ γ−β Λ−α ∂ a, 1 1 β

γ ∈ Nd .

β≤γ β≤α

From this, Example VI.3.3.9, and (2.6.1) we infer X −α p ω−(γ−β) p ω (α−β) p ω pω pω |∂ γ (Λ−α a)| ≤ c Λ1 Λ1 ≤ cΛ−γ 1 1 β≤γ β≤α

for γ q ω ≤ k(ν). Hence Λ−α 1 sition 2.4.4 and a(D) = Dα .

pω ¥

a ∈ M(Rd ). Now the assertion follows from Propo-

VII.2 Besov Spaces

121

Equivalent Norms Next we prove renorming theorems for Besov spaces. The first one is the basis for deriving ‘sandwich theorems’ between Besov spaces and classical function spaces. q ˚ b} and m ∈ ν N. The following assertions are 2.6.2 Theorem Suppose B ∈ {B, B, equivalent: (s+m)/ν

(α) u ∈ Bq,r (β)

.

s/ν ∂ u ∈ Bq,r , α q ω m/ω s/ν u, ∂j j u ∈ Bq,r , α

≤ m.

(γ) Furthermore, set

k·k

k·k

k·k k·k Then k·k

(k) (s+m)/ν

Bq,r

1 ≤ j ≤ d.

(1) (s+m)/ν Bq,r

(2) (s+m)/ν Bq,r

(3) (s+m)/ν

Bq,r

:= k·kBs/ν + q,r

m/ωj

k∂j

q,r

` X

(s+m)/ν Bq,r

:=

q,r

i=1

` m/ν X Xi

X

α p ω≤m

q,r

i k∇xm/ν · kBs/ν , i

k∇xji · kBs/ν , q,r

i=1 j=1

(4)

· kBs/ν ,

j=1

:= k·kBs/ν + :=

d X

k∂ α · kBs/ν . q,r

∼ k·kB(s+m)/ν for 1 ≤ k ≤ 4. q,r

s/ν

Proof For abbreviation, B s/ν := Bq,r , etc. p (1) By Theorem 2.6.1, ∂ α maps B (s+m)/ν continuously into B (s+m−α ω)/ν . s/ν q The latter space is continuously embedded in B for α ω ≤ m. From this we get α qω ≤ m .

∂ α ∈ L(B (s+m)/ν , B s/ν ) , Writing k·k

(k)

for k·k k·k

(1)

(k) s/ν

Bq,r

(2.6.2)

we obtain

≤ k·k

(2)

≤ k·k

(3)

≤ c k·k

(4)

≤ c k·kB(s+m)/ν ,

the last inequality being due to (2.6.2). (2) We put H := R+ (cf. Remark VI.3.3.1(b)). Then a(ζ) := η 2m/ν1 +

d X

(ξ j )2m/ωj ,

ζ = (ξ, η) ∈ Z ,

j=1

¡ ¢ ¡ ¢ belongs to H2m Z, (0, ∞) . Hence a−1 ∈ H−2m Z, (0, ∞) by Lemma VI.3.3.8. This, Example VI.3.3.9, and Lemma VI.3.3.6 imply Λ−2m a, Λ2m a−1 ∈ H0 (Z).

122

VII Function Spaces

Consequently, by Remark VI.3.4.7(a), these functions belong to M(Rd ). Now Proposition 2.4.4 gives k·kB(s+2m)/ν ∼ ka1 (D) · kBs/ν . Hence, by Theorem 2.4.1, k·kB(s+m)/ν ∼ kJ −m · kB(s+2m)/ν ∼ ka1 (D)J −m · kBs/ν .

(2.6.3)

Since (ξ j )m/ωj Λ−m ∈ H0 (Z), we infer from Remark VI.3.4.7(a) and Theorem 2.4.2 m/ωj

J −m ∈ L(B s/ν ) ,

m/ωj

J −m Dj

Dj 2m/ωj

Thus Dj

J −m = Dj

m/ωj

1≤j≤d.

and

d d ³ ´ X X 2m/ωj m/ω m/ω a1 (D)J −m = 1 + Dj J −m = J −m + (Dj j J −m )Dj j j=1

j=1

and (2.6.3) imply d ³ ´ X m/ω kukB(s+m)/ν ≤ c ka1 (D)J −m ukBs/ν ≤ c kJ −m ukBs/ν + kDj j ukBs/ν j=1

≤

(1) c kukB(s+m)/ν

,

due to J −m ∈ L(B s/ν ). Now the assertion follows from step (1).

¥

The following theorem is of particular importance if s ≤ 0. It is an analogue of Theorem 1.5.3. s/ν ˚ b} and m ∈ νN. Then u ∈ Bq,r 2.6.3 Theorem Suppose B ∈ {B, B, iff there exist (s+m)/ν uα ∈ Bq,r for α p ω ≤ m such that X u= (−1)|α| ∂ α uα . (2.6.4) α p ω≤m

Furthermore, u 7→ inf

X α p ω≤m

kuα kB(s+m)/ν

(2.6.5)

q,r

s/ν

is a norm for Bq,r , the infimum being taken over all representations (2.6.4). Proof It follows from Theorem 2.6.1 that u, defined by the right side of (2.6.4), s/ν belongs to Bq,r and depends continuously on the uα . s/ν

(s+2m)/ν

Conversely, let v ∈ Bq,r . Then u := J −2m v ∈ Bq,r Hence

by Theorem 2.4.1.

d d ³ ´ X X 2m/ωj m/ω v = J 2m u = 1 + Dj u=u+ (−1)m/ωj ∂j j uj , j=1

j=1

VII.2 Besov Spaces

123 m/ω

(s+m)/ν

where uj := −(−1)m/ωj ∂j j u ∈ Bq,r for 1 ≤ j ≤ d, due once more to Theorem 2.6.1. P Let n := α p ω≤m 1. Then the above considerations show that (s+m)/ν n s/ν T : (Bq,r ) → Bq,r ,

(uα ) 7→

P

α p ω≤m (−1)

|α| α

∂ uα

is a continuous linear surjection. We define Tb by the commutativity of the diagram T

(s+m)/ν n

(Bq,r

)

@ @ @ R

s/ν - Bq,r

µ ¡¡

¡ T

(s+m)/ν n (Bq,r ) / ker(T )

where the unlabeled arrow denotes the canonical projection. Then Tb is a toplinear isomorphism by the open mapping theorem. This proves the assertion, since (2.6.5) is the quotient norm of the factor space. ¥ 2.6.4 Remark The theorem remains true if (2.6.4) is replaced by u = u0 +

d X

m/ωj

∂j

uj

j=1

or u = u0 +

` X

i ∇xm/ν ui i

i=1

and (2.6.5) is modified accordingly. Proof

Obvious by the above proof.

¥

Sandwich Theorems The following ‘sandwich theorem’ clarifies the relation between Besov and classical function spaces. 2.6.5 Theorem Suppose m ∈ νN. Then m/ν

d

m/ν

d

m/ν Bq,1 ,→ Wqm/ν ,→ Bq,∞ ,

q s. Then Wq and Theorem 2.2.4(iii). ¥

2.7

d

t/ν

d

s/ν

˚q0 ,∞ ,→ B ,→ B q0 ,1 by (i)

Interpolation

Throughout this subsection •

1 ≤ r0 , r 1 ≤ ∞ .

•

s0 , s1 ∈ R with s0 6= s1 .

•

0 s1 .

s /ν

1 Theorem 2.2.4 guarantees Bq,r1 ,→ Bq,r for s > s1 . Hence 1

d

t/ν s1 /ν Bq,∞ ,→ Bq,r , 1

t > s1 ,

r1 < ∞ .

(2.7.1)

We can assume s1 > s0 . The definition of the continuous interpolation functor and s0 /ν s1 /ν 0 s1 /ν sθ /ν assertion (i) guarantee that (Bq,r in Bq,∞ . 0 , Bq,r1 )θ,∞ is the closure of Bq,r1 Thus we deduce from (2.2.10) and (2.7.1) . θ /ν s0 /ν s1 /ν 0 (Bq,r , Bq,r )θ,∞ = bsq,∞ . (2.7.2) 0 1 We fix t0 < s0 < s1 < t1 and put θj := (sj − t0 )/(t1 − t0 ). Then (2.7.2) and the reiteration theorem for the continuous interpolation functor, more specifically (I.2.8.7), give ¢0 . ¡ t0 /ν t1 /ν 0 t0 /ν t1 /ν 0 0 /ν 1 /ν 0 (bsq,∞ , bsq,∞ )θ,∞ = (Bq,∞ , Bq,∞ )θ0 ,∞ , (Bq,∞ , Bq,∞ )θ1 ,∞ θ,∞ . . θ /ν t0 /ν t1 /ν 0 = (Bq,∞ , Bq,∞ )(1−θ)θ0 +θθ1 ,∞ = bsq,∞ . This proves (ii). (4) Claim (iii) is a consequence of step (1) and Theorem VI.2.3.4(ii). (5) Assertions (iv) and (v) follow from (2.2.8) and (2.2.9), respectively, and from step (1) and Theorem VI.2.3.4(iv). For (v) we use in addition the following ˚0 is the closure property of the complex interpolation functor: if E1 ,→ E0 and E of E1 in E0 , then ˚0 , E1 ]θ [E0 , E1 ]θ = [E (cf. [BeL76, Theorem 4.2.2(b)] or [Tri95, Theorem 1.9.3(g)]).

¥

Interpolation with Different Target Spaces In the next theorem we complement these interpolation results by considering situations in which two Banach spaces E0 and E1 are involved. 2.7.2 Theorem Let (E0 , E1 ) be an interpolation couple and 1 ≤ q0 , q1 < ∞. Then ¡ s /ν ¢ . sθ /ν s /ν (i) Bq00,r0 (Rd , E0 ), Bq11,r1 (Rd , E1 ) θ,r(θ) = Bq(θ),r(θ) (Rd , Eθ,r(θ) ), provided r0 , r1 < ∞ and q(θ) = r(θ). £ s /ν ¤ . sθ /ν /ν d d ˚qs11,r (ii) Bq00,r0 (Rd , E0 ), B 1 (R , E1 ) θ = Bq(θ),r(θ) (R , E[θ] ), r0 < ∞. £ s /ν ¤ . sθ /ν s /ν (iii) Bq00,r0 (Rd , E0 ), Bq11,∞ (Rd , E1 ) θ = Bq(θ),r(θ) (Rd , E[θ] ). £ s0 /ν d ¤ . sθ /ν /ν d ˚q0 ,∞ (R , E0 ), B ˚qs11,∞ ˚ (iv) B (Rd , E1 ) θ = B q(θ),∞ (R , E[θ] ).

VII.2 Besov Spaces

127

Proof

From interpolation theory it is known that ¡ ¢ . ¡ ¢ Lq0 (Rd , E0 ), Lq1 (Rd , E1 ) θ = Lq(θ) Rd , (E0 , E1 )θ (2.7.3) © ª for (·, ·)θ ∈ (·, ·)θ,q(θ) , [·, ·]θ (e.g., [Tri95, Theorem 1.18.4]). Thus the assertions follow from step (1) of the preceding proof, Lemma 2.2.3, and Theorem VI.2.3.4. ¥ Embeddings of Intersection Spaces As an application of part (i) of this theorem and of the embedding results of Subsection 2.2, we can prove an embedding theorem for intersections of Besov spaces. A very simple special case thereof will be of use in Subsection 7.4 below. 2.7.3 Theorem Let (E0 , E1 ) be an interpolation couple. Suppose q, q0 , q1 ∈ [1, ∞], s, s0 , s1 ∈ R with s0 6= s1 , and 0 < θ < 1 satisfy s < sθ Then

and

s − |ω|/q < sθ − |ω|/q(θ) .

(2.7.4)

/ν /ν d Bqs11,r (Rd , E1 ) ∩ Bqs00,r (Rd , E0 ) ,→ bs/ν q,r (R , Eθ,p ) . 1 0

for 1 ≤ p ≤ ∞. Proof It follows from (2.7.4) that we can choose σj < sj and θ < ϑ < 1 sufficiently close to sj and θ, respectively, such that s < σϑ and s − |ω|/q < sϑ − |ω|/q(ϑ). Thus, by (2.2.2) and omitting Rd , /ν /ν Bqs11,r (E1 ) ∩ Bqs00,r (E0 ) ,→ Bqσ11,q/ν1 (E1 ) ∩ Bqσ00,q/ν0 (E0 ) . 1 0 σ /ν

The last intersection space embeds continuously into Bqjj,qj (Ej ) for j = 0, 1. From this we get by interpolation, due to Theorem 2.7.2(i), ¡ ¢ /ν /ν Bqs11,r (E1 ) ∩ Bqs00,r (E0 ) ,→ Bqσ00,q/ν0 (E0 ), Bqσ11,q/ν1 (E1 ) ϑ,q(ϑ) 1 0 . σϑ /ν = Bq(ϑ),q(ϑ) (Eϑ,q(ϑ) ) . Now we choose t0 < t1 in (s, σϑ ). Then (2.2.3) and (2.2.2) imply σ /ν

t /ν

1 ϑ t0 /ν Bq(ϑ),q(ϑ) (Eϑ,q(ϑ) ) ,→ Bq,q(ϑ) (Eϑ,q(ϑ) ) ,→ Bq,r (Eϑ,q(ϑ) ) .

Since Eϑ,q(ϑ) ,→ Eθ,p by (I.2.5.2), we infer from Theorem 2.2.5 and (2.2.10) t0 /ν t0 /ν Bq,r (Eϑ,q(ϑ) ) ,→ Bq,r (Eθ,p ) ,→ bs/ν q,r (Eθ,p ) .

This proves the theorem.

¥

Theorem 5.6.6 and Remark 5.6.7 deal with the case where the equality signs in (2.7.4) are permitted.

128

VII Function Spaces

Interpolation of Classical Spaces It is important to know that Besov spaces of positive order can be obtained from classical function spaces by interpolating. 2.7.4 Theorem Suppose k < s < m with k, m ∈ νN. If θ := (s − k)/(m − k), then s/ν . Bq,r = (Wqk/ν , Wqm/ν )θ,r ,

q m, then B∞,r ,→ BUC m/ν . d

s/ν

(ii) BC ∞ ,→ b∞,∞ . Proof

(1) Claim (i) is implied by (2.7.6). d

s/ν

(2) Since BUC m/ν ,→ b∞,∞ for m > s, and due to (2.7.7), assertion (ii) follows from (1.1.9). ¥ s/ν k/ν m/ν ˚∞,∞ In Theorem 3.7.6 it is shown that B = (C0 , C0 )0θ,∞ .

2.8

Besov Spaces on Corners

The basis for constructing function spaces on corners is contained in the following simple observation about image spaces of retractions.

VII.2 Besov Spaces

129

2.8.1 Lemma Let X0 , X , and Y be LCSs with X0 ,→ X . Suppose (r, rc ) is an r-c pair for (X , Y). Then the diagrams r

X

- Y

6 £¢

6 r

X0

c ¾ r

X

6

£¢ - rX0

£¢

X0 ¾

Y

6 rc

£¢ rX0

are commuting and (r, rc ) is an r-c pair for (X0 , rX0 ). If X0 is a Banach space, then k·krX0 ∼ krc ·kX0 . Proof The first part of the statement follows from Remarks VI.2.2.1. Let X0 = (X0 , k·k) be a Banach space. Since (r, rc ) is an r-c pair for (X0 , rX0 ), Lemma I.2.3.1 guarantees the direct sum decomposition X0 = pX ⊕ ker(r) ,

p := rc r .

Hence we can endow X0 with the equivalent norm x 7→ kyk + kzk ,

x=y⊕z ,

Then kykrX0 = for y ∈ rX0 .

inf

x∈r −1 (y)

kxk ∼

y = px ,

rz = 0 .

inf (krc yk + kzk) = krc yk

z∈ker(r)

¥

We assume

s/ν

• K is a corner in Rd . ˚ b} . • B = {B, B, s/ν

As before, Bq,r := Bq,r (Rd , E). s/ν s/ν ˚q,r It follows from (2.2.1) and the definitions of B and bq,r that s/ν S ,→ Bq,r ,→ S 0 . s/ν

Hence Theorem VI.1.2.3 and Lemma 2.8.1 imply that RK Bq,r is a well-defined Banach space satisfying s/ν S(K, E) ,→ RK Bq,r ,→ S 0 (K, E) .

Henceforth, s/ν s/ν Bq,r (K, E) := RK Bq,r .

(2.8.1)

s/ν s/ν ˚q,r This defines the Besov spaces Bq,r (K, E), the very little Besov spaces B (K, E), s/ν and the little Besov spaces bq,r (K, E) on corners.

130

VII Function Spaces s/ν

2.8.2 Theorem Bq,r (K, E) is a Banach space and (RK , EK ) is a universal r-e pair ¡ s/ν s/ν ¢ for Bq,r , Bq,r (K, E) . Proof

This is a restatement of the preceding considerations.

¥

Suppose m ∈ νN and X ∈

©

m/ν

BC m/ν , BUC m/ν , C0

, Wqm/ν ; q < ∞

ª

.

Then X (X, E) has been defined for X ∈ {Rd , K} in Subsections 1.2 and 1.1. It follows from Theorem 1.3.1 that . X (K, E) = RK X ,

(2.8.2)

where X = X (Rd , E) on the right side. s/ν

2.8.3 Theorem 9 The Besov spaces Bq,r (K, E) possess the same embedding and s/ν interpolation properties as the spaces Bq,r (Rd , E). Proof This is a consequence of Theorem 2.8.2, Lemma 2.2.1, Proposition I.2.3.2, and (2.8.2). ¥ s/ν

The next theorem shows that the duality assertions for Bq,r also apply to Besov spaces on corners. 2.8.4 Theorem Suppose E is reflexive or has a separable dual. Then ¡

˚s/ν (K, E) B q,r

¢0

−s/ν

= Bq0 ,r0 (K∗ , E 0 ) s/ν

with respect to h·, ·i. If 1 < q, r < ∞ and E is reflexive, then Bq,r (K, E), too, is reflexive. ¡ s/ν d ¢ s/ν ˚q,r (R ), B ˚q,r Proof Since (RK , EK ) is an r-e pair for B (K) , we deduce from Theorem 2.8.2, and Remark VI.1.2.1(c) that (E∗K , R∗K ) is an r-e pair for ¡

¢ s/ν s/ν ˚q,r ˚q,r (B (Rd , E))0 , (B (K, E))0 .

Thus, by Theorem 2.3.1, (E∗K , R∗K ) is an r-e pair for ¡ 9 For

¢ −s/ν s/ν ˚q,r Bq0 ,r0 (Rd , E 0 ), (B (K, E))0 .

brevity, here we do not list all the embedding and interpolation results derived in the preceding subsections. However, whenever we refer to this theorem in the following, we always specify which embedding or interpolation fact is meant.

VII.2 Besov Spaces

131

¡ ¢ Lemma VI.1.2.2 implies (EK )∗ , (RK )∗ = (RK∗ , EK∗ ). Consequently, ¡ s/ν ¢ . −s/ν −s/ν ˚ (K, E) 0 = B RK∗ Bq,r (Rd , E 0 ) = Bq,r (K∗ , E 0 ) . q,r s/ν

This proves the first assertion. If 1 < q, r < ∞ and E is reflexive, then Bq,r is reflexive by Theorem 2.3.1. Hence we infer from Theorem 2.8.2 and Lemma I.2.3.1 s/ν that Bq,r (K, E) is isomorphic to a closed linear subspace of a reflexive Banach space. Thus it is reflexive as well. ¥ ¡ s/ν ¢ ˚∞,∞ (K, E) 0 = B −s/ν (K∗ , E) holds without any restriction on E. 2.8.5 Remark B 1,1 Proof

Remark 2.3.2.

¥

In the case where K is closed, distributional derivatives behave naturally and they can be used to introduce equivalent norms. 2.8.6 Theorem Suppose K is closed. Then: ¡ (s+α p ω)/ν ¢ s/ν (i) ∂ α ∈ L Bq,r (K, E), Bq,r (K, E) , α ∈ Nd . (ii) The assertions of Theorems 2.6.2 and 2.6.3 and Remark 2.6.4 apply also to s/ν Bq,r (K, E). Proof

2.9

This follows from Theorems VI.1.2.3(iii) and 2.8.2.

¥

Notes

As already mentioned in the notes to Section VI.3, in the isotropic case the definition of Besov spaces (2.1.17), (2.1.18) is the standard one originating in J. Peetre’s paper [Pee67]. An extension of this approach to anisotropic Besov spaces has been given by M. Yamazaki [Yam86], who employed the ν-quasinorm E of Example VI.3.2.1(b). Our technique, being based on ν-dyadic partitions of unity induced by (Ω, ψ), is an obvious modification thereof and works for distributions with values in arbitrary Banach spaces. Some results on vector-valued Besov spaces are also found in the work of H.-J. Schmeißer and W. Sickel [SS01], [SS05]. Also see R. Denk and M. Kaip [DK13]. It has been pointed out to the author by S.I. Hiltunen that the duality Theorem 2.3.1 can be generalized as follows: in the case q 6= ∞, the assumption that E has a separable dual can be replaced by the more general condition that E itself is separable. The corresponding proof relies on Lebesgue spaces whose elements take their values in certain locally convex spaces (see Remark 2 in [Hil18]). The fundamental Fourier multiplier theorem 2.4.2 is an anisotropic extension of the corresponding result presented, in the isotropic case, by the author in [Ama97] (also see the announcement of L. Weis [Wei97]). For the sake of simplicity, we do not insist on minimal differentiability requirements for the symbol

132

VII Function Spaces

but employ the universal multiplier space M. It follows from Remark VI.3.6.4(b) that, everywhere in this section, M can be replaced by M2 |ω| . In the homogeneous case it suffices to assume that the to class C d+1 (see [Ama97]), ¡ d ¢ symbol belongs 2 |ω| 2d whereas a ∈ M R , L(E, F ) requires a ∈ C . The order d + 1 is optimal as long as no restriction on the Banach space E is imposed. It has been shown by M. Girardi and L. Weis [GiW03] that we can get away with less smoothness of the symbol if E belongs to suitably restricted classes of Banach spaces. The C k(ν) regularity for the symbols is only needed in Subsection 5.3, where we show that M is a multiplier space for Triebel–Lizorkin spaces. The other material of this subsection consists of more or less straightforward amplifications of the corresponding results in [Ama09]. Of course, most of the theory of Besov spaces developed in this section reduces, in the classical isotropic scalar case, to well-known results, essentially contained in H. Triebel’s books, in particular in [Tri83]. In that book, as well as in [Tri95], there is also sketched the possibility of defining anisotropic Besov (and Triebel–Lizorkin) spaces by Fourier analytic techniques. A noteworthy exception is Theorem 2.6.3 whose isotropic scalar counterpart seems to appear for the first time in [Ama00b, Theorem 2.1]. Last but not least, we direct the reader’s attention to Section 2.2 of [Tri83] for a survey of the historical evolution of the theory of function spaces. Detailed references to the early developments, in particular by the work of S.M. Nikol 0 ski˘ı and his students, notably O.V. Besov, are found in [Tri95] and in A. Kufner, O. John, and S. Fuˇcik [KuJF77].

VII.3 Slobodeckii and H¨ older Spaces

3

133

Intrinsic Norms, Slobodeckii and H¨older Spaces s/ν

The one sided Besov space scales [ Bp,r ; s > 0 ], 1 ≤ p, r ≤ ∞, are of particular importance. Indeed, they can be characterized intrinsically and contain the (in the s/ν isotropic case) widely used Slobodeckii spaces Wr for 1 ≤ r < ∞ and s ∈ / νN, s/ν and the H¨older spaces BUC for r = ∞ and s ∈ / νN. For these reasons they are investigated in the present section in some detail. The main results are contained in Theorems 3.5.1 and 3.5.2. The former contains a significant intersection space representation and the latter a characterization by a most useful norm. In order to arrive at these results we have to establish some facts on pair-wise commuting strongly continuous contraction semigroups, translation semigroups in particular. This is done in Subsections 3.1–3.3. In the following three subsections we introduce (anisotropic) Besov, Besov–Slobodeckii and Besov–H¨older spaces and apply the foregoing results to deduce the theorems mentioned above. In connection with quasilinear parabolic equations, Besov–H¨older spaces are often most useful. However, this scale lacks the property of dense injection whose importance is manifest by the considerations in Volume I. In the next to last subsection we give an intrinsic characterization of the class of anisotropic little H¨older spaces which do not possess these ‘shortcomings’. Finally, in the last subsection, we characterize Besov–Slobodeckii and Besov– H¨older spaces on open corners.

3.1

Commuting Semigroups

Let X = (X, k·k) be a Banach space. Given linear operators A and B in X, the product AB is the linear operator in X defined by dom(AB) :=

©

x ∈ dom(B) ; Bx ∈ dom(A)

ª

,

ABx := A(Bx) .

Then A and B commute, if ABx = BAx ,

x ∈ dom(AB) ∩ dom(BA) .

If A, B ∈ C(X), that is, A and B are closed, then A and B are resolvent commuting if £ ¤ (λ + A)−1 , (µ + B)−1 = 0 , λ ∈ ρ(−A) , µ ∈ ρ(−B) , where [·, ·] denotes the commutator. 3.1.1 Lemma If 10 A, B ∈ G(X) are resolvent commuting, then they commute. 10 Recall that G(X) is the set of all negative infinitesimal generators of strongly continuous semigroups on X.

134

VII Function Spaces

Proof Lemma III.4.9.1 guarantees that Ae−tB ⊃ e−tB A for t > 0. Hence, given x ∈ dom(AB) ∩ dom(BA), t−1 A(e−tB − 1)x = t−1 (e−tB − 1)Ax .

(3.1.1)

Since Ax ∈ dom(B), the right side converges for t → 0 towards −BAx. Furthermore, x ∈ dom(B) so that t−1 (e−tB − 1)x → −Bx as t → 0. Now the closedness of A implies that the left side of (3.1.1) converges toward −ABx. This proves the claim. ¥ Now we suppose that A1 , . . . , Ad are pair-wise commuting linear operators in X and set A := (A1 , . . . , Ad ). For α ∈ Nd we define Aα by A0 := 1 = 1E and dom(Aα ) :=

d \

α

dom(Aj j ) ,

αd 1 Aα x := Aα 1 · · · Ad x .

j=1

Note

α

α

σ(1) σ(d) Aα = Aσ(1) · · · Aσ(d) = B1 · · · B|α| ,

where σ ∈ Sd and B1 , . . . , B|α| ∈ {A1 , . . . , Ad } with αj of the B1 , . . . , B|α| being equal to Aj . Suppose A1 , . . . , Aα ∈ −G(X) are pair-wise resolvent commuting.

(3.1.2)

Given m ∈ νN, we put m/ν

KX

(A) :=

³ \ α p ω≤m

´ dom(Aα ), k·kK m/ν (A) ,

where kxkK m/ν (A) := X

(3.1.3)

X

X α p ω≤m

kAα xk .

q 3.1.2 Lemma Let (3.1.2) be satisfied and m ∈ ν N. Then d

m/ν

(i) K = KX (A) is a Banach space such that K ,→ X. (ii) K is invariant under Uj := { etAj ; t ≥ 0 } for j = 1, . . . , d. (iii) The K-realizations Uj,K of Uj are pair-wise commuting strongly continuous semigroups on K, and the infinitesimal generator of Uj,K is the K-realization Aj,K of Aj . q (iv) Suppose n ∈ ν N. Then (m+n)/ν

KX

where AK := (A1,K , . . . , Ad,K ).

n/ν

(A) = KK (AK ) ,

VII.3 Slobodeckii and H¨ older Spaces

135

Proof (1) It is clear that k·kK is a norm on K and K ,→ X. Hence we have to show that K is complete. For this we proceed by induction on d. (2) Assume d = 1. Since A = A1 ∈ C(X) and ρ(A) 6= ∅, Theorem 2.16.4 in [HillP57] guarantees that Aj ∈ C(X) for j ∈ N. Thus the assertion is clear in this case. (3) Suppose d ≥ 2 and that it has already been shown that K is complete if it is constructed with d − 1 pair-wise resolvent commuting generators. We set A := A1 and B := (A2 , . . . , Ad ). Then, writing α = (j, β) with j ∈ N and β ∈ Nd−1 , and, analogously, ω = (ω1 , ω 0 ), \ K= dom(Aj B β ) 0 p jω1 +β ω ≤m and kxkK =

X jω1 +β p ω 0 ≤m

kAj B β xk .

Let (xk ) be a Cauchy sequence in K. Then (Aα xk ) = (Aj B β xk ) is a Cauchy sequence in X for each α = (j, β) with α q ω = jω1 + β q ω 0 ≤ m. Hence there exist xα ∈ X such that Aα x = Aj B β xk → xα in X as k → ∞. In particular, xk → x0 =: x in X. We have to show that x ∈ K and Aα xk → Aα x if α q ω ≤ m. For this we have to verify that x ∈ dom(Aα ) and Aα xk → Aα x for each α ∈ Nd with α q ω ≤ m. We fix β and proceed by induction on j. For abbreviation, y := xα and C := B β . Suppose j = 0. Then the assertion follows from the induction hypothesis. Assume the claim is true for 0 ≤ i ≤ j − 1 with jω1 + β q ω 0 ≤ m. Choose any λ ∈ ρ(A) and note that Aα = (λ − A)p(A) + r(A) ,

(3.1.4)

where p(A) := −

j−1 X

λj−1−i Ai C ,

r(A) := λj C .

(3.1.5)

i=0

Hence we get from xk ∈ K ⊂ dom(Aα ) ⊂ dom(Ai C), 0 ≤ i ≤ j, and (3.1.5) that ¡ ¢ p(A)xk = (λ − A)−1 Aα xk − r(A)xk . The induction hypothesis guarantees that x belongs to dom(Aj−1 C) and r(A)xk = λj Cxk → r(A)x . Thus

¡ ¢ p(A)xk → (λ − A)−1 xα − r(A)x

136

VII Function Spaces

¡ ¢ in X. Using (3.1.5) and once more the induction hypothesis, we find that p(A)xk converges in X towards p(A)x. Hence ¡ ¢ p(A)x = (λ − A)−1 xα − r(A)x . This shows that p(A)x ∈ dom(A) and (λ − A)p(A)x = xα − r(A)x. Hence ¡ ¢ x ∈ dom(Aα ) = (λ − A)−1 dom(Aj−1 C) and Aα x = xα . Since these arguments apply to each permutation of A1 , . . . , Ad , we have shown that K is well-defined and complete. (4) Assume A ∈ −G(X). Then, by (s 7→ esA x) ∈ C(R+ , X), Z 1 t sA lim e x ds = x , x∈X . (3.1.6) t→0 t 0 Moreover, Z t esA x ds ∈ dom(A) ,

Z A

0

Z

t

sA

e

t

x ds =

0

AesA x ds = etA x − x

(3.1.7)

0

for t > 0 and x ∈ X, as is well-known and readily seen. We set n := m/ω1 + · · · + m/ωd and let Bi ∈ {A1 , . . . , Ad }, 1 ≤ i ≤ n, with m/ωj of the B1 , . . . , Bn being equal to Aj , 1 ≤ j ≤ d. Then we put Z Z t n 1 t xt := n ··· e i=1 si Bi x ds1 · · · dsn , t>0, x∈X , t 0 0 where e

n i=1

si B i

= es1 B1 · · · esn Bn .

(3.1.8)

Since A1 , . . . , Ad are pair-wise resolvent commuting, Lemma III.4.9.1 guarantees that the semigroups { esBi ; s ≥ 0 }, 1 ≤ i ≤ n, are pair-wise commuting. Thus (3.1.8) is well-defined. It follows from (3.1.6) that xt → x in X. From (3.1.7) we infer that d \ m/ω xt ∈ dom(Aj j ) =: Y . j=1 d

Hence Y ,→ X. If α q ω ≤ m, then αj ≤ m/ωj . Thus Y ⊂ D(Aα ). Consequently, Y ⊂ K, which shows that K is dense in X. This proves claim (i). (5) Assertion (ii) is immediate from Lemma III.4.9.1 and (3.1.7). (6) Let j ∈ {1, . . . , d} and set A := Aj . By (ii), the K-realization of U := Uj is simply the restriction of U to K. Suppose x ∈ K. Then Aα etA x = etA Aα x → Aα x

in X

as

t→0

for α q ω ≤ m. Thus etA x → x in K. This shows that UK is strongly continuous.

VII.3 Slobodeckii and H¨ older Spaces

137

Let B be the infinitesimal generator of UK . Suppose x ∈ dom(B). Then t−1 (etA x − x) → Bx in K as t → 0. Hence K ,→ X implies that this convergence takes place in X as well. Thus x ∈ dom(A) and Ax = Bx. This proves AK ⊃ B. Conversely, assume x ∈ dom(AK ). Then x ∈ K, Ax ∈ K, and t−1 (etA x − x) → Ax

(3.1.9)

in X. From Lemma 3.1.1 we infer ¡ ¢ Aα t−1 (etA x − x) − Ax = t−1 (etA Aα x − Aα x) − A(Aα x) → 0 for α longs 1≤j

q ω ≤ m. Hence convergence (3.1.9) takes place in K. Consequently, x beto dom(B) and Ax = Bx, that is, AK ⊂ B. Clearly, the semigroups Uj,K , ≤ d, are pair-wise commuting. This proves (iii). n/ν

(7) It follows from (i) and (iii) that M := KK (AK ) is a well-defined Banach space. Furthermore, kxkM =

X β p ω≤n

kAβK xkK = =

implies assertion (iv).

X

X

β p ω≤n α p ω≤m

X

X

α p ω≤m β p ω≤n

kAα AβK xk kAα+β xk = kxkK (m+n)/ν (A) X

¥

3.1.3 Remark Let A ∈ C(X). A linear subspace Y of dom(X) is a core for A, if A equals the closure of its restriction to Y . Suppose A ∈ −G(X). Then the core theorem guarantees that Y ⊂ dom(A) is a core for A, if Y is dense in X and invariant under the semigroup { etA ; t ≥ 0 } (e.g., [Dav80, Theorem 1.9]). Thus m/ν Lemma 3.1.2 shows that KX (A) is a core for each Aj . ¥

3.2

Semigroups and Interpolation

In this subsection we present equivalent norms for the real interpolation spaces (X, K)θ,r . To do this, we need some preparation. Thus we assume: A1 , . . . , Ad are pair-wise resolvent commuting generators of © ª strongly continuous contraction semigroups Uj (t) ; t ≥ 0 on X.

(3.2.1)

138

VII Function Spaces

Preliminary Estimates q 3.2.1 Lemma Suppose αj ∈ N and βj ∈ N. Set γ := α1 β1 + · · · + αd βd . Then d Y ¡

¢αj Uj (tβj ) − 1

j=1

= 2−γ

d Y ¡

Uj ((2t)βj ) − 1

¢αj

+

j=1

d Y d X ¡ ¢α ¡ ¢ U` (tβ` ) − 1 ` Uj (tβj ) − 1 p(t) j=1 `=1

¡ ¢ for t ∈ R+ , where p(t) := p U1 (tβ1 ), . . . , Ud (tβd ) with a polynomial p in d indeterminates. Proof (1) In this step we denote by q polynomials qin one indeterminate, not necessarily the same at different occurrences. Let r ∈ N. Then ¡ ¢r 2r = z + 1 − (z − 1) = (z + 1)r + (z − 1) q(z) , z∈C. By multiplying this relation by 2−r (z − 1)r we arrive at (z − 1)r = 2−r (z 2 − 1)r + (z − 1)r+1 q(z) , q Now we claim that, given s ∈ N,

z∈C.

(3.2.2)

s

(z − 1)r = 2−rs (z 2 − 1)r + (z − 1)r+1 q(z) ,

z∈C. (3.2.3) q Indeed, suppose its validity has been shown for some s ∈ N. Then we replace z in s (3.2.2) by z 2 to get s

(z 2 − 1)r = 2−r (z 2

s+1

− 1)r + (z − 1)r+1 q(z) ,

z∈C.

By inserting this expression into (3.2.3) we see that this identity applies with s replaced by s + 1. This proves the claim. (2) We apply (3.2.3) with r = αj and s = βj to Uj (tj ). This gives ¡ ¢αj ¡ ¢αj ¡ ¢αj +1 ¡ ¢ Uj (tj ) − 1 = 2−αj βj Uj (2βj tj ) − 1 + Uj (tj ) − 1 qj Uj (tj ) . Now we replace tj by tβj and multiply the resulting identities. Then we get the assertion. ¥ 3.2.2 Lemma Let αj , βj , and γ be as above and suppose rj ∈ N with rj ≥ αj . Then d Z ³ X kAα xk ≤ c kxk + j=1

for x ∈ D(Aα ).

∞ 0

´ °¡ ¢rj ° t−γ ° Uj (tβj ) − 1 x° dt/t

(3.2.4)

VII.3 Slobodeckii and H¨ older Spaces

Proof

139

(1) It follows from (3.1.6) and (3.1.7) that, given x ∈ D(Akj ), Akj x = lim t−k (e−tAj − 1)k x , t→0

Hence, if x ∈ D(Aα ),

k∈N.

¡ ¢α Aα x = lim t−α U (t) − 1 x , t→0

(3.2.5)

where t := (t1 , . . . , td ) ∈ (0, ∞)d and U (t) := U1 (t1 ) · · · Ud (td ). Putting tj := tβj in (3.2.5), it follows d Y ¡ ¢αj −γ Ax = lim t Uj (tβj ) − 1 x . t→0

j=1

We write ϕ(t) for the argument of limt→0 . Then we get from ¡ ¢ ¡ ¢ ϕ(t) + ϕ(2−1 t) − ϕ(t) + · · · + ϕ(2−k t) − ϕ(2−k+1 t) = ϕ(2−k t) that, given any t > 0, Aα x = ϕ(t) +

∞ X ¡ ¢ ϕ(2−k t) − ϕ(2−k+1 t) k=1

=t

−γ

Y¡ ¢αj Uj (tβj ) − 1 x j

∞ h X Y¡ ¢αj + (2−k t)−γ Uj ((2−k t)βj ) − 1 x

(3.2.6)

j

k=1

− (2−k+1 t)−γ

Y¡ ¢αj i Uj ((2−k+1 t)βj ) − 1 x . j

Next we consider the difference between the term on the left side of the identity in Lemma 3.2.1 and the first one on the right side, replace t by 2−k t, and multiply the outcome by (2−k t)γ . The resulting expression, applied to x, then equals the one in the bracket of (3.2.6). Thus, letting 1/2 ≤ t ≤ 1, we arrive at the estimate kAα xk ≤ kxk + c

∞ X

(2−k t)−γ

d °Y d X ¡ ¢α ¡ ¢ ° ° ° U` ((2−k t)β` ) − 1 ` Uj ((2−k t)βj ) − 1 x° . ° j=1

k=1

`=1

¡ ¢ Here we used the commutativity of p(t) and Ui (2−k t)βi and the uniform boundedness of p(t). Integration with respect to dt/t over [1/2, 1] yields kAα xk

³

Z

1

≤ c kxk + 0

d Y d °X ´ ¡ ¢α ¡ ¢ ° ° ° t−γ ° U` (tβ` ) − 1 ` Uj (tβj ) − 1 x° dt/t , j=1 `=1

where we also enlarged the range of integration on the right side.

(3.2.7)

140

VII Function Spaces

(2) Suppose αj ∈ N satisfy γ := α1 β1 + · · · + αd βd > γ. Then we replace αj in Lemma 3.2.1 by αj , multiply the resulting identity by t−γ , and integrate to arrive at the estimate Z 1 d °Y ¡ ¢αj ° ° −γ ° t ° Uj (tβj ) − 1 x° dt/t 0

j=1

Z

1

−γ

≤2

d °Y ¡ ¢αj ° ° ° t−γ ° Uj ((2t)βj ) − 1 x° dt/t

0

+c

(3.2.8)

j=1

d Z X

1

d °Y ¡ ¢αj ¡ ¢ ° ° ° t−γ ° U` (tβ` ) − 1 Uj (tβj ) − 1 x° dt/t .

0

j=1

`=1

In the first integral on the right side we use the substitution t0 = 2t. Then we −(γ−γ) get the same term as on the left side, multiplied < 1, except that the R 2 by 2 integration ranges from 0 to 2. The integral 1 . . . dt/t can be estimated by c kxk. Consequently, if the left side is finite, we find Z

1

d °Y ¡ ¢αj ° ° ° t−γ ° Uj (tβj ) − 1 x° dt/t

0

j=1

³ ≤ c kxk +

d Z X j=1

³ = c kxk +

1 0

d Z 1 X j=1

d °Y ´ ¡ ¢α ¡ ¢ ° ° ° t−γ ° U` (tβ` ) − 1 ` Uj (tβj ) − 1 x° dt/t `=1 d °Y ´ ¡ ¢α ° ° ° t−γ ° U` (tβ` ) − 1 j,` x° dt/t ,

0

`=1

where αj,` := αj if ` 6= j, and αj,j := αj + 1. Iteration leads to Z

1

d °Y ¡ ¢αj ° ° ° t−γ ° Uj (tβj ) − 1 x° dt/t

0

j=1

³ ≤ c kxk +

d Z X j=1

1

d °Y ´ ¡ ¢α0 ° ° ° t−γ ° U` (tβ` ) − 1 j,` x° dt/t ,

0

`=1

¡ ¢rj 0 where αj,j ≥ rj for 1 ≤ j ≤ d. Thus, retaining the factor Uj (tβj ) − 1 and majorizing the remaining ones by constants, we obtain Z

1

−γ

t 0

d °Y ¡ ¢αj ° ° ° Uj (tβj ) − 1 x° dt/t ° j=1

³ ≤ c kxk +

d Z X j=1

1 0

´ °¡ ¢rj ° t−γ ° Uj (tβj ) − 1 x° dt/t .

VII.3 Slobodeckii and H¨ older Spaces

141

Now we use this estimate to majorize the right side of (3.2.7). More precisely, in the j-th summand we put α`0 = α` for ` 6= j, and αj0 := αj + 1. Then the assertion follows, provided the left side of (3.2.8) is finite. However, if the left side of (3.2.8) is infinite (with the above choice of αj ), then the right side of (3.2.4) is infinite as well. This proves the lemma. ¥ Renorming Intersections of Interpolation Spaces After these preparations we can establish the main result of this subsection. We recall that [t] is the largest integer less than or equal to t ∈ R. For abbreviation, I := (0, ∞) and q L∗r (I n ) := Lr (I n , dy/|y|n ) , n∈N, 1≤r≤∞. (3.2.9) 3.2.3 Proposition Let assumption (3.2.1) be satisfied. Suppose 0 < s < m with m ∈ νN, and 1 ≤ r ≤ ∞. Then: m/ν (X, KX )s/m,r

(i)

d . \¡ m/ω ¢ = X, D(Aj j ) s/m,r . j=1

(ii) The function x 7→ kxk +

d X ° −s/ω ° j °t k(Uj (t) − 1)m/ωj xk ° j=1 m/ν

is an equivalent norm for (X, KX

L∗ r (I)

(3.2.10)

)s/m,r .

(iii) Assume ω = ω1 and put U (t) := U1 (t1 ) · · · Ud (td ) with t := (t1 , . . . , td ). Then ° ° x 7→ kxk + ° |t|−s/ω k(U (t) − 1)m/ω xk °L∗ (I d ) (3.2.11) r

is also an equivalent norm. Proof (1) Let n1 , . . . , nd ∈ N and 0 < θ < 1. An interpolation result of P. Grisvard [Gri66, Th´eor`eme 7.1] guarantees that d d ³ \ ´ \ ¡ ¡ . n ¢ n ¢ X, D(Aj j ) θ,r = X, D(Aj j ) j=1

θ,r

j=1

.

Furthermore, for k ∈ N and 0 < σ < k, ° ° x 7→ kxkX + °t−σ k(Uj (t) − 1)k xk °L∗ (I) r

¡ ¢ is a norm for X, D(Akj ) σ/k,r (e.g., [Tri95, Theorems 1.15.5 and 1.13.2]).

(3.2.12)

(3.2.13)

142

VII Function Spaces

(2) We assume α ∈ Nd satisfies α q ω ≤ m. Then we set m m aωj ··· , βj := , γ := α1 β1 + · · · + αd βd . ω1 ωd m q Moreover, for k ∈ N, rj := km/ωj and σj := γ/βj . Then it follows from (3.2.4) that d Z ∞ ³ ´ X °¡ ¢ rj ° α kA xk ≤ c kxk + t−σj ° Uj (t) − 1 x° dt/t , a :=

0

j=1

due to rj ≥ m/ωj ≥ αj . We infer from (3.2.13) that the right side is a norm for d \ ¡ ¢ X, D(Arj ) σj /rj ,1 .

j=1

Note that σj /rj = α q ω/km ≤ 1/k. Hence the monotonicity of the interpolation spaces with respect to θ (cf. (I.2.5.2)) implies d \ ¡ ¢ X, D(Akm/ωj ) 1/k,1 ,→ D(Aα ) . j=1

Since this is true for every α ∈ Nd with α q ω ≤ m, we obtain from (3.1.3) d \ ¡

km/ωj

X, D(Aj

)

¢

m/ν

1/k,1

,→ KX

q k∈N.

,

j=1

(3) From interpolation theory it is known that ¡ m/ω km/ωj ¢ D(Aj j ) ,→ X, D(Aj ) 1/k,∞ (e.g., [Tri95, Theorem 1.14.3]). Consequently, d \

m/ωj

D(Aj

) ,→

j=1

d \ ¡

km/ωj

X, D(Aj

)

¢ 1/k,∞

.

j=1

From this embedding, the result of step (2), and (3.2.12) we deduce ³ X,

d \ j=1

km/ωj

D(Aj

´ )

m/ν

1/k,1

,→ KX

,→

d \

m/ωj

D(Aj

)

j=1

³ ,→ X,

d \ j=1

km/ωj

D(Aj

´ )

1/k,∞

q for k ∈ N. Now (i) is a consequence of the reiteration theorem (I.2.8.2).

VII.3 Slobodeckii and H¨ older Spaces

143

(4) Assertion (ii) follows from (i), (3.2.12), and (3.2.13). (5) Suppose ω = ω1 and set σ := s/ω and k := m/ω. Note that U (t) − 1 =

d X

¡ ¢ U1 (t1 ) · · · Uj−1 (tj−1 ) Uj (tj ) − 1 ,

j=1

where the empty product is given the value 1. From this and the multinomial theorem we estimate d X Y °¡ °¡ ° ¢ ° ¢ ° U (t) − 1 k x° ≤ c ° Uj (tj ) − 1 αj x° .

(3.2.14)

|α|=k j=1

Now we apply Lemma 3.2.1 with β1 = · · · = βd := ω, replace tβ by t, and take the L∗r (I d ) norm. Then d X ° °° ¡ ¢ ° −σ Y ° ° Uj (tj ) − 1 αj x° ° ° |t| ° j=1

|α|=k

d X ° °° ¡ ¢ ° −σ Y ° ° Uj (2ω tj ) − 1 αj x° ° ° |t| °

≤ 2−m

j=1

|α|=k

+c

d L∗ r (I )

d L∗ r (I )

d ° d X X ¡ ¢ ¡ ¢ °° ° −σ Y ° ° U` (t` ) − 1 αj Uj (tj ) − 1 x° ° ° |t| ° |α|=k j=1

`=1

d L∗ r (I )

.

In the first term on the right side we use the substitution 2ω t → τ . Then we get the same expression as the one on the left, but multiplied by 2−(m−s) < 1. Hence the left side can be estimated by the second term on the right. By iterating this estimate we find, similarly as in step (2) of Lemma 3.2.2, d X ° °° ¡ ¢ ° −σ Y ° ° Uj (tj ) − 1 αj x° ° ° |t| ° j=1

|α|=k

d L∗ r (I )

d ° X ¡ ¢k ° ° ° −σ ° ° ≤c ° |t| ° Uj (tj ) − 1 x° ° j=1

(3.2.15) d L∗ r (I )

.

¡ ¢ Assume r < ∞. Fix j and write ξ := tj and η := t1 , . . . , b tj , . . . , td . Then ° ¡ ¢k ° ° ° −σ ° ° ° |t| ° Uj (tj ) − 1 x° ° ∗ d Lr (I ) ³Z ∞ Z °¡ ¢k °r ´1/r = (ξ 2 + |η|2 )−(σr+d)/2 dη ° Uj (ξ) − 1 x° dξ 0

I d−1

144

VII Function Spaces

and

Z

Z 2

2 −(σr+d)/2

(ξ + |η| )

dη = ξ

−(σr+d)+d−1

I d−1

(1 + |ζ|2 )−(σr+d)/2 dζ Id

= c ξ −σr−1 . Consequently, ° ¡ ¢k ° ° ° −σ ° ° ° |t| ° Uj (tj ) − 1 x° °

d L∗ r (I )

° °¡ ¢k ° ° ° ° ≤ c °t−σ ° Uj (t) − 1 x° °

L∗ r (I)

.

(3.2.16)

If r = ∞, then it is obvious that (3.2.16) applies as well. By combining estimates (3.2.14)–(3.2.16), we get ° °° ¡ ¢ ° −s/ω ° ° U (t) − 1 m/ω x° ° ° |t| °

d L∗ r (I )

≤c

d ° X °° ¡ ¢ ° −s/ω ° ° Uj (t) − 1 m/ω x° ° °t ° j=1

L∗ r (I)

(6) Retaining the notations of step (5), we find k X

(−1)j

j=0

k k ³ ´ ³ ´X ³ ´ X k k k j` k−` (z j − w)k = (−1)j (−1)k−` z w j j ` j=0

=

k X

`=0

(−1)k−`

j=0

`=0

=

k X `=0

k ³ ´ ³ ´ k k−` X k j` w (−1)j z ` j

(−1)k−`

³ ´ k k−` w (1 − z ` )k `

for z, w ∈ C. Leaving only the term with j = 0 on the left, we obtain (1 − w)k =

k X j=0

(−1)j

³ ´¡ ¢ k (z j − 1)k wk−j − (z j − w)k . j

Here we replace z by U1 (t)U (t) and w by U1 (t). Then ¡

1 − U1 (t)

¢k

k ³ ´³¡ X ¢k ¡ ¢ k = (−1)j U1 (jt)U (jt) − 1 U1 (k − j)t j j=1

´ ¡ ¢k − U1 ((j − 1)t)U (jt) − 1 U1 (kt) .

Hence k ³ X °¡ °¡ ¢ ° ¢ ° ° U1 (t) − 1 k x° ≤ c ° U (j(t + te1 ) − 1)k x° j=1

°¡ ¢ k °´ + ° U (jt + (j − 1)te1 ) − 1 x° ,

.

VII.3 Slobodeckii and H¨ older Spaces

145

where e1 := (1, 0, . . . , 0) ∈ Rd . We integrate this inequality over [0, t]d to obtain k Z ³° ¡ X °¡ ¢ ° ¢ ° ° U1 (t) − 1 k x° ≤ c ° U (jt + jte1 ) − 1 k x° d t j=1 [0,t]d (3.2.17) °¡ ¢k °´ + ° U (jt + (j − 1)te1 ) − 1 x° dt .

We set U 0 (t0 ) := U2 (t2 ) · · · Ud (td ) for t0 = (t2 , . . . , td ). Then a substitution of variables gives Z °¡ ¢ ° ° U (jt + jte1 ) − 1 k x° dt [0,t]d

Z tZ

= Z

0

[0,t]d−1 2kt

Z

kt

≤

Z

kt

··· 0

0

0

°¡ ¢ ° ° U (t) − 1 k x° dt .

0

From this we get °¡ ¢k ° t−σ ° U1 (t) − 1 x° Z 2kt Z kt Z ≤c ··· 0

°¡ ¢ ° ° U1 (j(t1 + t))U 0 (jt0 ) − 1 k x° dt0 dt1

kt ³ 0

¡ ¢k ° dt |t| ´d+σ −σ ° |t| ° U (t) − 1 x° d . t |t|

(3.2.18)

Consider the integral operator Kf , defined for measurable functions f : R → C by Z 2kt Z kt Z kt ³ ´d+σ |t| dt Kf (t) := ··· f (t) d . t |t| 0 0 0 If f ∈ L∗∞ (I d ), then

kKf kL∗∞ (I) ≤ c kf kL∗∞ (I d ) .

Suppose f ∈ L∗1 (I d ). Then Z ∞ Z ∞ Z 2kt Z ∞ Z ∞ dt 1 ³ |t| ´d+σ+1 dt |Kf (t)| ≤ ··· |f (t)| d t |t| t |t| 0 0 0 Z0 Z 0∞ 1 ³ |t| ´d+σ+1 dt = dt |f (t)| d t |t| I d |t|/2k |t| Z Z ∞ dt = τ −(d+σ+1) dτ |f (t)| d |t| I d 1/2k Z dt =c |f (t)| d . |t| Id This shows that K is a continuous map¢from L∗1 (I d ) into L∗1 (I), and from ¡ ∗ linear ∗ d ∗ d L∞ (I ) into L∞ (I). Thus K ∈ L Lr (I ), L∗r (I) by the Riesz–Thorin interpolation

146

VII Function Spaces

theorem (e.g., [BeL76, Theorem 1.1.1] or [Tri95, Theorem 1.18.7]). Consequently, °¡ ¢k ° letting f (t) := t−σ ° U1 (t) − 1 x°, we obtain from (3.2.18) ° ¡ ¢k ° ° ° −σ ° ° °t ° U1 (t) − 1 x° °

L∗ r (I)

° °¡ ¢k ° ° ° ° ≤ c ° |t|−σ ° U (t) − 1 x° °

d L∗ r (I )

.

Of course, analogous estimates apply to U2 , . . . , Ud . This implies d ° X °° ¡ ¢ ° −s/ω ° ° Uj (t) − 1 m/ω x° ° °t ° j=1

L∗ r (I)

° °¡ ¢m/ω ° ° ° ° ≤ c ° |t|−σ ° U (t) − 1 x° °

Together with step (5), this proves assertion (iii).

3.3

d L∗ r (I )

.

¥

Translation Semigroups

Let K be a closed corner in Rd and X ∈ {Rd , K}. Recall that assumption (VI.3.1.20) applies. Since X is a convex cone, it is invariant under addition: X + X ⊂ X. Hence the (d-parameter) left translation semigroup { λy ; y ∈ X } is well-defined on L1,loc (X, E) by λy u(x) := u(x + y) , a.a. x, y ∈ X . As usual, we employ the same symbol for its restriction to any of its invariant subspaces. q For a ∈ X we consider the one-parameter semigroup La := { λta ; t ≥ 0 }. We also denote by d X ¡ ¢ ∂a := aj ∂j ∈ L S 0 (X, E) j=1

the distributional directional derivative in the direction of a = (a1 , . . . , ad ). 3.3.1 Theorem Let K be a closed corner in Rd and X ∈ {Rd , K}. (i) La is a strongly continuous semigroup on S(X, E) and on S 0 (X, E). (ii) Suppose

© ª X ∈ C0 (X, E), BUC(X, E), Lq (X, E) .

Then La is a strongly continuous contraction semigroup on X . Its infinitesimal generator is the X -realization of the distributional directional derivative ∂a . (iii) In either case La is a strongly continuous group if X = Rd .

VII.3 Slobodeckii and H¨ older Spaces

Proof

147

(1) Let u ∈ S = S(X, E) and k, m ∈ N. Then, given y ∈ X, qk,m (λy u) = sup max hxik |∂ α (λy u)(x)|E x∈X |α|≤m

≤ c(k, y) sup max hx + yik |∂ α u(x + y)|E = c(k, y) qk,m (u) , x∈X |α|≤m

¡ ± ¢k where c(k, y) := supx∈X hxi hx + yi . Since K is closed, it follows from the definition of S that λy ∈ L(S). Suppose |y| ≤ 1. Then we infer from ³ ´1/2 hxi hxi 1 + |x|2 ≤ ≤ ≤c hx + tyi 1 + |x|2 − 2 |x| + 1 (1 + (|x| − t |y|)2 )1/2 for x ∈ X and 0 ≤ t ≤ 1, and the mean-value theorem, Z 1 ¡ ¢ (λy − 1)u (x) = u(x + y) − u(x) = ∂u(x + ty)y dt ,

(3.3.1)

0

that ¯ ¡ ¢ ¯ hxik ¯∂ α (λy − 1)u (x)¯E ≤ c |y| qk,m+1 (u) ,

x∈X,

k, m ∈ N .

Hence ¡ ¢ qk,m (λy − 1)u ≤ c |y| qk,m+1 (u) ,

y∈X,

|y| ≤ 1 ,

k, m ∈ N .

(3.3.2)

This implies that La is a strongly continuous semigroup on S. For u ∈ S and x ∈ X, ¡ ¢ t−1 (λta − 1)u(x) = t−1 u(x + ta) − u(x) → ∂a u(x) in E as t → 0. Hence, using the mean-value theorem once more, Z 1 ¡ −1 ¢ t (λta − 1) − ∂a u = (λτ ta − 1)∂a u dτ , t>0.

(3.3.3)

0

From this and (3.3.2) we get, for k, m ∈ N, ¡ ¢ qk,m (t−1 (λta − 1) − ∂a )u ≤ ctqk,m+1 (u) ,

t>0.

Thus t−1 (λta − 1)u → ∂a u in S as t → 0. Hence ∂a ∈ L(S) is the infinitesimal generator of La on S. © ª (2) Suppose X ∈ C0 (X, E), Lq (X, E) and u ∈ X . It is obvious, due to the closedness of K, that λta u ∈ X and kλta ukX ≤ kukX for t ≥ 0. Thus La is a contraction semigroup on X . Hence, given u ∈ X and v ∈ S, kλta u − ukX ≤ kλta (u − v)kX + kλta v − vkX + kv − ukX ¡ ¢ ≤ 2 ku − vkX + q0,0 (λta − 1)v

148

VII Function Spaces d

for t > 0. Let ε > 0. Since S ,→ X , there¡ exists v ∈¢S with ku − vkX < ε/3. By step (1) there exists tε > 0 such that q0,0 (λta − 1)v < ε/3 for 0 < t < tε . Hence k(λta − 1)ukX < ε for 0 < t < tε . This proves that La is strongly continuous on X . By step (1), S is invariant under La and contained in the domain of its infinitesimal generator A. Hence, by the core theorem, S is a core for A. Thus, given u ∈ dom(A), there exists a sequence (uk ) in S such that uk → u ,

∂a uk = Auk → Au in X .

From (VI.1.2.6), which holds for X = Rd also, we get ∂a ∈ L(S 0 ) with S 0 := S 0 (X, E). Theorem VI.1.2.3(i) guarantees that S is dense in S 0 . From this and (1.2.1) we infer that ∂a uk → ∂a u and ∂a uk → Au in S 0 . Hence Au = ∂a u for u ∈ dom(A), which shows that A is the X -realization of ∂a . (3) Suppose X = BUC(X, E). Set Y := BUC 2 (X, E). It is again obvious that Y is an invariant linear subspace for La . Given u ∈ Y, it follows from (3.3.1) and (3.3.3) that °¡ −1 ¢ ° t (λta − 1) − ∂a uk∞ ≤ t k∂a2 uk∞ , t>0. Thus Y ⊂ dom(A) and Au = ∂a u for y ∈ Y. Theorem 1.6.4 implies that Y is dense in X . Thus Y is a core for A. Similarly as in the preceding step, we now deduce from the core theorem that A is the X -realization of ∂a . ¥ 3.3.2 Remark If K is not closed, then neither S(K, E) nor C0 (K, E) nor BUC(K, E) is invariant under La since the ‘boundary conditions’ u | ∂j K = 0 for j ∈ J ∗ are not preserved. ¥ We denote by e1 , . . . , ed the standard basis of Rd . Then Uj := Lej , 1 ≤ j ≤ d, are pair-wise commuting strongly continuous contraction semigroups on X . The infinitesimal generator of Uj is the X -realization of the distributional derivative ∂j on X. 3.3.3 Lemma Suppose

© ª X ∈ C0 (X, E), BUC(X, E), Lq (X, E) . q Set ∂ := (∂1 , . . . , ∂d ). Then, given m ∈ ν N,  m/ν  C0 (X, E) , if X = C0 (X, E) ,   m/ν KX (∂) = BUC m/ν (X, E) , if X = BUC(X, E) ,    Wqm/ν (X, E) , if X = Lq (X, E) . Proof It follows from Theorem 3.3.1 and Lemma III.4.9.1 that −∂1 , . . . , −∂d are pair-wise resolvent commuting elements of G(X ). Hence the assertion is a consequence of (3.1.3) and definitions (1.1.4), (1.1.7) and Theorem 1.1.4(i), due to the fact that S(X, E) is a core for ∂j if X = C0 (X, E). ¥

VII.3 Slobodeckii and H¨ older Spaces

3.4

149

Renorming Besov Spaces

Now we apply the general results of the preceding subsections to Besov spaces of positive order. This leads to important characterizations by intrinsic norms. Recall definition (3.2.9) of L∗q . Suppose K is a corner in Rd and X ∈ {Rd , K}. For u ∈ L1,loc (X, E) we put 4y u(x) := u(x + y) − u(y) = (λy − 1)u(x) ,

a.a. x, y ∈ X .

(3.4.1)

Moreover, 4k+1 := 4y 4ky for k ∈ N with 40y = id. Given s > 0 and p, r ∈ [1, ∞], y ° −s ° [s]+1 ° [u]s,p,r,j = [u]E k4tej ukLp (X,E) °L∗ (I) s,p,r,j := t

(3.4.2)

° −s ° ° [u]s,p,r = [u]E k4[s]+1 ukLp (X,E) °L∗ (I d ) . s,p,r := |y| y

(3.4.3)

r

and r

Furthermore, [·]s,p,j := [·]s,p,p,j ,

[·]s,p := [·]s,p,p . s/ν

By means of these seminorms we can characterize Bp,r (X, E) intrinsically. In the following theorem we set, for m ∈ νN, ( Wpm/ν (X, E) , if p < ∞ , m/ν Xp := BUC m/ν (X, E) , if p = ∞ , 0/ν

and Xp := Xp

s/ν

s/ν

. Moreover, Bp,r = Bp,r (X, E).

3.4.1 Theorem Let K be a closed corner in Rd and X ∈ {Rd , K}. Suppose s > 0 and 1 ≤ p, r ≤ ∞. (i) It holds s/ν u ∈ Bp,r

iff

u ∈ Xp

and

[u]s/ωj ,p,r,j < ∞ for 1 ≤ j ≤ d .

Moreover, ∗

k·ks/ν,p,r := k·kp +

d X

[·]s/ωj ,p,r,j

(3.4.4)

j=1 s/ν

is a norm for Bp,r . (ii) Let the weight system be homogeneous, that is, ω = ν1, and assume m ∈ νN. (α) If m < s, then s/ν u ∈ Bp,r

iff u ∈ Xpm/ν

and [∂ α u](s−m)/ν,p,r < ∞ for |α| ≤ m/ν .

150

VII Function Spaces

Furthermore,

X

(m)

u 7→ |||u|||s/ν,p,r := |||u|||m/ν,p +

[∂ α u](s−m)/ν,p,r

|α|≤m/ν s/ν

is a norm for Bp,r . (β) If m ∈ νN and m < s < m + ν, then s/ν u ∈ Bp,r

iff u ∈ Xpm/ν

and [∂ α u](s−m)/ν,p,r < ∞ for |α| = m/ν ,

and

X

u 7→ |||u|||s/ν,p,r := |||u|||m/ν,p +

[∂ α u](s−m)/ν,p,r

|α|=m/ν

is a norm for

s/ν Bp,r .

Proof (1) We fix m ∈ νN with s < m. Then Lemma 3.3.3 and Theorems 2.7.4 and 2.8.3 imply ¡ ¢ m/ν s/ν . Bp,r = Xp , KXp (∂) s/m,r . (3.4.5) Thus it follows from Proposition 3.2.3(ii) that u 7→ kukp +

d X ° −s/ω ° m/ω j °t k4tej j ukLp °L∗ (I) r

j=1 s/ν

is a norm for Bp,r . Due to (3.2.13), we can replace m/ωj by [s/ωj ] + 1. This proves (i). (2) Suppose ω = ν1 and m ∈ νN satisfies m < s. Assume n − ν ≤ s < n with m/ν n ∈ νN. Set θ := (s − m)/(n − m) and Yp := KXp (∂). Then Lemma 3.1.2(iv) implies ¡ ¢ ¡ m/ν ¢ (n−m)/ν n/ν Yp , KYp (∂) θ,r = KXp (∂), KXp (∂) θ,r . m/ν

n/ν

By Lemma 3.3.3, the space on the right equals (Xp , Xp from Theorems 2.7.4 and 2.8.3 ¡ ¢ (n−m)/ν s/ν . Bp,r = Y p , K Yp (∂) θ,r .

)θ,r . Thus we obtain

Now Proposition 3.2.3(iii) implies that ° ° u 7→ |||u|||m/ν,p + ° |y|−(s−m)/ν k4(n−m)/ν ukYp °L∗ (I d ) y r

(3.4.6)

s/ν

is an equivalent norm for Bp,r . Note that k4(n−m)/ν ukYp = k4[(s−m)/ν]+1 ukX m/ν = y y

X

p

k∂ α 4[(s−m)/ν]+1 ukp . (3.4.7) y

|α|≤m/ν (m)

Since ∂ α and 4y commute, we thus see that (3.4.6) equals |||·|||s/ν,p,r . This shows that (ii.α) is true.

VII.3 Slobodeckii and H¨ older Spaces

151

£ ¤ (3) Assume that m < s < m + ν = n. Then (s − m)/ν = 0. Hence (3.4.7) reads as X k4y ukYp = k∂ α 4y ukp . |α|≤m/ν 1

If u ∈ Xp ∩ C , then the mean-value theorem implies Z 1 Z 1 4y u(x) = ∇u(x + ty)y dt = λty ∇u(x) dt y , 0

a.a. x, y ∈ X .

(3.4.8)

0

Thus kλz ukL(Xp ) ≤ 1 for z ∈ X gives k4y ukp ≤ |y| k∇ukp d

and

k4y ukp ≤ 2 kukp

d

for y ∈ I . Set M := { y ∈ I ; |y| ≤ 1 }. Then we deduce from these estimates Z ³³Z ´¡ ¢r dy ´1/r [∂ α u](s−m)/ν,p,r = + |y|−(s−m)/ν k4y ∂ α ukp |y|d M I d \M ¡ ¢ ≤ c k∇∂ α ukp + k∂ α ukp if r < ∞, and, similarly,

¡ ¢ [∂ α u](s−m)/ν,p,∞ ≤ c k∇∂ α ukp + k∂ α ukp ,

m/ν

provided u ∈ Xp

and |α| < m/ν. Consequently, X [∂ α u](s−m)/ν,p,r ≤ c |||u|||m/ν,p,r . |α| 0. By (3.7.3) and (3.7.8) we find δξ > 0 such that £

∇k (ϕε ∗ u − u)

¤ δξ ϑ,∞

≤ ξ/2 ,

ε>0.

(3.7.8)

VII.3 Slobodeckii and H¨ older Spaces

161

Thus, by (3.7.1), ° £ k ¤ ∇ (ϕε ∗ u − u) ϑ,∞ ≤ ξ/2 + 4δξ−ϑ k∇k (ϕε ∗ u − u)°∞ for ε > 0. Now it follows from (3.7.7) that there exists εξ > 0 such that £ k ¤ ∇ (ϕε ∗ u − u) ϑ,∞ < ξ , 0 < ε ≤ εξ . From this and (3.7.7) we see that ϕε ∗ u converges in BUC s towards u. This proves X ⊂ bs∞ , that is, condition (3.7.3) is sufficient if X = Rd . (3) Suppose X = K, s 6= k + 1, and u ∈ X (K, E). Then (3.7.1) implies [∇k u]ϑ,∞ ≤ [∇k u]1ϑ,∞ + 4 k∇k uk∞ . Thus u ∈ BUC s (K, F ). Hence v := EK u ∈ BUC s (Rd , E) by Theorem 2.8.2. and

We write K = X1 × · · · × Xd with Xi ∈ {R+ , R} and set Fi := BUC(Xˆı , E) ¡ ¢ wi := xi 7→ w(xi ; ·) ∈ BUC(Xi , Fi )

for w ∈ BUC(X, E). Suppose i is such that Xi = R+ . If xi < yi ≤ 0, then, recalling the notations of Subsection VI.1.1, Z ∞ |vi (xi ) − vi (yi )|Fi |ui (−txi ) − ui (−tyi )|Fi ≤ tϑ |h(t)| dt . (3.7.9) |xi − yi |ϑ |(−txi ) − (−tyi )|ϑ 0 Let ε > 0 be given and suppose s < 1. We fix τi > 0 such that Z ∞ tϑ |h(t)| dt [ui ]ϑ,∞ < ε/2 . τi

Then we choose δi = δi (ε) > 0 such that Z ∞ iδ tϑ |h(t)| dt [ui ]τϑ,∞ < ε/2 ,

0 < δ < δi .

0

Now we infer from (3.7.9) that |vi (xi ) − vi (yi )|Fi 0 } be a mollifier on Rd . Then the arguments of step (2) of the preceding proof show that (ϕε ∗ u)i → ui in BUC s/νi (Xi , Fi ) for 1 ≤ i ≤ ` as ε → 0. Thus, by Theorem 3.5.1, ϕε ∗ u → u in BUC s/ν (X) as ε → 0. Hence u ∈ bucs/ν (X). This proves X (Rd ) ,→ bucs/ν (Rd ). If X = K, then we find similarly X (K) ,→ bucs/ν (K) by appropriately modifying the extension and restriction arguments of step (3) of the preceding proof. ¥ 3.7.3 Corollary Define ki ∈ N by νi ki < s < νi (ki + 1) for 1 ≤ i ≤ `. Then u belongs to bucs/ν (X, E) iff u∈

` \

¡ ¢ BUC ki Xi , BUC(Xˆı , E)

i=1

and lim

δ→0

Proof

` X

[∇ki ui ]δ(s−νi ki )/νi ,∞ = 0 .

i=1

This follows from (3.7.11) and Theorem (3.7.1).

¥

Lastly, we prove a partial analogue of Corollary 3.6.3. 3.7.4 Theorem Let K be a closed corner in Rd and X ∈ {Rd , K}. Suppose s belongs to R+ \N. Using (3.5.1), write q ν := (ν2 , . . . , ν` )

and

q X := X2 × · · · × X` .

Then ¡ ¢ ¡ ¢ p q q bucs/ν (X, E) ,→ BUC X1 , bucs/ν (X , E) ∩ bucs/ν1 X1 , BUC(X , E) . Proof

¡ ¢ Again we omit E and write ui := xi 7→ u(xi ; ·) . Theorem 3.7.2 implies ¡ ¢ p¡ q q¢ . bucs/ν (X) = bucs/ν1 X1 , BUC(X ) ∩ bucs/ν X , BUC(X1 ) .

(3.7.12)

164

VII Function Spaces

Let ε > 0 and u ∈ bucs/ν (X) be given. There exists v ∈ BC ∞ (X) satisfying ∗ |||u − v|||s/ν,∞ < ε/3. Consequently, ∗

|||u1 (x1 ) − v1 (x1 )|||s/ν p ,∞ < ε/3 ,

x1 ∈ X1 .

Thus we deduce from ∗

∗

|||u1 (x1 ) − u1 (y1 )|||s/ν p ,∞ ≤ 2ε/3 + |||v1 (x1 ) − v1 (y1 )|||s/ν p ,∞ and the smoothness of v that ¡ p q¢ u1 ∈ BUC X1 , BUC s/ν (X ) .

(3.7.13)

Assume ki is as in the preceding corollary and set ϑi := (s − νi ki )/νi . Then £ ¤ [∇ki ui ]δϑi ,∞ ≤ ∇ki (ui − vi ) ϑi ,∞ + [∇ki vi ]δϑi ,∞ ≤ ε/3 + [∇ki vi ]δϑi ,∞ for 2 ≤ i ≤ `. Hence there exists δε > 0 such that [∇ki ui ]δϑi ,∞ < ε for δ ≤ δε and 2 ≤ i ≤ `. From this, (3.7.13), and Corollary 3.7.3 we deduce that u1 belongs to ¡ p q¢ BUC X1 , bucs/ν (X ) . Now the assertion follows from (3.7.12). ¥ 3.7.5 Example (Parabolic weight systems) Let [m, ν] be a parabolic weight system and use the notation of Example 3.6.5. Then ¡ ¢ ¡ ¢ . buc(s,s/ν) (Y × J, E) = bucs Y, BUC(J, E) ∩ bucs/ν J, BUC(Y, E) ¡ ¢ ¡ ¢ ,→ BUC J, bucs (Y, E) ∩ bucs/ν J, BUC(Y, E) . Let k, n ∈ N satisfy n < s < n + 1 and kν < s < (k + 1)ν. It follows that u belongs to buc(s,s/ν) (Y × J, E) iff ¡ ¢ ¡ ¢ u ∈ BUC n Y, BUC(J, E) ∩ BUC k J, BUC(Y, E) and

£

∇n u(·, t)

¤δ s−n,∞

£ ¤δ + ∂ k u(y, ·) (s−kν)/ν,∞ → 0

as

δ→0,

uniformly with respect to (y, t) ∈ Y × J. Proof

Theorems 3.7.2 and 3.7.4 and Corollary 3.7.3.

¥

Very Little H¨older Spaces Finally, we characterize the very little Besov–H¨older spaces by continuous interpolation.

VII.3 Slobodeckii and H¨ older Spaces

165

3.7.6 Theorem Let K be a closed corner in Rd and X ∈ {Rd , K}. (i) Suppose k < s < m with k, m ∈ νN and set θ := (s − k)/(m − k). Then ¢0 . ¡ k/ν m/ν s/ν ˚∞ B (X, E) = C0 (X, E), C0 (X, E) θ,∞ . (ii) Let 0 < s0 < s1 and 0 < θ < 1. Then ¢ . ¡ ˚s0 /ν ˚sθ /ν (X, E) = ˚s1 /ν (X, E) 0 . B B∞ (X, E), B ∞ ∞ θ,∞ Proof

s/ν s/ν ˚∞ ˚∞ We write B for B (X, E), etc. k/ν

m/ν

j/ν

(1) Set Xθ := (C0 , C0 )θ,∞ . It follows from C0 ,→ BUC j/ν , (2.7.6), and s/ν Theorem 2.8.3 that Xθ ,→ B∞ . From Proposition 3.2.3 and Lemma 3.3.3 we infer s/ν that the norm of B∞ induces the one of Xθ . Thus, by the completeness of Xθ , we s/ν see that Xθ is a closed linear subspace of B∞ . k/ν

Let Xθ0 := (C0

m/ν 0 )θ,∞ .

, C0

m/ν

Then Xθ0 is the closure of C0

m/ν density of S in C0 , it s/ν ˚∞ This proves Xθ0 = B ,

in Xθ . By the s/ν

Xθ0

follows that is the closure of S in Xθ , hence in B∞ . that is, assertion (i). sj /ν . m/ν ˚∞ (2) Fix 0 < s0 < s1 < m with m ∈ νN. Then B = (C0 , C0 )0sj /m,∞ for j = 0, 1. Thus claim (ii) follows from the reiteration theorem (I.2.8.7). ¥ Lastly, we briefly consider corners which are not necessarily closed. 3.7.7 Theorem Let M be a corner in Rd , s > 0, and 1 ≤ r ≤ ∞. Then Brs/ν (M, E) ,→ Brs/ν (M, E) Proof

and

s/ν bs/ν ∞ (M, E) ,→ b∞ (M, E) .

We omit E.

(1) Suppose r < ∞. Let mi ∈ νN satisfy m0 < s < m1 . By definition (1.2.2) m /ν m /ν and S(M) ,→ S(M) we see that Wr i (M) is a closed subspace of Wr i (M). s/ν s/ν From this, (2.7.5), and Theorem 2.8.3 we obtain Br (M) ,→ Br (M). (2) Definitions (1.1.6) and (1.1.7) imply BUC mi /ν (M) ,→ BUC mi /ν (M). Thus we get the assertions by invoking (2.7.6) and (2.7.7), similarly as in step (1). ¥

3.8

Notes

In the scalar-valued case, S.M. Nikol0 ski˘ı presents in his book [Nik75] (also see s [BesIN78], [BesIN79]) a detailed study of anisotropic Banach spaces Bp,r on Rd

166

VII Function Spaces

(and subsets thereof). Here s = (s1 , . . . , sd ) with si ∈ (0, ∞) and p = (p1 , . . . , pd ) with 1 ≤ pi ≤ ∞. Using our notations, these spaces are defined by s Bp,r (Rd ) :=

d \

¡ ¢ Bpsii ,r Xi , Lpi (Xˆı ) ,

i=1

with Xi := R for 1 ≤ i ≤ d. In particular, he establishes optimal embedding theorems of the form 0

s Bp,r ,→ Bps0 ,r

(3.8.1)

with s > s0 . s/ν

s The spaces Bp,r are much more general than the spaces Bp,r , considered here. Due to this great generality, the conditions guaranteeing the validity of embedding s/ν (3.8.1) are rather complicated. On the other hand, the spaces Bp,r can be studied systematically, as is demonstrated in this and the preceding section. Moreover, besides of being defined for all s ∈ R, they are closed under interpolation. This is s not true for the more general spaces Bp,r . In fact, let 0 < σ < τ and set s := σ/ν and t := τ /ν. Then t = (τ /σ)s. In Theorem 4 of [SchT76] it is shown that, given si with sij > 0 for 1 ≤ j ≤ d and i = 0, 1,

¡ s0 ¢ (1−θ)s0 +θs1 s1 d B2·1, 2 (Rd ), B2·1, (Rd ) 2 (R ) θ,2 = B2·1, 2 iff there exists µ > 0 such that s0 = µs1 . This indicates that the restriction to the s/ν anisotropic Besov spaces Bp,r is a natural one. Fortunately, this class is broad enough to cover the needs of most relevant applications. Although it is possible to extend much of the theory expounded here and s/ν in the preceding section to ‘mixed norm spaces’ Bp,r , we refrain from doing this (see, for example, [JoS07], [JoMHS14]). In the literature known to us, it is always claimed that the closedness of A1 , . . . , Ad obviously implies that K m/ν (A) is complete. Since we could not see this, we included a demonstration of this fact in steps (2) and (3) of the proof of Lemma 3.1.2. It is inspired by the argument for the case d = 1 given in [HillP57]. Proposition 3.2.3, the basic result of Subsection 3.2, is due to H.-J. Schmeißer and H. Triebel [SchT76]. Our exposition of assertions (i) and (ii) is an elaboration of their proof. In step (5) of its proof we follow the proof of [Tri95, Lemma 1.13.4]. The reasoning in step (6) is taken from S.G. Kre˘ın, J.U. Petunin, and E.M. Semenov [KrPS82, Lemma V.1.11]. (The latter authors attribute it to T. Muramatu [Mur70].) A fundamental consequence of Proposition 3.2.3 are the renorming statements and intersection space characterizations of Subsections 3.4–3.6. They extend corresponding statements given in [Ama09].

VII.3 Slobodeckii and H¨ older Spaces

167

In the isotropic case, the characterization of little H¨older spaces, contained in Theorem 3.7.1, is well-known (see, for example, A. Lunardi [Lun95] for the one-dimensional case). A similar characterization holds for the little Nikol 0 ski˘ı s/ν spaces bq,∞ (see G. Simonett [Simo92]).

168

4

VII Function Spaces

Bessel Potential Spaces s/ν

In the preceding sections we have shown that the Slobodeckii spaces Wp can m/ν be obtained by real interpolation from the Sobolev spaces Wp with m ∈ νN. In this section we investigate the scale of Bessel potential spaces which can be obtained from the Sobolev spaces by complex interpolation. Unlike in the case of Besov spaces, we now have to restrict the class of Banach spaces to (subclasses of) UMD spaces and q to the interval (1, ∞) in order to get interesting and useful results. Given these restrictions, we are then rewarded by the fact that the Sobolev spaces are included in the scale of Bessel potential spaces, which is not true, in general, for Besov space scales. The first subsection contains the definition of Bessel potential spaces as well as immediate consequences thereof for embeddings and real interpolation. As in the foregoing sections, a fundamental role is played by Fourier multiplier theorems. It is this place where the restrictions on E and q, alluded to above, come into play. In Subsection 4.2 we present suitable extensions of the well-known classical Fourier multiplier theorems due to Mikhlin and Marcinkiewicz, respectively, where we restrict ourselves to scalar symbols. The main result is the fact that M(Rd ) is an admissible symbol class in either case. As a consequence, obvious modifications of the proofs of Section 2, which are based on the scalar-valued version of the Fourier multiplier theorem 2.4.2, lead to corresponding results for Bessel potential spaces. This applies, in particular, in Subsection 4.3 to the theorem about renorming by derivatives. In Subsection 4.4 we prove a duality theorem. Subsection 4.5 is devoted to complex interpolation. Notably, we establish a complex interpolation theorem for Bessel potential spaces with different target spaces. Lastly, using the Fourier multiplier theorems once more, we derive in the last subsection an intersection space characterization similar to the one for BesovSlobodeckii spaces. Once more, we remind the reader of assumption (VI.3.1.20).

4.1

Basic Facts, Embeddings, and Real Interpolation

Let s ∈ R and 1 ≤ q < ∞. It follows from Lq = Lq (Rd , E) ,→ S 0 = S 0 (Rd , E) and Proposition VI.3.4.2 that J −s ∈ L(Lq , S 0 ). Hence Remarks VI.2.2.1 imply that Hqs/ν = Hqs/ν (Rd , E) := J −s Lq (Rd , E) ,

(4.1.1)

the image space of Lq under J −s in S 0 , is a well-defined Banach space, an E-valued anisotropic Bessel potential space on Rd . Note that ¡ ¢ Hqs/ν = { u ∈ S 0 ; J s u ∈ Lq }, k·kH s/ν , (4.1.2) q

VII.4 Bessel Potential Spaces

169

where kukH s/ν = kJ s ukq .

(4.1.3)

q

Proposition VI.3.4.2 and Remarks VI.2.2.1 imply d

d

S ,→ Hqs/ν ,→ S 0 . 0/ν

Furthermore, Hq

(4.1.4)

= Lq and J s ∈ Lis(Hq(s+t)/ν , Hqt/ν ) ,

s, t ∈ R ,

(4.1.5)

with (J s )−1 = J −s . In fact, the isomorphism (4.1.5) is isometric. Let K be a corner in Rd . It follows from (4.1.4) that we can define Hqs/ν (K, E) := RK Hqs/ν ,

s∈R,

1≤q s0 − |ω|/q0 , s /ν

d

1/q1 > 1/q0 ,

(4.1.9)

s /ν

then Hq11 (X, E) ,→ Hq00 (X, E). (ii) Suppose either s ≥ t + |ω|/q and t ∈ R+ \νN, or s > t + |ω|/q. Then d ˚t (X, E) . Hqs/ν (X, E) ,→ C

Proof We can assume that X = Rd . (1) Set t := s0 + |ω| (1/q1 − 1/q0 ). Then s1 > t > s0 . Hence we infer from Theorems 4.1.3(i) and 2.2.2 Hqs11 /ν ,→ Hqs/ν ,→ Hqs00 /ν . s/ν

This proves (i), since the density assertion follows from the density of S in Hp (2) Similarly, using also (2.2.3),

.

s/ν t/ν Hqs/ν ,→ Bq,∞ ,→ B∞,∞ , t/ν

t/ν

where B∞,∞ can be replaced by B∞,1 if t ∈ N. Now the second assertion follows s/ν

from (2.6.6), (3.6.6), and the density of S in Hq

.

¥

Below, in Theorem 5.6.5(ii), it is shown that in the first part of (4.1.9) the equality sign can be admitted if q1 > 1.

VII.4 Bessel Potential Spaces

4.2

171

A Marcinkiewicz Multiplier Theorem

In Section III.4 we have presented an extension of the Mikhlin multiplier theorem to vector-valued Banach spaces. In connection with anisotropic Bessel potential spaces we need a variant of the Marcinkiewicz multiplier theorem. For this purpose we have to consider a subclass of UMD spaces which satisfy the following additional restriction. The Banach space E has property (α) if Z

1 0

Z

1 0

n ¯X ¯ ¯ ¯ ri (s)rj (t)αij eij ¯ ds dt ¯ i,j=1

E

Z

1

Z

1

≤c 0

0

(4.2.1)

n ¯X ¯ ¯ ¯ ri (s)rj (t)eij ¯ ds dt ¯ i,j=1

E

q for each n ∈ N and (eij , αij ) ∈ E × C with |αij | ≤ 1, where (rj ) is the sequence of Rademacher functions rj (t) := sign(sin 2j πt) for 0 < t < 1. 4.2.1 Examples (a) Every finite-dimensional Banach space has property (α). (b) If E has property (α), then each closed linear subspace thereof has property (α). (c) Every Banach space isomorphic to one with property (α) possesses property (α) also. (d) If E is a UMD space with property (α), then E 0 too is a UMD space with property (α). (e) Let (X, µ) be a σ-finite measure space and 1 ≤ q < ∞. If E has property (α), then Lq (X, µ; E) too possesses property (α). Proof

See Section 4 in P.Ch. Kunstmann and L. Weis [KW04].

¥

Henceforth, we write Mi(Rd , E), resp. Ma(Rd , E), for the Banach space of ¡ q ¢ all m ∈ C d (Rd ) , E satisfying kmkMi := max sup p |ξ||α| |∂ α m(ξ)|E < ∞ , α≤1 ξ∈(Rd ) resp. kmkMa := max |ξ α ∂ α m(ξ)|E < ∞ , α≤1

endowed with the norm k·kMi , resp. k·kMa . (Cf. Section III.4; Mi and Ma should remind the reader of Mikhlin and Marcinkiewicz, respectively. Also note that Mi(Rd ) has been denoted in Section III.4 by MM (Rd ).)

172

VII Function Spaces

4.2.2 Lemma (i) M(Rd , E) ,→ Ma(Rd , E). (ii) If ν = ν1, then M(Rd , E) ,→ Mi(Rd , E). Proof

(1) Note j

|(ξ j )α | = (|ξ j |2ω/ωj )α

j

ωj /2ω

≤ (|ξ j |2ω/ωj + 1)α

j

ωj /2ω

≤ Λ1 (ξ)α

j

ωj

.

pω Hence |ξ α | ≤ Λα (ξ). Moreover, α ≤ 1 implies α q ω ≤ |ω| ≤ k(ν). From this and 1 (VI.3.4.7) we get k·kMa ≤ k·kM , hence (i). (2) Let ν = ν1. Then |ξ||α| =

` ³X

|ξi |2

i=1

´|α|/2

≤

` ³X

|ξi |2 + 1

´|α| ν/2ν

= Λα 1

pω

(ξ) .

i=1

Consequently, as above, k·kMi ≤ k·kM . This proves (ii).

¥

4.2.3 Lemma If E1 × E2 → E3 is a multiplication, then its point-wise extension is a multiplication F(Rd , E1 ) × F(Rd , E2 ) → F(Rd , E3 ) for F ∈ {Ma, Mi}. Proof

Leibniz’ rule.

¥

After these preparations we can formulate the following vector-valued extension of the Marcinkiewicz multiplier theorem. 4.2.4 Theorem Let E be a UMD space with property (α).¡ Then m 7→ ¢ m(D) is a continuous algebra homomorphism from Ma(Rd ) into L Lp (Rd , E) , provided 1 < p < ∞. Proof

It follows from [KW04, 5.2.a] that ¡ ¢ ¡ ¢ m 7→ m(D) ∈ L Ma(Rd ), L(Lp (Rd , E)) .

Hence the assertion is a consequence of F −1 m1 m2 F = F −1 m1 FF −1 m2 F for m1 , m2 ∈ Ma(Rd ).

¥

Using the definition of Hps (Rd , E), it is now easy to extend this theorem to a Fourier multiplier theorem for Bessel potential spaces. Since in the homogeneous case, that is, if ν = ν1, property (α) is not required, it is convenient to introduce

VII.4 Bessel Potential Spaces

173

the following definition: The Banach space E is ν -admissible if it is a UMD space which has property (α) if ν 6= ν1. 4.2.5 Theorem Let E be ν-admissible, 1 < p < ∞, and s ∈ R. Then m 7→ m(D) ¡ s/ν ¢ is a continuous algebra homomorphism from M(Rd ) into L Hp (Rd , E) . Proof (1) Suppose s = 0. Then the assertion follows from Lemma 4.2.2 and Theorem III.4.4.3 if ν = ν1, resp. Theorem 4.2.4 otherwise. (2) It is a consequence of s −s m(D) = F −1 mF = F −1 Λ−s m(D)J s 1 mΛ1 F = J

and (4.1.5) that the diagram s/ν

Hp

Js

(Rd , E)

m(D)

-

∼ =

?

Lp (Rd , E)

s/ν

Hp

∼ = m(D)

-

(Rd , E) Js

?

Lp (Rd , E)

is commuting. Hence the assertion follows from step (1) and (4.1.5).

4.3

¥

Renorming by Derivatives

Due to Theorem 4.2.5, the proofs of the preceding sections, which make use of the Fourier multiplier theorem 2.4.2 with scalar symbols, apply verbatim to Bessel potential spaces too. As a first application of this observation we get the following analogues of the theorems of Subsection 2.6. 4.3.1 Theorem Let K be a closed corner in Rd and X ∈ {Rd , K}. Suppose E is ν-admissible, 1 < p < ∞, and s ∈ R. ¡ (s+α p ω)/ν ¢ s/ν (i) If α ∈ Nd , then ∂ α ∈ L Hp (X, E), Hp (X, E) . q (ii) Suppose m ∈ ν N. The following assertions are equivalent. (s+m)/ν

(α) u ∈ Hp

s/ν

(β) ∂ α u ∈ Hp m/ωj

(γ) u, ∂j

(X, E).

(X, E), α q ω ≤ m. s/ν

u ∈ Hp

(X, E), 1 ≤ j ≤ d.

174

VII Function Spaces

Furthermore, set k·k

k·k

k·k k·k

Then k·k

(1) (s+m)/ν

Hp

(2) (s+m)/ν

Hp

(3) (s+m)/ν

Hp

(s+m)/ν

Hp

(s+m)/ν

Hp

p

m/ωj

k∂j

p

` X

i k∇xm/ν · kH s/ν , i p

i=1

` m/ν X Xi

:=

X

α p ω≤m

· kH s/ν , p

j=1

:= k·kH s/ν + :=

d X

k∇xji · kH s/ν , p

i=1 j=1

(4)

(k)

:= k·kH s/ν +

k∂ α · kH s/ν . p

∼ k·kH (s+m)/ν for 1 ≤ k ≤ 4. p

Proof (1) Let X = Rd . Then the assertions follow from the proofs of Theorems 2.6.1 and 2.6.2 and the preceding remarks. (2) If X = K, then we get the claims from Theorems VI.1.2.3(iii) and 4.1.1.

¥

As an immediate consequence of these renorming results we obtain the following fundamental fact. 4.3.2 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. Suppose that E is . m/ν m/ν ν-admissible and 1 < p < ∞. If m ∈ νN, then Hp (X, E) = Wp (X, E). Proof

Note that k·k

(4) m/ν

Hp

rems 1.4.1(ii) and 4.3.1.

4.4

= |||·|||m/ν,p . Hence the assertion is implied by Theo-

¥

Duality

First we show that the Bessel potential spaces on Rd form an interpolationextrapolation scale in the sense of Section V.1. 4.4.1 Theorem Let E be ν-admissible and 1 < p < ∞. £ ¤ The interpolation-extrapolation scale (Xα , Aα ) ; α ∈ R , generated by ¡ ¢ (X0 , A) := Lp (Rd , E), J . α/ν satisfies Xα = Hp (Rd , E).

and

[·, ·]θ , 0 < θ < 1,

VII.4 Bessel Potential Spaces

175

Proof An obvious modification of the proof of Theorem 2.5.5, by invoking Theorem 4.2.4 instead of Theorem 2.4.2, shows that J ∈ BIP(Hps/ν ) .

(4.4.1)

The assertion follows by changing the proof of Theorem 2.5.6 analogously, taking also Remark 2.5.7 into account. ¥ A ν-admissible Banach space is reflexive. Using this fact and Theorem 4.4.1, we can easily prove the following analogue of Theorems 2.3.1 and 2.8.4. 4.4.2 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. Suppose 1 < p < ∞ and s/ν E is ν-admissible. Then Hp (X, E) is reflexive and ¡ s/ν ¢0 −s/ν Hp (X, E) = Hp0 (X∗ , E 0 ) with respect to the Lp duality pairing h·, ·i. Proof

(1) First we assume that X = Rd .

We set E0 := Lp (Rd , E) and E0] := Lp0 (Rd , E 0 ). Then E0 is reflexive and E00 = E0] with respect to h·, ·i. We also denote by A0 , resp. A]0 , the E0 -, resp. E0] -, realization of J. Then A0 ∈ C(E0 ) and A]0 ∈ C(E0] ) by Theorem 4.4.1. Moreover, since Λ1 is even, that is, Λ1 = Λ1 , we get from (III.4.2.2) ­ ® hv, Jui = hv, F −1 Λ1 F ui = (2π)−d v, F (Λ1 u b) ­ ® ­¡ ¢ ® b i = (2π)−d Λ1 vb b , u = (2π)−d v, F (Λ1 u b) = (2π)−d hΛ1 vb, u ­¡ ¢ ® = (2π)−d (Λ1 vb)b , u = hF −1 Λ1 F v, ui = hJv, ui b

b

b

b

b

b

s/ν

for u ∈ S(Rd , E) and v ∈ S(Rd , E 0 ). This and the density of S in Hp hv, A0 ui = 1/ν

hA]0 v, ui

,

(v, u) ∈ 1/ν

since D(A0 ) = Hp (Rd , E) and D(A]0 ) = Hp lows from (4.4.2) that A00 := (A0 )0 ⊃ A]0 .

D(A]0 )

× D(A0 ) ,

imply (4.4.2)

(Rd , E 0 ) by Theorem 4.4.1. It fol-

(2) Suppose u0 ∈ dom(A00 ). Since A]0 ∈ Lis(E1] , E0] ), there exists u] ∈ dom(A]0 ) such that A00 u0 = A]0 u] = A00 u] . 1/ν

(4.4.3)

From dom(A0 ) = Hp (Rd , E) and (4.1.4) it follows that A0 has a dense range in E0 . Hence A00 is injective, so that we get u0 = u] from (4.4.3). Together with step (1), this proves dom(A00 ) = dom(A]0 ). Hence A00 = A]0 . Now Theorem V.1.5.12 implies the assertion if X = Rd . (3) If X = K, the claim follows by an obvious modification of the proof of Theorem 2.8.4. ¥

176

VII Function Spaces −m/ν

4.4.3 Corollary If m ∈ νN, then Hp Proof

4.5

Theorem 4.3.2 and (1.2.4).

. −m/ν (X, E) = Wp (X, E).

¥

Complex Interpolation

Theorem 4.1.3 shows that real interpolation of Bessel potential spaces results in Besov spaces. By contrast, Bessel potential spaces are stable under complex interpolation, as the next theorem shows. Recall that sθ := (1 − θ)s0 + θs1 for s0 6= s1 and 0 < θ < 1. 4.5.1 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. Suppose 1 < p < ∞ and E is ν-admissible. Assume −∞ < s0 < s1 < ∞ and 0 < θ < 1. Then . [Hps0 /ν (X, E), Hps1 /ν (X, E)]θ = Hpsθ /ν (X, E) . Proof

This is a consequence of (4.4.1) and Theorem V.1.5.4.

¥

It is the purpose of the following considerations to extend this theorem to the case of different target spaces. For this we need some preparation. A Holomorphic Semigroup Recall definition (VI.3.4.7) of the multiplier space M(Rd , E) and that J = Λ1 (D). 4.5.2 Lemma The map (z 7→ Λz1 ) : [Rez < 0] → M(Rd ) = M(Rd , C) is holomorphic, M (z) := Λz1 log Λ1 ∈ M(Rd ), and ∂Λz1 = M (z), where ∂ := ∂z . Proof

We write M := M(Rd ).

(1) Example VI.3.3.9 and Remark VI.3.4.7(a) imply Λz1 ∈ M ,

kΛz1 kM ≤ c (1 + |z|k ) ,

Rez ≤ 0 ,

(4.5.1)

where k := k(ν). (2) In the point-wise sense, ∂Λz1 = Λz1 log Λ1 = M (z) ,

z∈C,

(4.5.2)

and |M (z)| = ΛRez log Λ1 . 1

(4.5.3)

VII.4 Bessel Potential Spaces

177

From (VI.3.3.6) we infer |∂ξα M (z)| ≤ c(|z|)ΛRez−α 1

pω

(1 + log Λ1 )

for 0 < α q ω ≤ k and z ∈ C. Let Rez ≤ −r < 0. Then it follows from (4.5.3) and this estimate that Λα 1

pω

|∂ξα M (z)| ≤ c(|z|)Λ−r 1 (1 + log Λ1 ) ≤ c(|z|, r) ,

α qω ≤ k .

In other words, M (z) ∈ M ,

|M (z)|M ≤ c(|z|, r) ,

Rez ≤ −r < 0 .

(4.5.4)

(3) Now suppose z, h ∈ C satisfy Rez ≤ −r < 0, Re(z + h) ≤ −r, and |h| ≤ 1. The mean-value theorem gives, in the point-wise sense, Z Λz+h 1

−

Λz1

=

Λz1 (Λh1

− 1) =

Λz1

1 0

∂Λth 1 dth .

We differentiate this equation with respect to z and use (4.5.2) to obtain Z

1

M (z + h) − M (z) = M (z)

M (th) dth . 0

Thus, by (4.5.4), kM (z + h) − M (z)kM ≤ c(|z|, r) |h| .

(4.5.5)

In the point-wise sense, Z Λz+h − Λz1 − ∂Λz1 h = 1 that is,

0

Z Λz+h − Λz1 − M (z)h = 1

1

(∂Λz+th − ∂Λz1 ) dth , 1

1¡

¢ M (z + th) − M (z) dth .

0

Hence we infer from step (1) and (4.5.5) that kΛz+h − Λz1 − M (z)hkM ≤ c(|z|, r) |h|2 . 1

(4.5.6)

This shows that z 7→ Λz1 is differentiable on [Rez < 0] in the topology of M, hence analytic. Furthermore, its derivative equals M (z). The lemma is proved. ¥

178

VII Function Spaces

4.5.3 Theorem Suppose E is ν-admissible , 1 < p < ∞, and s ∈ R. Then: (i) The map ¡ ¢ [Rez ≤ 0] → L Hps/ν (Rd , E) , z 7→ J z is strongly continuous and its restriction to [Rez < 0] is holomorphic. s/ν

(ii) { J −t ; t ≥ 0 } is a strongly continuous analytic semigroup on Hp Its infinitesimal generator is − log J = − log Λ1 (D). s/ν

(iii) { J i t ; t ∈ R } is a strongly continuous group on Hp kJ i t kL(H s/ν ) ≤ ce|t| ,

(Rd , E) satisfying (4.5.7)

t∈R.

p

s/ν

(Rd , E).

(s+1)/ν

d

s/ν

Proof (1) We know from (4.4.1) that J ∈ BIP(Hp ). Since Hp ,→ Hp by Theorem 4.1.3(i), J is densely densely defined. Theorem III.4.7.1 guarantees that s/ν { J −t ; t ≥ 0 } is a strongly continuous semigroup on L(Hp ), and { J i t ; t ∈ R } s/ν is a strongly continuous group on L(Hp ). (2) We deduce from (4.5.1), (4.5.6), Lemma VI.3.4.5, and Theorems 2.4.2 s/ν and 4.2.5 that (4.5.7) is true, M (z)(D) ∈ L(Hp ), and kh−1 (J z+h − J z ) − M (z)(D)kL(H s/ν ) ≤ c(|z|, r) |h| p

(4.5.8)

z

for Rez, Re(z + h) ≤ r < 0. From (4.5.8) we infer that z → 7 J is differentiable on [Rez < 0], hence holomorphic, and that its derivative equals M (z)(D). The theorem is proved. ¥ 4.5.4 Remark Let E be an arbitrary Banach space, 1 ≤ q ≤ ∞, and s ∈ R. Then s/ν assertions (i)–(iii) of this theorem are valid if we replace Hp (Rd , E) everywhere s/ν d by the little Besov spaces bp (R , E). Proof It suffices to use Theorems 2.5.6, 2.2.4(iii), and 2.4.2 instead of Theorems 4.4.1, 4.1.3(i), and 4.2.5, respectively. ¥ Interpolation with Different Target Spaces We need a slightly more general version of the complex interpolation method than the one presented in Example I.2.4.2. We use the notation introduced there. In particular, S = [0 < Rez < 1] and Sx = [Rez = x] for 0 ≤ x ≤ 1. Let (E0 , E1 ) be an interpolation couple and γ ∈ R. We denote by F(E0 , E1 , γ) the set of all f ∈ C(S, E0 + E1 ) such that f | S is holomorphic and ¡ ¢ z 7→ e−γ |Imz| f (z) ∈ C0 (Sj , Ej ) , j = 0, 1 . It is a Banach space with the norm kf kF (E0 ,E1 ,γ) := max sup |e−γ |t| f (j + it)|Ej . j=0,1 t∈R

Note that F (E0 , E1 , 0) = F (E0 , E1 ) in the notation of Example I.2.4.2.

VII.4 Bessel Potential Spaces

179

Given θ ∈ (0, 1), put © ª [E0 , E1 ]θ,γ := e ∈ E0 + E1 ; e = f (θ) for some f ∈ F (E0 , E1 , γ) . It is a Banach space with the norm © ª kekθ,γ := inf kf kF (E0 ,E1 ,γ) ; f ∈ F (E0 , E1 , γ), f (θ) = e . In fact, . [E0 , E1 ]θ,γ = [E0 , E1 ]θ,0 = E[θ] ,

γ∈R.

(4.5.9)

In other words, [E0 , E1 ]θ,γ is the complex interpolation space [E0 , E1 ]θ , except for equivalent norms. For a proof of these facts we refer to H. Triebel [Tri95, Subsections 1.9.1 and 1.9.2]. The following theorem is the counterpart of Theorem 2.7.2(ii). Recall that 1/p(θ) = (1 − θ)/p0 + θ/p1 . 4.5.5 Theorem Let K be a corner and X ∈ {Rd , K}. Suppose (E0 , E1 ) is an interpolation couple of ν-admissible Banach spaces. If s0 , s1 ∈ R with s0 6= s1 and p0 , p1 ∈ (1, ∞), then £

Hps00 /ν (X, E0 ), Hps11 /ν (X, E1 )

¤ . sθ /ν = Hp(θ) (X, E[θ] ) θ

for 0 < θ < 1. Proof

(1) Due to Corollary 4.1.2 we can assume that X = Rd . For abbreviation, Xpt (E) := Hpt/ν (Rd , E) for

t∈R

and 1 < p < ∞ .

It follows from (2.7.3) that £ 0 ¤ . 0 Xp0 (E0 ), Xp01 (E1 ) θ = Xp(θ) (E[θ] ) .

(4.5.10)

From Ej ,→ E0 + E1 we get Xpsjj (Ej ) ,→ S 0 (Rd , E0 + E1 ) ,

j = 0, 1 .

¡ ¢ Hence Xps00 (E0 ), Xps11 (E1 ) is an interpolation couple. Without loss of generality, s1 > s 0 . (2) We put ¡ ¢ ¡ ¢ F0 := F Xp00 (E0 ), Xp01 (E1 ), 0 , F1 := F Xps00 (E0 ), Xps11 (E1 ), 1 . Suppose f ∈ F0 and set g(z) := (F f )(z) := J −s0 −z(s1 −s0 ) f (z) ,

z∈S .

180

VII Function Spaces

Then

g(j + it) = J −i t(s1 −s0 ) J −sj f (j + i t) , j = 0, 1 ∈ R . ¡ ¢ Since f | Sj ∈ C0 R, Xj0 (Ej ) , it follows from (4.1.3) and (4.5.7) that g | Sj belongs ¡ ¢ s to C0 R, Xpjj (Ej ) and e−|t| kg(j + i t)kXpsj (Ej ) = e−|t| kJ sj g(j + it)kXp0

j

j

≤ c kf (j + i t)kXp0

j

(Ej )

(Ej )

≤ c sup kf (j + i t)kXp0

j

t∈R

(Ej )

for j = 0, 1 and t ∈ R. Consequently, sup e−|t| kF f (j + it)kXpsj (Ej ) ≤ c sup kf (j + i t)kXp0 j

t∈R

j

t∈R

We can write f = f0 + f1 , where ¡ ¢ fj ∈ B S, Xp0j (E0 ) with

(Ej )

,

j = 0, 1 . (4.5.11)

¡ ¢ fj (j + i·) ∈ C0 R, Xp0j (Ej )

and fj being holomorphic from S into Xp0j (Ej ), j = 0, 1 (see the proof of Theorem 1.10.3.1 in [Tri95]). From this, gj (z) := F fj (z) = J −s0 −z(s1 −s0 ) fj (z) ,

z∈S ,

and Theorem 4.5.3(i) we deduce that f : S → Xps00 (E0 ) + Xps11 (E1 ) is holomorphic. Together with (4.5.11) it thus follows that F ∈ L(F0 , F1 ) .

(4.5.12)

sθ (3) Suppose u ∈ Xp(θ) (E[θ] ). Then

¤ . £ 0 J sθ u ∈ Xp(θ) (E[θ] ) = Xp00 (E0 ), Xp11 (E1 ) θ,1 by (4.5.9) and (4.5.10). Hence there exists f ∈ F0 with f (θ) = J s0 u. Moreover, kukX sθ

p(θ)

(E[θ] )

0 = kJ sθ ukXp(θ) (E[θ] ) © ª = inf kf kF0 ; f ∈ F0 , f (θ) = J sθ u .

(4.5.13)

It follows from (4.5.12) that © ª kuk[Xps0 (E0 ),Xps1 (E1 )]θ,1 = inf kgkF1 ; g ∈ F1 , g(θ) = u 0 1 © ª ≤ inf kF f kF1 ; f ∈ F0 , F f (θ) = J −sθ f (θ) = u © ª ≤ c inf kf kF0 ; f ∈ F0 , f (θ) = J sθ u . From this and (4.5.13) we obtain kuk[Xps0 (E0 ),Xps1 (E1 )]θ,1 ≤ c kukX sθ 0

1

p(θ)

(E[θ] )

,

sθ u ∈ Xp(θ) (E[θ] ) .

VII.4 Bessel Potential Spaces

Thus, due to (4.5.9),

181

£ ¤ sθ Xp(θ) (E[θ] ) ,→ Xps00 (E0 ), Xps11 (E1 ) θ .

(4.5.14)

¡ ¢ (4) Assume g ∈ F Xps00 (E0 ), Xps11 (E1 ), 0 =: Fe0 . Put f (z) := Gg(z) := J s0 +z(s1 −s0 ) g(z) , Then

f (j + it) = J i t(s1 −s0 ) J sj g(j + it) ,

z∈S .

j = 0, 1 ,

t∈R.

Using (4.1.3) and (4.5.7), we get e−|t| kGg(j + i t)kXp0

j

(Ej )

≤ c kg(j + it)kXpsj (Ej ) ,

j = 0, 1 ,

t∈R.

j

We fix r > s1 . Then

¡ ¢ Re −r + s0 + z(s1 − s0 ) < 0 ,

We write

z∈S .

Gg(z) = J −r+s0 +z(s1 −s0 ) J r g(z) ,

z∈S .

r

Note that J is a continuous linear map ¡ ¢ from F Xps00 (E0 ), Xps11 (E1 ), 0 into

¡ ¢ F Xps00 −r (E0 ), Xps11 −r (E1 ), 0 .

Hence we deduce, similarly as in step (2), that ¡ ¢ Gg ∈ F Xp00 (E0 ), Xp01 (E1 ), 1 =: Fe1 and G ∈ L(Fe0 , Fe1 ) .

(4.5.15)

sθ (5) Assume u ∈ Xp(θ) (E[θ] ). Then

¤ . £ 0 J sθ u ∈ Xp(θ) (E[θ] ) = Xp00 (E0 ), Xp01 (E1 ) θ,1

by (4.1.1), (4.5.9), and (4.5.10). Hence we find an f ∈ Fe1 with f (θ) = J sθ u. Furthermore, similarly as in the preceding step, we deduce from Gg(θ) = J sθ g(θ) and (4.5.15) that © ª sθ e 0 kukX sθ (E[θ] ) = kJ sθ ukXp(θ) (E[θ] ) ≤ c inf kf kF1 ; f ∈ F1 , f (θ) = J u p(θ) © ª ≤ c inf kGgkF1 ; g ∈ Fe0 , Gg(θ) = J sθ u © ª ≤ c inf kgkF0 ; g ∈ Fe0 , g(θ) = u = c kuk[Xps0 (E0 ),Xps1 (E1 )]θ,1 . 0

This implies

£

Xps00 (E0 ), Xps11 (E1 )

¤ θ

which, due to (4.5.14), proves the theorem.

1

sθ ,→ Xp(θ) (E[θ] ) , ¥

182

VII Function Spaces

Note that this theorem contains Theorem 4.5.1 as a particular case. Thus it provides an alternative proof for that theorem.

4.6

Intersection-Space Characterizations

Let K be a corner in Rd and X ∈ {Rd , K}. Recall that (VI.3.3.2) applies. We consider the decomposition X = X1 × · · · × X` , associated with the weight system [`, d, ν] and use the notation introduced in the beginning of Subsection 3.5. s/ν

In the following, we show that Hp (X, E) possesses intersection space chars/ν acterizations analogous to the ones for Bp (X, E). 4.6.1 Theorem Suppose K is a corner in Rd and X ∈ {Rd , K}. Assume E is ν-admissible and 1 < p < ∞. Let M be a subset of L := {1, . . . , `}. Then \ ¡ ¢ ¡ ¢ . \ Hps/ν (X, E) = Lp Xµ , Hps/νµ (Xµ , E) ∩ Hps/νλ Xλ , Lp (Xλ , E) µ∈M

λ∈L\M

for s > 0. Proof

(1) Assume X = Rd . Given 1 ≤ i ≤ `, ¡ ¢ Λ(i) (ξi , ηi ) := Λ (ξi ; 0), (ηi ; 0) ,

(ξi , ηi ) ∈ Xi × Hi ,

(4.6.1)

is the natural νi -quasinorm on Zi := Xi × Hi (see Remark VI.3.3.1(a)). Hence Λs(i) belongs to H(Zi ) by Example VI.3.3.9. We set ai (ξi ) := Λs(i) (ξi , 1) , Then

ξ i ∈ Xi .

¡ ¢ ai (Dxi ) ∈ Lis Hps/νi (Xi , E), Lp (Xi , E)

by (4.1.5). Denoting by Ai the point-wise extension of ai (Dxi ) over Xˆı , defined by ¡ ¢ ai (Dxi )u (xˆı ) := ai (Dxi )u(·; xˆı ), it follows that ¡ ¢ ¢ Ai ∈ Lis Lp (Xˆı , Hps/νi (Xi , E) , Lp (Xˆı , Lp (Xi , E)) .

(4.6.2)

(2) We assume X = Rd and set Λs(1) + · · · + Λs(`) (ξ, η) := Λs(1) (ξ1 , η1 ) + · · · + Λs(`) (ξ` , η` ) for (ξ, η) ∈ Z := Rd × H. Then Λs(1) + · · · + Λs(`) belongs to Hs (Z) and is strictly q positive on Z. From this and Lemmas VI.3.3.6 and VI.3.3.8 it follows (Λs(1) + · · · + Λs(`) )/Λs , Λs /(Λs(1) + · · · + Λs(`) ) ∈ H0 (Z) .

(4.6.3)

VII.4 Bessel Potential Spaces

183

Note that a(ξ) := a1 (ξ1 ) + · · · + a` (ξ` ) = (Λs(1) + · · · + Λs(`) )(ξ, 1) for ξ ∈ X and 1 = 1` . Hence we infer from (4.6.3) and Remark VI.3.4.7(a) that a/Λs1 , Λs1 /a ∈ M(Rd ). Thus ¡ ¢ (a/Λs1 )(D), (Λs1 /a)(D) ∈ L Lp (X, E) by Theorem 4.2.5. Consequently, by that theorem, (3.5.5), and step (1), kukH s/ν (X,E) = kJ s u kLp (X,E) ≤ c kΛs1 /akM ka(D)ukLp (X,E) p

≤c

` X

kAi ukLp (X,E) ≤ c

i=1

` X i=1

for u ∈ X (X, E) :=

` \

kukL

s/νi (Xi ,E)) p (Xı ,Hp

ˆ

¡ ¢ Lp Xˆı , Hps/νi (Xi , E) .

i=1 s/ν Hp (X, E).

This proves X (X, E) ,→ (3) Let X = Rd . Similarly as above, we find ai /Λs1 ∈ M(Rd ) for 1 ≤ i ≤ `. Thus the arguments of the last step imply kAi ukLp (X,E) ≤ c kJ s ukLp (X,E) = c kukH s/ν (X,E) p

for u ∈

s/ν Hp (X, E) ` X

kukL

i=1 s/ν

for u ∈ Hp

and 1 ≤ i ≤ `. Hence, by (4.6.2) and (3.5.5),

s/νi (Xi ,E)) p ((Xı ,E),Hp

ˆ

≤c

` X

s/ν

(X, E). Consequently, Hp Hps/ν (X, E)

kAi ukLp (X,E) ≤ c kukH s/ν (X,E)

i=1

p

(X, E) ,→ X (X, E). This proves

` ¢ . \ ¡ = Lp Xˆı , Hps/νi (Xi , E) i=1

in this case. (4) Still assuming X = Rd , Fubini’s theorem gives Z Z p kuk = |ai (Dxi )u(·; xˆı )|pE dxi dxˆı s/ν Lp (Xı ,Hp i (Xi ,E)) Xı Xi ˆ ˆ Z Z = |ai (Dxi )u(·; xˆı )|pE dxˆı dxi Xi Xı ˆ = kukH s/νi (X ,L (X ,E)) . i p p ˆı

(4.6.4)

184

VII Function Spaces

¡ ¢ ¢ s/ν s/ν ¡ Thus we can replace Lp Xˆı , Hp i (Xi , E) in (4.6.4) by Hp i Xi , Lp (Xˆı , E) if i ∈ L\M. This proves the assertion if X = Rd . (5) We write K = I1 × · · · × Id , where Ii ∈ {R, R+ }. Using the notation of Subsection VI.1.1, we set Ri := Ei := idR if Ii ∈ R. Then RK = Rd ◦ · · · ◦ R1 and EK = E1 ◦ · · · ◦ Ed . We also set dˆı := d − di for 1 ≤ i ≤ `. We put ci × · · · × Id . Kˆı := I1 × · · · × K ¡ t/ν ¢ t/ν Then, by Theorem 4.1.1, (RKi , EKi ) is an r-e pair for Hp i (Rdi , E), Hp i (Ki , E) for t ∈ {0, s}. Theorem 4.1.1 implies also that (RKı , EKı ) is an r-e pair for ˆ ˆ Ki := Id(i−1)+1 × · · · × Id(i) ,

¡

d Lp (R ˆı , Hpt/νi (Ki , E)), Lp (Kˆı , Hpt/νi (Ki , E))

¢

for t ∈ {0, s}. From this we infer that (RKı ◦ RKi , EKi ◦ EKı ) is an r-e pair for ˆ ˆ ¡

¢ d Lp (R ˆı , Hps/νi (Rdi , E)), Lp (Kˆı , Hps/νi (Ki , E)) .

Similarly, (RKi ◦ RKı , EKı ◦ EKi ) is an r-e pair for ˆ ˆ ¡

¢ d Hps/νi (Rdi , Lp (R ˆı , E)), Hps/νi (Ki , Lp (Kˆı , E)) .

We can identify (RKi ◦ RKı , EKı ◦ EKi ) and (RKı ◦ RKi , EKi ◦ EKı ) with (RK , EK ) ˆ ˆ ˆ ˆ by means of the canonical¡ identifications based on (3.5.3). This implies that ¢ (RK , EK ) is an r-e pair for F(Rd , E), F(K, E) , where F(X, E) is the intersection space on the right side of (4.6.4). Hence it follows from the validity of the assertion for X = Rd and u = RK EK u that EK RK . Hps/ν (K, E) −→ Hps/ν (Rd , E) = Fs (Rd , E) −→ Fs (K, E) , s/ν

Thus Hp

u 7→ u .

(K, E) ,→ Fs (K, E). Similarly,

EK RK . Fs (K, E) −→ Fs (Rd , E) = Hps/ν (Rd , E) −→ Hps/ν (K, E) ,

v 7→ v ,

s/ν

implies Fs (K, E) ,→ Hp (K, E). Together with the analogue of step (4), this proves the assertion for X = K. ¥ q q 4.6.2 Corollary We write ν := (ν2 , . . . , ν` ) and X := X2 × · · · × X` . Then ¡ ¢ ¡ ¢ p q q . Hps/ν (X, E) = Lp X1 , Hps/ν (X , E) ∩ Hps/ν1 X1 , Lp (X , E) , s>0. qq If m ∈ ν N, then ¡ ¢ ¡ ¢ p q q . Wpm/ν (X, E) = Lp X1 , Wpm/ν (X , E) ∩ Wpm/ν1 X1 , Lp (X , E) .

VII.4 Bessel Potential Spaces

Proof

185

(1) Note that ¡ ¢ ¡ ¢ q Lp Xˆı , Hps/νi (Xi , E) = Lp X1 , Lp (Xˆı , Hps/νi (Xi , E))

for 2 ≤ i ≤ `. Hence ` \ i=2

` ³ \ ¡ ¢ . ¡ q ¢´ Lp Xˆı , Hps/νi (Xi , E) = Lp X1 , Lp Xˆı , Hps/νi (Xi , E) . i=2

Using the theorem with L := {2, . . . , `} and M := L, we get ` \ i=2

¡ q ¢ . p q Lp Xˆı , Hps/νi (Xi , E) = Hps/ν (X , E) .

From this we get the first assertion by applying the theorem once more, this time with L := {1, . . . , `} and M := {2, . . . , `}. (2) The second claim is now a consequence of Theorem 4.4.2. ¥ 4.6.3 Example (Parabolic weight systems) Suppose E is a UMD space with property (α), K a corner in Rd , and X ∈ {Rd , K}. Let [m, ν] be a parabolic weight system with ν > 1 and write X = Y × J with J ∈ {R, R+ }. Then ¡ ¢ ¡ ¢ . Hp(s,s/ν) (Y × J, E) = Lp J, Hps (Y, E) ∩ Hps/ν J, Lp (Y, E) , s>0. In particular, ¡ ¢ ¡ ¢ . Wp(ν,1) (Y × J) = Lp J, Wpν (Y, E) ∩ Wp1 J, Lp (Y, E) . Proof

4.7

This follows from the corollary by relabeling coordinates.

¥

Notes

In contrast to the theory of Besov spaces, where no restrictions on the Banach space E have to be imposed, for most of the above theory of Bessel potential spaces we require that E be ν-admissible. This is due to the fact that this condition guarantees the validity of the Mikhlin, respectively Marcinkiewicz, Fourier multiplier theorem on Lp (Rd , E). Indeed, it is known that Mikhlin’s theorem is not true for Banach spaces E not possessing the UMD property, and the Marcinkiewicz theorem may fail to hold if the UMD space E does not possess property (α). The attentive reader will have noticed that we restricted ourselves in the Fourier multiplier theorems to the case of scalar symbols. In fact, Theorems III.4.4.3 and 4.2.4 are not valid, in general, if m is an operator-valued symbol. In this case we have to impose an additional condition on the symbol, namely R-boundedness.

186

VII Function Spaces

A subset T of L(E, F ) is R-bounded if there exists a constant c such that, given any n ∈ N, T0 , . . . , Tn ∈ T , x0 , . . . , xn ∈ E, n °X ° ° ° rj Tj xj ° ° j=0

L2 ([0,1],F )

n °X ° ° ° ≤ c° rj x j ° j=0

L2 ([0,1],E)

,

(4.7.1)

where the rj are the Rademacher functions on [0, 1]. The smallest constant c is the R-bound, R(T ), of T . 4.7.1 Remarks (a) The L2 spaces in definition (4.7.1) can be replaced by Lq spaces for any q ∈ [1, ∞). (b) If Ti ⊂ L(E0 , E1 ) are R-bounded for i = 1, 2, then T1 + T2 , too, is R-bounded and R(T1 + T2 ) ≤ R(T1 ) + R(T2 ). (c) Let T ∈ L(E0 , E1 ) and S ∈ L(E1 , E2 ) be R-bounded. Then ST := { ST ; S ∈ S, T ∈ T } is R-bounded and R(ST ) ≤ R(S)R(T ). (d) If T ⊂ L(E0 , E1 ) is R-bounded, then it is bounded by R(T ). (e) If E0 and E1 are Hilbert spaces, then T ⊂ L(E0 , E1 ) is R-bounded iff it is bounded. (f ) Let E0 and E1 be finite-dimensional. Then T ⊂ L(E0 , E1 ) is R-bounded iff it is bounded. Proof For (a) see P.Ch. Kunstmann and L. Weis [KW04, Remark 2.6]. (b) and (c) follow directly from the definition. By choosing n = 1 in (4.7.1), we get (d). For a demonstration of (e) we refer to R. Denk, M. Hieber, and J. Pr¨ uss [DHP03, Remark 3.2(3)]. Lastly, (f) is obvious by (e), since E0 and E1 can be identified with Euclidean spaces. ¥ Now we can formulate operator-valued extensions of the multiplier theorems of Mikhlin and Marcinkiewicz. 4.7.2 Theorem Let E¢ and F be UMD spaces and 1 < p < ∞. Suppose m belongs ¡ q to C d (Rd ) , L(E, F ) . © ª q (i) If Mi(m) := |ξ|α ∂ α m(ξ) ; ξ ∈ (Rd ) , α ≤ 1 is R-bounded in L(E, F ), then ¡ ¢ m(D) ∈ L Lp (Rd , E), Lp (Rd , F ) (4.7.2) and km(D)k ≤ cR(Mi(m)).

VII.4 Bessel Potential Spaces

187

(ii) Let E and F possess property (α) and ª q ξ α ∂ α m(ξ) ; ξ ∈ (Rd ) , α ≤ 1 ¡ ¢ be R-bounded. Then (4.7.2) applies and km(D)k ≤ cR Ma(m) . Ma(m) :=

©

This theorem is due to L. Weis [Wei01], if d = 1. The extension to d ≥ 2 ˇ Strkalj ˇ ˇ has been obtained independently by Z. and L. Weis [SW07] and R. Haller, H. Heck, and A. Noll [HHN02]. Detailed expositions of these and related results, historical remarks, and alternative proofs can be found in [KW04], [DHP03], and [PS16]. We refer to these works for discussions of the necessity of assumptions. As for the necessity of condition (α), see T. Hyt¨onen and L. Weis [HW08]. A rather comprehensive historical overview of the development of vector-valued Fourier multiplier theorems, as well as extensions and sharpenings, can be found in Hyt¨onen’s thesis [Hyt03]. Although condition (α) cannot be dispensed with, in general, in the multiplier theorem of Marcinkiewicz¡ type, it ¢follows from a result ¡ of T. Hyt¨ ¢ onen [Hyt07, Corollary 1] that m ∈ H0 Rd , L(E) implies m(D) ∈ L Lp (Rd , E) under the sole assumption that E is a UMD space. Unfortunately, most symbols of interest to us are not homogeneous. Thus that nice result is of no use for us. Consequently, we cannot avoid to require condition (α), even if we restrict ourselves to scalar symbols. A detailed and comprehensive presentation of the theory of R-boundedness is contained in the recent books by T. Hyt¨onen, J. van Neerven, M. Veraar, and L. Weis ([HvNVW16] and [HvNVW17]). Theorem 4.5.5 is new. Subsection 4.6 generalizes the corresponding results of [Ama09] to corners.

188

5

VII Function Spaces

Triebel–Lizorkin Spaces

If we interchange the roles of Lq and `r in the definition of Besov spaces, then we arrive at Triebel–Lizorkin spaces. They are closely related to Besov spaces and possess similar properties. In this section we introduce these spaces in the anisotropic vector-valued setting and discuss some of their properties. In our setup, Triebel–Lizorkin spaces play an auxiliary role only. Hence we restrict our considerations essentially to those properties which we need for our purposes. In the present section we employ these spaces and their properties to prove sharp Sobolev type embedding theorems for Sobolev and Bessel potential spaces and anisotropic vector-valued extensions of the classical Gagliardo–Nirenberg inequalities. We stress the fact –– and this is the main reason for introducing Triebel– Lizorkin spaces –– that these results apply to arbitrary Banach spaces E. Unless explicitly stated otherwise, all spaces of distributions are over Rd and sequence spaces over N. Thus we write Lq (E) := Lq (Rd , E) and `q (E) := `q (N, E), where we omit E if it equals C. Similar conventions apply to S 0 (E), etc. Recall the standing assumption (VI.3.1.20).

5.1

Maximal Inequalities

Let Q be a ν-quasinorm on Rd . Then BQ (x, t) :=

©

y ∈ Rd ; Q(x − y) ≤ t

ª

is the Q-ball of radius t centered at x ∈ Rd . We write |BQ (x, t)| for its volume. Let v be a measurable function on Rd . Then the anisotropic Q-maximal function MQ v of v (of Hardy–Littlewood type) is defined by

1 MQ v(x) := sup |B (x, t)| t>0 Q

Z |v| dy ∈ [0, ∞] BQ (x,t)

for x ∈ Rd . The following lemma shows that the particular choice of the ν-quasinorm is unimportant. 5.1.1 Lemma Let P be a second ν-quasinorm on Rd . Then MP u(x) ∼ MQ u(x), uniformly with respect to u and x. Proof

We set BQ := BQ (0, 1). Then Z dx = t|ω| |BQ |

|BQ (x, t)| = |BQ (0, t)| = [Q≤t]

(5.1.1)

VII.5 Triebel–Lizorkin Spaces

189

for x ∈ Rd and t > 0. By Remark VI.3.3.3 there exists a constant β ≥ 1 such that β −1 P ≤ Q ≤ βP. Hence BQ (0, 1/β) ⊂ BP ⊂ BQ (0, β) . Thus

β −|ω| |BQ | = |BQ (0, 1/β)| ≤ |BP | ≤ |BQ (0, β)| = β |ω| |BQ | .

From this we deduce MP u(x) ≤ sup t>0

³ β |ω| Z |BQ |

´ |u(y)| dy = β 2 |ω| MQ u(x) . BQ (x,βt)

Similarly, β −2 |ω| MQ u(x) ≤ MP u(x) for x ∈ Rd .

¥

The next theorem is an anisotropic version of the classical Hardy–Littlewood maximal inequality. ° ° 5.1.2 Theorem Suppose 1 < q ≤ ∞. Then °MQ |u|°q ≤ c kukq for u ∈ Lq . Proof By the above lemma we can assume that Q is the Euclidean ν-quasinorm. Then the assertion follows from Corollary 1.8 in [CaT75] by choosing in Theorem 1.7 for ϕ the characteristic function of [ |x| < 1] and setting f := |u|. This choice is possible by Corollary 1.9 of [CaT75]. ¥ The next theorem is an anisotropic version of the vector-valued maximal inequality of C. Fefferman and E.M. Stein [FS71]. 5.1.3 Theorem Suppose 1 < p < ∞ and 1 < r ≤ ∞. Then ° ° °(MQ |uk |)° ≤ c k(uk )kLp (`r ) L (` ) p

r

for (uk ) ∈ Lp (`r ). Proof If Q = E, then this has been shown by M. Yamazaki [Yam86, Theorem 2.2]. Hence the claim follows from Lemma 5.1.1. ¥ Preliminary Estimates for Sequences In the following, we prove some technical results which will be needed in subsequent subsections. For this we fix a ν-quasinorm Q and set Ω := [Q < 1].

(5.1.2)

By Example VI.3.6.1, Ω is a ν-admissible 0-neighborhood of Rd . Moreover, ¡ ¢ we fix an Ω-adapted smooth function ψ. Then (Ωk ), (ψk ) is the (5.1.3) ν-dyadic partition of unity on Rd induced by (Ω, ψ).

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VII Function Spaces

Recall that Ωk = [2k−1 < Q < 2k+1 ]

Ω0 = [Q < 2] and

for

k≥1.

Henceforth, M := MQ . Furthermore, given v : Rd → E, |v(x − y)|E , 2 |ω| k q Λ (2 y)

Tk v(x) := sup

y∈Rd

x ∈ Rd ,

k∈N.

(5.1.4)

1

5.1.4 Lemma Let m ∈ N. Then ¡ ¢2 1/2 Tk (F −1 uk )(x) ≤ c M |F −1 uk |E (x) ,

x ∈ Rd ,

for uk ∈ D0 (Rd , E) with supp(uk ) ⊂ 2k+m q Ω0 and k ∈ N. Proof (1) We set vk := F −1 uk and fix ϕ ∈ S with ϕ(x) b = 1 for x ∈ 2m q Ω0 . Then −k |ω| ϕk := σ2k ϕ belongs to S and ϕ bk = 2 σ2−k ϕ b by Proposition VI.3.1.3(iii). Hence 2k |ω| ϕ bk (x) = 1 for x ∈ 2m+k q Ω0 . Thus, see Theorem 1.9.1 in the Appendix, vk = 2k |ω| F −1 (uk ϕ bk ) = 2k |ω| vk ∗ ϕk and, by Remark 1.9.6(g) of the Appendix, ∂ α vk = 2k |ω| vk ∗ ∂ α ϕk = 2k |ω|+α

pω

v k ∗ σ2 k ∂ α ϕ

for α ∈ Nd . From this we get (cf. Proposition 1.9.3 of the Appendix) Z α k |ω|+α p ω |∂ vk (x − z)|E ≤ 2 |vk (y)|E |σ2k ∂ α ϕ|(x − y − z) dy for x, y ∈ Rd . Here and below, integration and suprema are evaluated over Rd . We set t := 2 |ω| + d + 1. Since |Λt1 ∂ α ϕ| ≤ c for |α| = 1, Z |∂ α vk (x − z)|E ≤ c 2k |ω| |vk (y)|E σ2k Λ−t (5.1.5) 1 (x − y − z) dy for x, z ∈ Rd , |α| = 1, and k ∈ N. It follows from Λ1 = (1 + N2ν )1/2ν and N(ξ + η) ≤ N(ξ) + N(η) that 2 |ω|

Λ1

2 |ω|

(ξ) ≤ cΛ1

2 |ω|

(ξ − η)Λ1

(η) ,

ξ, η ∈ Rd .

(5.1.6)

By means of this estimate, setting ξ := 2k q (x − y) and η := 2k q z, we obtain from (5.1.5) Z |∂ α vk (x − z)|E |vk (y)|E k |ω| ≤ c 2 σ2k Λ−d−1 (x − y − z) dy 1 2 |ω| k q 2 |ω| k q Λ1 (2 z) Λ1 (2 (x − y)) (5.1.7) ≤ c Tk vk (x)

VII.5 Triebel–Lizorkin Spaces

191

for x, z ∈ Rd , |α| = 1, and k ∈ N. Here we used the transformation of variables ξ = x − y and, recalling Proposition VI.3.1.3(ii), Z Z −d−1 k |ω| 2 σ2 k Λ 1 (ξ − z) dξ = Λ−d−1 (y) dy < ∞ . 1 Since (5.1.7) holds for all z ∈ Rd , Tk (∂ α vk )(x) ≤ c Tk vk (x) ,

x ∈ Rd ,

(5.1.8)

for |α| = 1 and k ∈ N. (2) By Lemma 5.1.1, we can assume that Ω is convex (cf. Lemma VIII.1.2.2 below). Let g ∈ C 1 (Rd , E) and B := BQ . Then the mean-value theorem Z 1 ¡ ¢ g(y) = g(z) + ∂g z + t(y − z) (y − z) dt , y, z ∈ B , 0

implies, by choosing z such that |g(z)|E = minB |g|E , ³ ´ |g(y)|E ≤ c min |g|E + max sup |∂ α g|E B |α|=1 B Z ³³ ´2 ´ 1 1/2 α ≤c |g| dx + max sup |∂ g| E |2−k q B| 2−k p B E |α|=1 B for y ∈ B and k ∈ N. Here we replace g by σt g for t > 0. Then Z ³ ´2 1 p 1/2 |g(x)|E ≤ c max tα ω sup |∂ α g|E + ct−2 |ω| |g| dx |2−k q B| t p 2−k p B E |α|=1 t pB for x ∈ t q B and k ∈ N. We substitute g(y) := vk (x − y − z) to obtain p |vk (x − z)|E ≤ c max tα ω sup |∂ α vk (x − y − z)|E |α|=1 y∈t p B Z ³ ´2 1 1/2 −2 |ω| + ct |v (x − y − z)| dy k E |2−k q B| t p 2−k p B

(5.1.9)

for x, z ∈ Rd and k ∈ N. Now we assume 0 < t ≤ 1. Observe that y ∈ t2−k q B iff Q(y) ≤ t2−k if and only if y ∈ B(0, t2−k ). Hence ¡ ¢ ¡ ¢ Q(y + z) ≤ Q(y) + Q(z) ≤ 2−k 1 + 2k Q(z) = 2−k 1 + Q(2k q z) for y ∈ t2−k q B. Thus, setting R := R(k, z) := 1 + Q(2k q z), Z Z 1/2 1/2 |vk (x − y − z)|E dy ≤ |vk (x − ξ)|E dξ t2−k p B R2−k p B Z 1/2 1/2 = |vk (y)|E dy ≤ |B(x, R2−k )| M (|vk |E )(x) . B(x,R2−k )

192

VII Function Spaces

Since |B(x, R2−k )| = |B(0, R2−k )| = R|ω| |2−k q B|, we obtain from (5.1.9) |vk (x − z)|E ≤ ctmin ω max sup |∂ α vk (x − z − y)|E |α|=1 y∈B

+ ct

−2 |ω|

¡ ¢2 |ω| ¡ ¢2 1/2 1 + Q(2k q z) M |vk |E (x)

for x, z ∈ Rd , k ∈ N, and 0 < t ≤ 1. Using 1 + Q ∼ Λ1 , we can divide this inequality by Λ2 |ω| (2k q z) and then take the supremum over z ∈ Rd to arrive at ¡ ¢2 1/2 Tk vk (x) ≤ ctmin ω max Tk (∂ α vk )(x) + ct−2 |ω| M |vk |E (x) |α|=1

for x ∈ Rd , k ∈ N, and 0 < t ≤ 1. Due to (5.1.8) we can find a constant c1 ≥ 1 such that the first term on the right side of the last inequality is majorized by c1 tmin ω Tk vk (x) ,

x ∈ Rd ,

k∈N,

Now the assertion follows by setting t := (2c1 )−1/ min ω .

0 0]. We compute ω1 q q ∂j ht (x ) = − ht (x )(ω1 −2ω)/ω1 (xj )2(ω−ωj )/ωj xj , ωj

2≤j≤d.

q q q q Since ht (x ) = tω1 h1 (t−1 q x ), it follows ∂j ht (x ) = tω1 −ωj ∂j h1 (t−1 q x ). Hence ¡ q q ¯ q¢ q gt,jk (x ) := ∂j ht (x ) ¯ ∂k ht (x ) = t2ω1 −ωj −ωk g1,jk (t−1 q x ) q q for 2 ≤ j, k ≤ d. Set ajk := g1,jk (t−1 q x ), so that gt,jk (x ) = t2ω1 −ωj −ωk ajk . Then, denoting by Sd−1 the group of permutations of d − 1 elements, X £ q¤ det gt,jk (x ) = t2ω1 (d−1) sign(σ)t−ωj −ωσ(j) ajσ(j) σ∈Sd−1

=t

p

2ω1 (d−1)−2 |ω |

det[ajk ] .

Consequently, √

p√ q q gt (x ) = tω1 (d−1)−|ω | g1 (t−1 q x ) ,

t>0,

q q Q (x ) < t .

From this we get Z

Z p q q q q ω1 (d−1) −|ω | √ g (x ) dx = 2t t g1 (t−1 q x ) dx t p p [Q 0

for all measurable u : Rd → E for which ϕ ∗ |u| exists a.e. Proof (1) By Fubini’s theorem and the preceding lemma, Z Z ∞ Z ∞ ϕ dx = f (s) vol(Ss−1 Q ) ds = s|ω| (1−1/d) f (s) ds vol(SQ ) < ∞ . 0

0

Hence ϕ is integrable and kϕt k1 = t|ω| kσt ϕk1 = kϕk1 ,

t>0. Pn

(2) First we P assume that fPis of the form i=1 ai χ[0,ξi ] with ai ≥ 0 and ξi > 0. Then ϕ = ai χ[Q≤ξi ] = ai χBQ (0,ξi ) and X kϕk1 = ai |BQ (0, ξi )| . Furthermore, |χ[Q(t p x)0.

(5.1.14)

Observe that hm ≤ hm+1 ≤ f and the sequence converges point-wise towards f . Thus, if u is as in the assertion, by Lebesgue’s dominated convergence theorem

196

VII Function Spaces

we can pass to the limit in (5.1.14) to obtain |ϕt ∗ u(x)| ≤ kϕk1 M |u|(x) for each t > 0 and a.a. x ∈ Rd . This proves the lemma. ¥ Now we can derive the desired estimate for a single Lp function. 5.1.8 Lemma Suppose 1 < p < ∞. Then °¡ ¢° ° ψk (D)u ° ≤ c kukp , Lp (`∞ (E))

u ∈ Lp (E) .

Proof (1) We set f (s) := (1 + s2ν )−|ω|/ν for s ≥ 0. Then f is decreasing and s|ω| (1−1/d) f (s) ≤ s−|ω| (1+1/d) for s > 0. Hence f satisfies the hypotheses of the −2 |ω| preceding lemma, and ϕ := f ◦ N = Λ1 . e ⊂ Ω0 . Hence we infer from (2) Recall that ψk = σ2−k ψe for k ≥ 1 and supp(ψ) Proposition VI.3.1.3 that 2 |ω|

k(σ2k Λ1

2 |ω|

)F −1 ψk k∞ = 2k |ω| kΛ1

e ∞ = c 2k |ω| F −1 ψk

for k ≥ 1. Consequently, Z 2 |ω| −2 |ω| |ψk (D)u(x)| ≤ (σ2k Λ1 |F −1 ψk |)(y)(σ2k Λ1 )(y) |u(x − y)| dy Z −2 |ω| ≤ c 2k |ω| (σ2k Λ1 )(y) |u(x − y)| dy = c ϕ2k ∗ |u|(x) ≤ cM |u|(x) for k ∈ N, due to Lemma 5.1.7. Now Theorem 5.1.2 implies the assertion.

5.2

¥

Definition and Basic Embeddings

Throughout the rest of this section •

1 ≤ q < ∞, 1 ≤ r ≤ ∞ ,

•

s∈R,

(5.2.1)

unless explicit restrictions are imposed. Let conditions (5.1.2)¢ and (5.1.3) be satisfied. Since ψk (D)u is an element ¡ of OM (E), 2sj ψj (D)u(x) is for each x ∈ Rd a sequence in E. Thus the following definition is meaningful. s/ν

s/ν

d The anisotropic¡Triebel–Lizorkin space ¢ ¡ s Fq,r¢ (E) = Fq,r (R , E) consists of all 0 u ∈ S (E) for which ψk (D)u k∈N ∈ Lq `r (E) . We endow it with the norm ° °¡ ° °¡ ¢° ¢° ° ° u 7→ ° ψk (D)u °L (`s (E)) = °° 2ks ψk (D)u °` (E) ° . (5.2.2) q

r

r

q

Clearly, this definition depends on the choice of Q and ψ. The next lemma shows, s/ν however, that the topology of Fq,r (E) is independent of (Q, ψ).

VII.5 Triebel–Lizorkin Spaces

197

Equivalent Norms 5.2.1 Lemma Let Q1 and Q2 be ν-quasinorms, define Ωj := [Qj < 1], and choose j Ωj -adapted smooth functions ψ j for j = 1, 2. Denote by k·k the norm (5.2.2) with 1 2 j ψk replaced by ψk . Then k·k ∼ k·k . Proof We use the notations of the proof of Lemma 2.1.1. In particular, ψ−i := 0 for i ≥ 0. Then we can find m ∈ N such that ψk2 = ψk2 χ1k,m for k ∈ N. Thus ψk2 (D)u = F −1 (ψk2 χ1k,m u b) = F −1 ψk2 ∗ χ1k,m (D)u m+1 X

=

1 F −1 ψk2 ∗ ψk+i (D)u

i=−m−1

for u ∈ S(E). Consequently, 2ks ψk2 (D)u =

m+1 X

1 2−is F −1 ψk2 ∗ 2(k+i)s ψk+i (D)u .

(5.2.3)

i=−m−1

It follows from Remark VI.3.6.4(a) that 2 |ω|

F −1 (σ2k ψk2 )k1 ≤ c , k∈N. (5.2.4) ¡ −1 1 ¢ 1 Since supp F (ψk+i (D)u) = supp(ψk+i u b) ⊂ 2m+1 q Ω for k ∈ N and |i| ≤ m + 1, we can apply Lemma 5.1.5 with E0 = E1 = E2 = E and ak = 1 to the convolution 1 2−is F −1 ψk2 ∗ ψk+i (D)u to obtain °¡ −is −1 2 °¡ 1 ¢ ° ¢ ° 1 ° 2 F ψk ∗ 2(k+i)s ψk+i (D)u k °L (` (E)) ≤ c ° ψk+i (D)u k °L (`s (E)) kΛ1

q

r

q

r

1

≤ c kuk for |i| ≤ m + 1. Thus (5.2.3) implies °¡ ¢° kuk2 = ° 2ks ψk2 (D)u °L (` (E)) q

≤

m+1 X

r

°¡ −is −1 2 ¢ ° 1 ° 2 F ψk ∗ 2(k+i)s ψk+i (D)u °

k Lq (`r (E))

i=−m−1 1

≤ c kuk1 .

2

This proves that k·k is stronger than k·k . The assertion follows by interchanging the roles of (Q1 , ψ 1 ) and (Q2 , ψ 2 ). ¥ P∞ We know from Lemma VI.3.6.2 that u = k=0 ψk (D)u in S 0 (E). The follows/ν ing lemma shows that this is also true in Fq,r (E) if r < ∞. s/ν

5.2.2 Lemma If u ∈ Fq,r (E) and r < ∞, then u =

P∞ k=0

s/ν

ψk (D)u in Fq,r (E).

198

VII Function Spaces s/ν

Proof Let u ∈ Fq,r (E). Set un := u − it follows ψk (D)un = ψk (D)

∞ X

Pn j=0

ψj (D)u =

j=n+1

ψj (D)u for n ∈ N. Since u ∈ S 0 (E),

1 X

ψk+i (D)ψk (D)u

(5.2.5)

i=−1

for k ≥ n, and ψk (D)un = 0 for k ≤ n − 1. From (5.2.4), (5.2.5), and Lemma 5.1.5 we infer (setting wk := ψk+i , uk := ψk (D)u for k ≥ n, and uk := 0 otherwise) that °¡ ¢ ° kun kF s/ν = ° ψk (D)un k °L q,r

s q (`r (E))

∞ °³ X ¡ ks ¢r ´1/r ° ° ° ≤ c° 2 |ψk (D)u|E ° . q

k=n

s/ν

Since u ∈ Fq,r (E), it follows that kun kF s/ν → 0 as n → ∞. This shows that the q,r claim is true. ¥ Embeddings In the following theorem we collect the basic embedding properties of Triebel– Lizorkin spaces. s/ν

s/ν

5.2.3 Theorem We write Fq,r for Fq,r (E), etc. Suppose 1 ≤ q < ∞. (i) If 1 ≤ r0 < r1 ≤ ∞, then d

s/ν

d

d

s/ν s/ν s/ν S ,→ Bq,1 ,→ Fq,r ,→ Fq,r ,→ Bq,∞ ,→ S 0 . 0 1 s/ν

s/ν

Moreover, Fq,r0 is dense in Fq,r1 if r1 < ∞. s/ν

s/ν

s/ν

(ii) Let 1 ≤ r0 ≤ q ≤ r1 ≤ ∞. Then Fq,r0 ,→ Fq,q = Bq s/ν

s/ν

,→ Fq,r1 .

t/ν

(iii) Suppose s > t and r0 , r1 ∈ [1, ∞]. Then Fq,r0 ,→ Fq,r1 . (iv) If E1 ,→ E0 and 1 ≤ r ≤ ∞, then s/ν s/ν Fq,r (E1 ) ,→ Fq,r (E0 ) .

(5.2.6)

d

This embedding is dense if E1 ,→ E0 . Proof (1) The first and the last embedding of (i) follow from Theorem 2.2.2. The middle one is an immediate consequence of (5.2.2), Theorem VI.2.2.2(i), and (1.6.2). In order to show the second embedding (without the density assertion), it s/ν s/ν thus suffices to prove Bq,1 ,→ Fq,1 . This follows from °X ° X ° ° kukF s/ν (E) = ° |2ks ψk (D)u|E ° ≤ 2ks kψk (D)ukLq (E) = kukB s/ν (E) . q,1

k

q

q

k

VII.5 Triebel–Lizorkin Spaces

199

Similarly, kukB s/ν (E) = sup 2ks kψk (D)ukLq (E) q,∞ k ° ° ° ≤ ° sup 2ks |ψk (D)u|E °q = kukF s/ν ≤ c kukF s/ν . q,∞

k

q,r1

This proves the validity of the next to last embedding.

s/ν . s/ν s/ν (2) The Fubini–Tonelli theorem guarantees Fq,q = Bq,q = Bq , due to the s/ν fact that, by Lemma 2.1.1, we can assume that Bq,q is given the norm based on the ν-dyadic partition of unity induced by (Q, ψ). Now (ii) follows from (i). (3) Assertion (iii) is immediate by (VI.2.2.5) and (1.6.2). (4) Claim (5.2.6) is a consequence of Theorem VI.2.2.2(ii) and the trivial observation (1.6.2). s/ν

s/ν

(5) We show that S is dense in Fq,r if r 0. There exists nε such that kuk − uj kF s/ν ≤ ε for k, j ≥ nε . It folq,r

s/ν

lows from step (2) with vj := uk − uj , where k ≥ nε is fixed, that uk − u ∈ Fq,r (E) and kuk − ukF s/ν ≤ ε .

(5.2.8)

q,r

s/ν

Thus u = uk + (u − uk ) ∈ Fq,r (E) and, (5.2.8) being true for each k ≥ nε , we see s/ν that uk → u in Fq,r (E). This proves the theorem. ¥ s/ν

5.2.5 Remark In step (2) of this proof we have shown that Fq,r (E) enjoys the s/ν Fatou property: If (vj ) is a sequence in Fq,r (E) such that kvj kF s/ν ≤ c0 and vj → v q,r s/ν in S 0 (E), then v ∈ Fq,r (E) and kvkF s/ν ≤ c c0 . ¥ q,r

VII.5 Triebel–Lizorkin Spaces

5.3

201

Fourier Multiplier Theorems

For Triebel–Lizorkin spaces we can prove a Fourier multiplier theorem which is analogous to Theorem 2.4.2. 5.3.1 Theorem Let (5.2.1) be true. Suppose E0 × E1 → E2 ,

(e0 , e1 ) 7→ e0 e1

is a multiplication and denote its point-wise extension again by juxtaposition. Then ¡ ¢ ¡ ¡ s/ν ¢¢ s/ν a 7→ a(D) ∈ L M(E0 ), L Fq,r (E1 ), Fq,r (E2 ) . Proof

s/ν

Given u ∈ Fq,r (E1 ) and a ∈ M(E0 ), we infer from ψk = ψk χk that ¡ ¢ 2sk ψk (D) a(D)u = 2sk F −1 (ψk ab u) = 2sk F −1 (ψk a) ∗ χk u b =

1 X

2−is F −1 (ψk a) ∗ 2(k+i)s ψk+i (D)u .

i=−1

Remark VI.3.6.4(a) guarantees that (5.1.10) is satisfied with wk := ψk and ak = a. Since ψk+i u b has its support in 2 q Ωk , we can apply Lemma 5.1.5 to obtain °¡ ¢° ka(D)ukF s/ν (E2 ) = ° ψk (D)(a(D)u) °Lq (`s (E2 )) q,r °¡ ¢°r ≤ c(s) kakM ° ψk (D)u °L (`s (E )) = c(s) kakM kukF s/ν (E1 ) . q

This proves the theorem.

r

1

q,r

¥

It is obvious that the analogue of Theorem 2.4.2(ii) holds in this case also. The next theorem shows that, similarly as for Besov spaces, J t is an isomorphism between Triebel–Lizorkin spaces. 5.3.2 Theorem Let assumption (5.2.1) be satisfied and t ∈ R. Then ¡ (s+t)/ν ¢ s/ν J t ∈ Lis Fq,r (E), Fq,r (E) and (J t )−1 = J −t . Proof

The proof of (2.4.2) shows that ak := 2−kt Λt1 ψk ∈ M ,

kak kM ≤ c ,

k∈N.

(5.3.1)

Moreover, using ψk = ψk χ2k , 2ks ψk (D)J t u = 2ks F −1 (ψk Λt1 u b) = 2(s+t)k F −1 (χk ψk 2−kt Λt1 χk u b) =

1 1 X X i=−1 j=−1

2−(s+t)j F −1 (ψk+i ak ) ∗ 2(s+t)(k+j) ψk+j (D)u .

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VII Function Spaces

From this, (5.3.1), Remark VI.3.6.4(a), and Lemma 5.1.5 we infer that kJ t ukF s/ν ≤ c kukF (s+t)/ν q,r

(s+t)/ν

for u ∈ Fq,r

. Now the assertion is obvious.

q,r

¥

The following analogue of Proposition 2.4.4 is a consequence of the two preceding theorems. ¡ ¢ 5.3.3 Proposition Let t ∈ R. Suppose Λ−t 1 a ∈ M L(E0 , E1 ) . Then ¡ (s+t)/ν ¢ s/ν (i) a(D) ∈ L Fq,r (E0 ), Fq,r (E1 ) and ka(D)k ≤ c kΛ−t 1 akM . ¡ ¢ ¡ ¢ t −1 (ii) If, in addition, a ∈ C L(E0 , E1 ) and Λ1 a ∈ B L(E1 , E0 ) , then ¡ (s+t)/ν ¢ s/ν a(D) ∈ Lis Fq,r (E0 ), Fq,r (E1 ) , a(D)−1 = a−1 (D) , ¡ ¢ t −1 and ka(D)−1 k ≤ c kΛ−t k∞ . 1 akM , kΛ1 a Proof

Use obvious substitutions in the proof of Proposition 2.4.4.

¥

5.3.4 Remark It is clear that, by the results of this subsection, we can carry over to Triebel–Lizorkin spaces all results which we proved for Besov spaces by using Theorems 2.4.1 and 2.4.2 and Proposition 2.4.4. ¥

5.4

Interpolation

As a first application of Remark 5.3.4 we show that Triebel–Lizorkin spaces are compatible with complex interpolation. 5.4.1 Theorem Suppose 1 ≤ q < ∞ and r < ∞. Then ¡ 0/ν ¢ J ∈ BIP Fq,r (E) , £ ¤ and the interpolation-extrapolation scale (Xα , Aα ) ; α ∈ R generated by ¡ 0/ν ¢ (X0 , A0 ) := Fq,r (E), J and [·, ·]θ , 0 < θ < 1 , . α/ν satisfies Xα = Fq,r (E). Proof

See the proof of Theorem 2.5.6 and Remark 2.5.7.

¥

5.4.2 Corollary Assume 1 ≤ q < ∞, r < ∞, and 0 < θ < 1. Then £ s0 /ν ¤ . s /ν s1 /ν θ Fq,r (E), Fq,r (E) θ = Fq,r (E) , −∞ < s0 < s1 < ∞ . Proof

Theorem V.1.5.4.

¥

VII.5 Triebel–Lizorkin Spaces

5.5

203

Renorming by Derivatives

A further implementation of Remark 5.3.4 proves the following analogue of Theorem 2.6.2. 5.5.1 Theorem Let (5.2.1) be satisfied. ¡ (s+α p ω)/ν ¢ s/ν (i) ∂ α ∈ L Fq,r (E), Fq,r (E) . q (ii) Let m ∈ ν N. The following assertions are equivalent: (s+m)/ν

(α) u ∈ Fq,r (β) (γ)

Furthermore, define k·k k·k

(k) (s+m)/ν

Fq,r

Proof

(E).

s/ν ∂ u ∈ Fq,r (E), α q ω m/ω s/ν u, ∂j j u ∈ Fq,r (E), α

(k)

≤ m. 1 ≤ j ≤ d.

· · by replacing Bq,r in Theorem 2.6.2 by Fq,r . Then

(s+m)/ν

Fq,r

∼ k·kF (s+m)/ν for 1 ≤ k ≤ 4. q,r

Cf. the proof of Theorem 2.6.2.

¥

Sandwich Theorems Due to Theorem 5.2.3(i), the following ‘sandwich theorem’ sharpens (2.6.6) and puts Bessel potential spaces in relation to Triebel–Lizorkin spaces. 5.5.2 Theorem Suppose 1 < p < ∞. m/ν

m/ν

(i) If m ∈ νN, then Fp,1 (E) ,→ Wp s/ν

s/ν

(ii) Fp,1 (E) ,→ Hp

m/ν

(E) ,→ Fp,∞ (E).

s/ν

(E) ,→ Fp,∞ (E) for s ∈ R.

P 0/ν Proof (1) We assume m = 0. Let u ∈ Fp,1 (E) ,→ S 0 (E). Then u = k ψk (D)u in S 0 (E) by Lemma VI.3.6.2. Hence, denoting by δx the Dirac distribution supported at {x}, ¯ X ¯ ¯X ¡ ¯X ¯ ¢¯¯ ¯ ¯ ¯ ¯ ¯ ψk (D)u¯ = ¯ δx ψk (D)u ¯ = ¯ ψk (D)u(x)¯ ¯δ x k

E

E

k

≤

X

k

E

|ψk (D)u(x)|E

k

for x ∈ Rd . From this we get

°X ° ° ° kukp ≤ ° |ψk (D)u|E ° k

Lp

= kukF 0/ν , p,1

0/ν

that is, Fp,1 (E) ,→ Lp (E). 0/ν

On the other hand, Lp (E) ,→ Fp,∞ (E) follows from Lemma 5.1.8. Thus (i) applies if m = 0.

204

VII Function Spaces

q (2) Assume m ∈ ν N. It follows from step (1) and Theorem 5.5.1 that kukF m/ν ≤ c p,∞

X α p ω≤m

k∂ α ukLp (E) ≤ c

X α p ω≤m

k∂ α ukF 0/ν ≤ c kukF m/ν . p,1

p,1

Now (i) is implied by (1.1.4) and Theorem 1.4.1. (3) Assertion (ii) is immediate from (4.1.5), Theorem 5.3.2, and step (1).

5.6

¥

Sobolev Embeddings and Related Results

With the help of Triebel–Lizorkin spaces we can extend the classical Sobolev embedding theorem to the anisotropic vector-valued setting. In addition, we can show that the equality signs in (4.1.9) can be admitted too. For this we need some preparation. Multiplicative Inequalities Henceforth, we use the notation introduced in (VI.2.3.1). Recall that sθ := (1 − θ)s0 + θs1

and

1/q(θ) = (1 − θ)/q0 + θ/q1 .

5.6.1 Lemma Let s0 , s1 ∈ R and 0 < θ < 1. Then ks1 k(2ksθ ak )k`r ≤ c k(2ks0 ak )k1−θ ak )kθ`∞ `∞ k(2

for (ak ) ∈ CN . Proof Without loss of generality, s0 < s1 . Set cj := k(2ksj ak )k k`∞ for j = 0, 1. Then c0 ≤ c1 . We can assume c1 > 0. Since s0 < s1 , there is some j0 > 0 such that ( c0 2−s0 j , if j ≤ j0 , −s0 j −s1 j min{c0 2 , c1 2 }= c1 2−s1 j , if j > j0 . From c0 2−s0 j0 ≤ c1 2−s1 j0 and c1 2−s1 (j0 +1) ≤ c0 2−s0 (j0 +1) we get c1 ∼ c0 2(s1 −s0 )j0 . Therefore s1 k k(2s0 k ak )k1−θ ak )kθ`∞ ∼ c0 2(s1 −s0 )j0 θ . `∞ k(2

On the other hand, |ak | ≤ min{c0 2−s0 k , c1 2−s1 k } so that ( c0 2−s0 k for 0 ≤ k ≤ j0 , |ak | ≤ c1 2−s1 k for j0 < k < ∞ .

(5.6.1)

VII.5 Triebel–Lizorkin Spaces

205

Hence, if r < ∞, ³X ´1/r X k(2sθ k ak )k`r ≤ cr0 2(sθ −s0 )kr + cr1 2(sθ −s1 )kr k≤j0

≤

³X

k>j0

X

cr0 2θ(s1 −s0 )kr +

k≤j0

≤ c0 2(s1 −s0 )θj0

³X

cr0 2−(1−θ)(s1 −s0 )kr+(s1 −s0 )j0 r

k>j0

2θ(s1 −s0 )(k−j0 )r +

k≤j0 (s1 −s0 )θj0

≤ c c0 2

´1/r

X

2−(1−θ)(s1 −s0 )(k−j0 )r

´1/r

k>j0

.

Similarly, k(2sθ k ak )k`∞ ≤ c0 2(s1 −s0 )θj0 . Now the assertion follows from (5.6.1).

¥

5.6.2 Lemma Suppose s0 , s1 ∈ R, 0 < θ < 1, and r0 , r1 ∈ [1, ∞]. (i) Let 1 ≤ q0 , q1 < ∞. Then θ kukF sθ /ν ≤ c kuk1−θ s0 /ν kuk s1 /ν Fq0 ,r0

q(θ),r

s /ν

Fq1 ,r1

s /ν

for u ∈ Fq00,r0 ∩ Fq11,r1 (E). (ii) If 1 ≤ q0 < ∞ and q1 = ∞, then θ kukF sθ /ν ≤ c kuk1−θ s0 /ν kuk s1 /ν Fq0 ,r0

q(θ),r

s /ν

B∞

s /ν

for u ∈ Fq00,r0 ∩ B∞1 (E). Proof (1) We apply the preceding lemma to ak := |ψk (D)u(x)|E inequality. Then °¡ ¢° kukF sθ /ν = ° 2ksθ ψk (D)u °L (` ) q(θ) r q(θ),r °° ¡ ¢°1−θ °¡ ks1 ¢° °° ks0 ° 2 ψk (D)u °θ ≤ c ° 2 ψk (D)u °

° ° `∞ °

`∞

≤

c kuk1−θ s /ν Fq00,∞

kukθ s1 /ν Fq1 ,∞

and use H¨older’s

q(θ)

.

Claim (i) follows now from Theorem 5.2.3(i). (2) Similarly as in step (1), ° °¡ ¢°1−θ °¡ ¢°θ ° ° ° kukF sθ /ν ≤ c °° 2ks0 ψk (D)u °` ° 2ks1 ψk (D)u °` ° ∞ ∞ q(θ) q(θ),r ° °¡ °°¡ ¢°1−θ ° ¢°θ ° ° ° °° ks1 ° ≤ c °° 2ks0 ψk (D)u °`∞ ° ° 2 ψk (D)u °`∞ ° q0 /(1−θ)

≤ This proves (ii).

¥

c kuk1−θ s /ν Fq00,∞

°¡ ks ¢° ° 2 1 ψk (D)u °θ

`∞ (L∞ )

.

∞

206

VII Function Spaces

Optimal Sobolev-Type Embeddings First we consider Triebel–Lizorkin spaces. 5.6.3 Theorem Let s0 , s1 ∈ R and 1 ≤ q0 , q1 < ∞ satisfy s1 − |ω|/q1 ≥ s0 − |ω|/q0 s /ν s /ν and s1 ≥ s0 . If 1 ≤ r0 , r1 < ∞, then Fq11,r1 (E) ,→ Fq00,r0 (E). Proof Due to Theorem 5.2.3(iii) we can assume, by making s1 smaller if necessary, that s1 − |ω|/q1 = s0 − |ω|/q0 and s1 > s0 . Thus q1 < q0 and we can define θ ∈ (0, 1) by 1/q0 = (1 − θ)/q1 . Suppose σ0 6= σ1 , 1 ≤ p0 < ∞, and 1/p := (1 − θ)/p0 . Given ρ, ρ0 ∈ [1, ∞], Lemma 5.6.2(ii) implies θ kukF σθ /ν ≤ c kuk1−θ σ0 /ν kuk σ1 /ν .

(5.6.2)

B∞

Fp0 ,ρ0

p,ρ

We now set σ1 := s1 − |ω|/q1 , σ0 := s1 , and p0 := q1 . Then p = q0 and σθ = (1 − θ)s1 + θ(s1 − |ω|/q1 ) = s1 − θ |ω|/q1 = s1 − |ω|/q1 + (1 − θ) |ω|/q1 = s1 − |ω|/q1 + |ω|/q0 = s0 . Consequently, letting ρ0 := r1 and ρ := r0 in (5.6.2), θ kukF s0 /ν ≤ c kuk1−θ s1 /ν kuk (s1 −|ω|/q0 )/ν . q0 ,r0

Fq1 ,r1

(5.6.3)

B∞

We infer from Theorem 5.2.3(i) and (2.2.3) that /ν (s1 −|ω|/q1 )/ν Fqs11,r/ν1 (E) ,→ Bqs11,∞ (E) ,→ B∞ (E) .

The assertion follows now from (5.6.3).

¥

5.6.4 Corollary Let K be a corner in Rd and X ∈ {Rd , K}. Suppose s0 , s1 ∈ R and q0 , q1 ∈ [1, ∞] satisfy s1 − |ω|/q1 ≥ s0 − |ω|/q0 and s1 ≥ s0 . Then Bqs11 /ν (X, E) ,→ Bqs00 /ν (X, E) . s/ν

Proof Due to Theorem 2.8.3, we can assume that X = Rd . By Bq s > t, it suffices to consider the case s1 − |ω|/q1 = s0 − |ω|/q0 .

s/ν

If q0 < ∞, then the claim follows from the theorem and Bq s /ν s0 /ν s0 /ν erwise, Bq11 ,→ B∞,q by Theorem 2.2.2. ¥ 1 ,→ B∞

t/ν

,→ Bq

for

s/ν

= Fq,q . Oth-

Now we can prove the Sobolev embedding theorems alluded to above. Observe that there is no restriction whatsoever on E.

VII.5 Triebel–Lizorkin Spaces

207

5.6.5 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. (i) Suppose m0 , m1 ∈ νN, 1 < p1 < p0 < ∞, and m1 − |ω|/p1 ≥ m0 − |ω|/p0 . d

m /ν

m /ν

Then Wp1 1 (X, E) ,→ Wp0 0 (X, E). (ii) Let −∞ < s0 < s1 < ∞, 1 < p0 , p1 < ∞, and s1 − |ω|/p1 ≥ s0 − |ω|/p0 . s /ν

Then Hp11 Proof

d

s /ν

(X, E) ,→ Hp00

(X, E).

Let X = Rd . Then we get from Theorems 5.5.2 and 5.6.3 m /ν

0 1 /ν 1 /ν 0 /ν Wpm (E) ,→ Fpm1 ,∞ (E) ,→ Fp0 ,1 (E) ,→ Wpm (E) 1 0

and

s /ν

/ν Hps11 /ν (E) ,→ Fps11,∞ (E) ,→ Fp00,1 (E) ,→ Hps00 /ν (E) .

If X = K, the assertion follows now by extension and restriction, due to Theorems 1.3.1 and 4.1.1. The density is a consequence of the density of S(X, E) in these spaces. ¥ Sharp Embeddings of Intersection Spaces By means of Corollary 5.6.4 we can improve the embedding given in Theorem 2.7.3 by admitting the limiting case s − |ω|/q = sθ − |ω|/q(θ). Furthermore, there is an analogous result for Bessel potential spaces. 5.6.6 Theorem Let (E0 , E1 ) be an interpolation couple, s, s0 , s1 ∈ R with s0 6= s1 , 1 ≤ q, q0 , q1 < ∞, and 0 < θ < 1. Suppose K is a corner in Rd , X ∈ {Rd , K}, 1/q(θ) ≥ 1/q ≥ 0

and

sθ − |ω|/q(θ) ≥ s − |ω|/q .

(i) It holds Bqs00 /ν (X, E0 ) ∩ Bqs11 /ν (X, E1 ) ,→ Bqs/ν (X, Eθ,q(θ) ) . (ii) If E0 and E1 are ν-admissible and 1 < q, q0 , q1 < ∞, then Hqs00 /ν (X, E0 ) ∩ Hqs11 /ν (X, E1 ) ,→ Hqs/ν (X, E[θ] ) . Proof

(1) Due to Bqs00 /ν (X, E0 ) ∩ Bqs11 /ν (X, E1 ) ,→ Bqsjj /ν (X, Ej ) ,

j = 0, 1 ,

interpolation gives

¡ ¢ Bqs00 /ν (X, E0 ) ∩ Bqs11 /ν (X, E1 ) ,→ Bqs00 /ν (X, E0 ), Bqs11 /ν (X, E1 ) θ,q(θ) .

Hence, using Theorems 2.7.2(i) and 2.8.3, s /ν

θ Bqs00 /ν (X, E0 ) ∩ Bqs11 /ν (X, E1 ) ,→ Bq(θ) (X, Eθ,q(θ) ) .

Now assertion (i) follows from Corollary 5.6.4.

208

VII Function Spaces

(2) Since Hqs00 /ν (X, E0 ) ∩ Hqs11 /ν (X, E1 ) ,→ Hqsjj /ν (X, Ej ) ,

j = 0, 1 ,

complex interpolation and Theorem 4.5.5 imply £ ¤ Hqs00 /ν (X, E0 ) ∩ Hqs11 /ν (X, E1 ) ,→ Hqs00 /ν (X, E0 ), Hqs11 /ν (X, E1 ) θ . sθ /ν = Hq(θ) (X, E[θ] ) . Now the assertion follows from Theorem 5.6.5(ii).

¥

5.6.7 Remark Given the hypotheses of the preceding theorem, s /ν

θ Bqs00 /ν (X, E0 ) ∩ Bqs11 /ν (X, E1 ) ,→ Bq(θ) (X, E[θ] ) .

Proof It suffices to use part (ii) of Theorem 2.7.2 instead of (i) in the above proof. ¥

5.7

Gagliardo–Nirenberg Type Estimates

By means of the preceding multiplicative inequalities and various embedding theorems for Besov, Triebel–Lizorkin, and Sobolev spaces we prove now far reaching generalizations of the well-known Gagliardo–Nirenberg inequalities ([Gag59], [Nir59]). Throughout this subsection • K is a corner in Rd and X ∈ {Rd , K}. We also suppose •

− ∞ < s0 ≤ s < s1 < ∞ ,

1 ≤ p, p0 , p1 ≤ ∞

(5.7.1)

satisfy s−

³ ³ |ω| |ω| ´ |ω| ´ = (1 − θ) s0 − + θ s1 − , p p0 p1

(5.7.2)

where s − s0 ≤θ≤1. s1 − s0

(5.7.3)

Nonhomogeneous Inequalities First we prove multiplicative inequalities involving complete norms of the spaces under consideration.

VII.5 Triebel–Lizorkin Spaces

209

5.7.1 Theorem Let (5.7.1)–(5.7.3) be satisfied. Then θ kukB s/ν ≤ c kuk1−θ s0 /ν kuk s1 /ν s /ν

for u ∈ Bp11

s /ν

∩ Bp00

Bp1

Bp0

p

(5.7.4)

(X, E).

Proof It suffices to consider X = Rd . Then we get the assertions for X = K by means of extension and restriction, due to Theorems 1.3.1 and 2.8.2. (1) We define sθ and p(θ) by (VI.2.3.1), admitting now the endpoint values θ = 0 or θ = 1 also. Then (5.7.2) can be written as s− or θ−

|ω| |ω| = sθ − p p(θ)

(5.7.5)

s − s0 |ω| ³ 1 1´ = − . s1 − s0 s1 − s0 p(θ) p

From this we read off that (5.7.3) is equivalent to 1 1 ≤ . p p(θ)

(5.7.6)

Note that θ = 0 implies s = s0 and, consequently, p = p0 . Thus (5.7.4) is trivial in this case. Hence we can assume that θ > 0. (2) Suppose min{p0 , p1 } < ∞ ,

θ 1 if sj ∈ νN;

(β)

θ < 1 unless either s, s1 ∈ νN and 1 < p, p1 < ∞, or s, s1 ∈ / νN.

5.7.4 Remarks (a) The assertions of Theorem 5.7.1 remain valid, given any of the substitutions: Bps/ν Ã Hps/ν , Bpsjj /ν Ã Hpsjj /ν , provided p, pj ∈ (1, ∞). Proof

This follows from the proof of Theorem 5.7.2 by using Theorem 5.5.2(ii).

¥

212

VII Function Spaces

(b) If (5.7.2) is satisfied, then s − s0 ≤θ≤1 s1 − s0 Proof

iff

1 1 ≥ ≥0. p(θ) p

See step (1) of the proof of Theorem 5.7.1.

¥

(c) The somewhat cumbersome formulation of Theorem 5.7.2 and its corollary cannot be avoided, since the restrictions on the integrability parameters are essential if the regularity indices are integer multiples of ν. Indeed, it has been observed by J. Bourgain, H. Brezis, and P. Mironescu [BoBM00, Remark D.1] 1/2 that W11 ∩ L∞ (R) is not contained in W2 (R), although (s, p) = (1/2, 2) , satisfy (5.7.2) for any θ ∈ [0, 1].

(s0 , p0 ) = (0, ∞) ,

(s1 , p1 ) = (1, 1)

¥

Let E1 ,→ E ,→ E0 and 0 < θ < 1. Then E is of class J(θ, E0 , E1 ) if θ kxkE ≤ c kxk1−θ E0 kxkE1 ,

x ∈ E1 .

It is known (e.g., [BeL76, Theorem 3.5.2] or [Tri95, Lemma 1.10.1]) that E is of class J(θ, E0 , E1 )

iff

Eθ,1 ,→ E ,

(5.7.19)

where Eϑ,r := (E0 , E1 )ϑ,r for 0 < ϑ < 1 and 1 ≤ r ≤ ∞. 5.7.5 Remark Suppose 0 < θ < 1 and m0 , m1 , mθ ∈ νN with m0 < m1 . Then it follows from Corollary 5.7.2 that ¡ ¢ mθ /ν Wp(θ) (X, E) is of class J θ, Wpm0 0 /ν (X, E), Wpm1 1 /ν (X, E) , provided 1 < p0 , p1 ≤ ∞. Thus 1−θ

θ

|||u|||mθ /ν,p(θ) ≤ c |||u|||m0 /ν,p0 |||u|||m1 /ν,p1 m /ν

for u ∈ Wp1 1 (X, E). This shows, in particular, that the multiplicative inequalities (5.7.4) cannot be obtained by interpolation, in general, even if p = p(θ). Indeed, in the case where p0 = p1 = ∞, BUC mθ (Rd ) cannot be obtained by interpolation between BUC m0 (Rd ) and BUC m1 (Rd ). ¥ Homogeneous Estimates The multiplicative inequalities of Theorems 5.7.1 and 5.7.2 can be considerably improved, inasmuch as the norms can be replaced by homogeneous seminorms.

VII.5 Triebel–Lizorkin Spaces

213

This depends on a scaling argument which, in turn, rests on the following simple observation. We use the seminorms [·]θ,q,r and [[·]]θ,q,r;i introduced in Subsection 3.5. Now it is convenient to set ( k·kp , if s = 0 , [·]s,p,r := (5.7.20) [·]s,p,r , if s > 0 , for 1 ≤ p, r ≤ ∞. 5.7.6 Lemma Suppose ω = ν1, 1 ≤ p ≤ ∞, and s ≥ 0. Then [σt u]s/ν,p,r equals ts−|ω|/p [u]s/ν,p,r for t > 0. Proof If s = 0, then this is clear from Proposition VI.3.1.3. Thus assume s > 0. Note that t q y = tν y implies 4y σt u(x) = u(t q x + t q y) − u(t q x) = σt 4tν y u(x) for x, y ∈ X and t > 0. Hence, by induction, 4ky ◦ σt = σt ◦ 4ktν y for k ∈ N. Let r < ∞. Then ³Z ¡ ¢r dy ´1/r [σt u]s/ν,p,r = |y|−s/ν k4[s/ν]+1 σ uk t p y |y|d Id ³Z ¡ ¢r dy ´1/r [s/ν]+1 = t−|ω|/p |y|−s/ν k4tν y ukp |y|d Id = ts−|ω|/p [u]s/ν,p,r . If r = ∞, then we argue correspondingly.

¥

s/ν

Suppose s > 0 and 1 ≤ p, r ≤ ∞. Then we endow Bp,r (X, E) with the norm12 kukB s/ν := p,r

` [s/ν i ]− ³ X X i=1

j=0

´ k∇xji ukp + [[∇xji u]]s/νi −[s−νi ]− ,p,r;i .

Due to Theorem 3.5.2 and the remark following it, this is possible. Let s ≤ 0 and 1 ≤ p, r ≤ ∞. We denote by m = m(s) the smallest element of νN satisfying m + s > 0. Then Theorem 2.6.3 and Remark 2.6.4 guarantee that s/ν we can equip Bp,r (X, E) with the norm ` ³ ´ X i kukB s/ν := inf ku0 kB (s+m)/ν + k∇xm/ν uk , (s+m)/νi i B p,r

12 [·]

−

p,r

i=1

is the largest integer strictly smaller than t ∈ R.

p,r

214

VII Function Spaces

where the infimum is taken over all representations u = u0 +

` X

i ∇xm/ν ui , i

(s+m)/νi u0 , u1 , . . . , u` ∈ Bp,r .

(5.7.21)

i=1

Then we put ` X

kuk p s/ν := Bp,r

[s/νi ]−

[[∇xi

u]]s/νi −[s/νi ]− ,p,r;i ,

s>0

(5.7.22)

i=1

and kuk p s/ν := inf

` ³X

Bp,r

m/νi

[[∇xi

´ ui ]]s+m−[s+m]− ,p,r;i ,

s≤0,

i=1

where, again, the infimum is taken over all representations (5.7.21). Lastly, 0 k·kB s/ν := k·kB s/ν − k·k p s/ν , s/ν

If s ∈ νN, then Wp

s∈R.

Bp,r

p,r

p,r

(5.7.23)

is given the norm |||·|||s/ν,p introduced in (1.1.4). Then k·k p

s/ν

Wp,r

:=

X α p ω=s

k∂ α q kp

(5.7.24)

and 0 k·kW s/ν := |||·|||s/ν,p − k·k p

s/ν

Wp

p

if s ∈ νN. The following lemma shows that the seminorms (5.7.22) and (5.7.24) are the homogeneous part of the corresponding norms. 5.7.7 Lemma Suppose s ∈ R and 1 ≤ p, r ≤ ∞. Then kσt uk p s/ν = ts−|ω|/p kuk p s/ν , Bp,r

and t−s+|ω|/p kσt uk0

s/ν

Bp,r

→ 0 as t → 0. If s ≥ 0, then

kσt uk p

s/ν

Wp

and t−s+|ω|/p kσt uk0

s/ν

Wp

Proof

t>0,

Bp,r

= ts−|ω|/p kuk p

s/ν

Wp

,

t>0,

→ 0 as t → ∞.

This is a consequence of Proposition VI.3.1.3 and Lemma 5.7.6.

¥

VII.5 Triebel–Lizorkin Spaces

215

Now we can prove the main result of this subsection, the homogeneous version of estimate (5.7.4) and its variants. 5.7.8 Theorem Let (5.7.1)–(5.7.3) be satisfied. Then kuk p s/ν ≤ c kuk1−θ p s0 /ν kukθp s1 /ν Bp

s /ν

for u ∈ Bp11 is possible:

s /ν

∩ Bp00

Bp1

Bp0

(5.7.25)

(K, E). Furthermore, any one of the following substitutions k·k p s/ν Ã k·k p Bp

if

s/ν

Wp

s≥0,

and, for j = 0, 1, k·k p sj /ν Ã k·k p

s /ν

Wpjj

Bpj

if

sj ≥ 0

and

1 < pj ≤ ∞ ,

provided θ < 1 unless s, s1 ∈ νN and 1 < p, p1 < ∞ or s, s1 ∈ / νN. Proof We replace u in (5.7.4) by σt u. Then it follows from (5.7.5) and Lemma 5.7.7 that ³ ´ ts−|ω|/p kuk p s/ν ≤ ts−|ω|/p kuk1−θ p s0 /ν kukθp s1 /ν + c(t, u) , Bp

Bp 0

Bp1

where c(t, u) → 0 as t → ∞. Now we divide this inequality by ts−|ω|/p and let t → ∞. This proves (5.7.25). The other claims follow similarly. ¥ We illustrate this theorem by some prototypical examples, which, in particular, are related to results available in the literature (see Subsection 5.8 for references). Isotropic Multiplicative Inequalities All examples below follow from Theorem 5.7.8. First we consider isotropic E-valued spaces. More precisely, we assume ω = 1. Recall definition (5.7.20). 5.7.9 Examples (a)(Classical estimates) Assume j, m ∈ N with j < m and p0 , p1 belong to (1, ∞]. Set ³1 1 j m´ 1 − θ := + θ − + (5.7.26) π d p1 d p0 for j/m ≤ θ ≤ 1. (α) Assume 1/π ≥ 0. Then m θ k∇j ukπ ≤ c kuk1−θ p0 k∇ ukp1 ,

provided θ < 1 if p1 = ∞.

u ∈ Wpm1 ∩ Wp00 (X, E) ,

216

VII Function Spaces

(β) Suppose 1/π < 0. Set k := [−d/π] and σ := −k − d/π ∈ [0, 1). Then m θ [∇j+k u]σ,∞ ≤ c kuk1−θ p0 k∇ ukp1 ,

u ∈ Wpm1 ∩ Wp00 (X, E) ,

for (j + k + σ)/m ≤ θ < 1. Proof Set s0 := m and s1 := m. (1) Letting s := j, it is immediate that (α) holds. (2) If 1/π < 0, we set s := k + j + σ and p := ∞. Then −1/π = (k + σ)/d and s/m ≤ θ < 1. This and (5.7.26) imply that (5.7.2) and (5.7.3) are satisfied. ¥ q (b)(Negative order estimates) Let s0 ≤ 0, m ∈ N, and p0 , p1 ∈ [1, ∞] with 1 < p1 . Assume 1 1 sθ := − ≥0, p p(θ) d

−

s0 ≤θ 1, then the claim follows from Example 5.7.9(a) by setting s0 = 0 = j, s1 = m, p1 = p, and θ = 1 in (5.7.26). (2) Suppose p = 1. By extension and restriction we can assume that X = Rd . Since D is dense in W1m , it suffices to prove (5.7.31) for u ∈ D. Using obvious notation, Z

Z

xj

|u(x)|E ≤

j

|∂j u|E dxj ,

|∂j u|E dx ≤

x ∈ Rd ,

1≤j≤d,

(5.7.32)

−∞

where we integrate over R if nothing else is indicated. Hence d/(d−1) |u(x)|E

≤

d Z ³Y

|∂j u|E dxj

´1/(d−1)

.

j=1

The generalized H¨older inequality implies Z d/(d−1) |u|E

Z ³Z

1

dx ≤

|∂1 u|E dx

1

d Z Y

|∂j u|E dxj

´1/(d−1)

dx1

j=2

Z ≤

|∂1 u|E dx

1

d Z ³Y j=2

R2

|∂j u|E dxj dx1

´1/(d−1)

.

218

VII Function Spaces

By integrating this inequality successively with respect to dx2 , dx3 , . . . , dxd over Rd and applying each time the generalized H¨older inequality, we arrive at kukd/(d−1) ≤

d ³Y

k∂j uk1

´1/d

.

j=1

Now the elementary inequality (ξ 1 q · · · q ξ d )1/d ≤ (ξ 1 + · · · + ξ d )/d ,

ξ j ∈ R+ ,

which is a consequence of the concavity of the logarithm, implies Z X d d 1X 1 kukd/(d−1) ≤ k∂j uk1 = |∂j u|E dx . d j=1 d Rd j=1 We define p∗ by 1/p∗ := 1/p − 1/d. Then kuk1∗ ≤ k∇uk1 /d. Consequently, k∇uk1∗ ≤ k∇2 uk1 /d . Hence, by step (1), kuk(1∗ )∗ ≤ c k∇2 uk1 , where 1/(1∗ )∗ = 1/1∗ − 1/d = 1 − 2/d. Now (5.7.31) follows by induction. ¥ 5.7.13 Remark If d = 1, then W11 (X, E) ,→ C0 (X, E) and kuk∞ ≤ k∂uk1 , Proof

u ∈ W11 (X, E) .

This is a consequence of (5.7.32) and the density of D in W11 and in C0 .

¥

Parabolic Estimates Next we give a simple application of Theorem 5.7.8 for the case of parabolic weight systems. 5.7.14 Examples We suppose that [m, ν] is a parabolic weight system and use the notations of Examples 3.6.5. (a) (Classical Estimates) Let 1 < p0 , p1 ≤ ∞. Set ³1 1 ν ´ 1−θ −θν 1 := θ − + = + . q p1 m+ν p0 m+ν p(θ) (α) Assume 1/q ≥ 0 and 0 < θ < 1. Then ¡ ν ¢θ kukq ≤ c kuk1−θ k∇ ukp1 + k∂t ukp1 , p0 (β) Suppose 1/q < 0 and 0 < θ ≤ 1. Set £ ¤ k := −(m + ν)/q and

u ∈ Wp(ν,1) ∩ Wp(0,0) (Y × J, E) . 1 0

σ := −k − (m + ν)/q .

Then 0 ≤ k < ν and 0 ≤ σ < 1. Assume (k + σ)/ν ≤ θ < 1.

VII.5 Triebel–Lizorkin Spaces

219

If σ > 0, then ¡ ν ¢θ [[∇k u]]σ,∞;t + [[u]]σ/ν,∞;x ≤ c kuk1−θ k∇ ukp1 + k∂t ukp1 . p0 If σ = 0 and k ≥ 1, then ¡ ν ¢θ [[∇k−1 u]]1,∞;t + kuk∞ ≤ c kuk1−θ k∇ ukp1 + k∂t ukp1 . p0 Proof In this situation, |ω| = m + ν. Put s0 := 0, s1 := ν. In case (α), set s := 0. In case (β), let s := k + σ and p := ∞. Then (5.7.1)–(5.7.3) are satisfied. Hence the assertion follows from Theorem 5.7.8. ¥ (b)(Negative order estimates) Suppose s0 ≤ 0, s1 := ν, and p0 , p1 ∈ [1, ∞] with p1 < ∞. Also assume −s0 /ν ≤ θ < 1 and 1 1 sθ := − ≥0. p p(θ) m + ν Then

¡ ¢θ kukp ≤ c kuk1−θ p (s0 ,s0 /ν) k∇ν ukp1 + k∂t ukp1 Bp0 ,∞

(ν,1)

for u ∈ Wp1

(s ,s /ν)

∩ Bp00,∞0

(Y × J, E). In particular,

¡ ν ¢θ kukp1 /θ ≤ c kuk1−θ p −θν (1,1/ν) k∇ ukp1 + k∂t ukp1 1−θ B∞

for 0 < θ < 1. Proof

5.8

Theorem 5.7.8 with s := 0.

¥

Notes

In the scalar-valued isotropic case, the basic references for Triebel–Lizorkin spaces are, of course, Triebel’s books, [Tri83] in particular. There many more results can be found, as well as detailed historical references. Quite a few of the proofs of [Tri83] can be adapted to our anisotropic vectorvalued setting. Notably, our demonstrations of Lemmas 5.1.4 and 5.1.5 are modifications and adaptions of similar results given in Chapter I of [Tri83]. The proof of Lemma 5.1.7 uses arguments of E.M. Stein [Ste93, II. §2.1]. The fact that Triebel–Lizorkin spaces (and Besov spaces too) posses the Fatou property has first been observed in the scalar-valued isotropic case by J. Franke [Fra86]. Our proof follows H.-J. Schmeißer and W. Sickel [SS01], who considered the isotropic vector-valued case (also see S. Dachkowski [Dac03]). In the classical isotropic case, there are various Fourier multiplier theorems for Triebel–Lizorkin spaces in [Tri83] (see, in particular, Theorem 2.3.7 therein).

220

VII Function Spaces

Theorem 5.5.2 is due to H.-J. Schmeißer and W. Sickel, as are the proofs of the Sobolev embedding theorems 5.6.3 and 5.6.5 –– in the isotropic vectorvalued case (see [SS01] and [SS05]). The demonstrations of the basic Lemmas 5.6.1 and 5.6.2 follow H. Brezis and P. Mironescu [BrM01]. That paper contains also inequality (5.7.18) in the particular (isotropic scalar) case, where either s0 ≥ 0, 1 < p0 , p1 < ∞, or s0 = 0, p0 = ∞, and, in either case, p = p(θ). In a more recent paper [BrM18] these authors find necessary and sufficient conditions on the parameters for (5.7.18) to be valid. Theorem 5.6.6 seems to be new. Related results have been obtained by R. Denk, J. Saal, and J.Seiler [DSS08], R. Denk and M Kaip [DK13], and M. Meyries and M. Veraar [MeV14]. Example 5.7.9(a) is precisely the E-valued version of the Gagliardo–Nirenberg inequality as it appears in L. Nirenberg’s paper [Nir59], except for some limiting cases. Most notably is our restriction p0 , p1 > 1. For this reason we have given –– using Nirenberg’s method –– a direct proof of the Sobolev inequality in the case p = 1. There are several extensions of the classical Gagliardo–Nirenberg inequality to fractional order spaces. Most of them consider the case s0 = 0 and p = p(θ) and use homogeneous Bessel potential spaces as the right endpoint space. A general study in the framework of homogeneous Triebel–Lizorkin and Bessel potential spaces has been carried out by H. Triebel [Tri14] for the case s0 ≥ 0. Inequalities valid in that setting require, of course, that p0 andq p1 belong to (1, ∞). The seminorm in the homogeneous Bessel potential space Hps (Rd ) is u 7→ k(−∆)s/2 ukp . This essentially restricts the applicability of such inequalities to Rd . In contrast, the fractional order inequalities of Subsection 5.7 are much more flexible and allow extensions to manifolds with boundary, for example. This is already witnessed by the fact that our results apply to distributions defined on corners. (Observe that, although we use homogeneous seminorms, the functions belong to inhomogeneous spaces.) It should also be mentioned that there is a number of variants of Gagliardo– Nirenberg inequalities involving BMO, Lorentz, Morrey, or Sobolev spaces with weights. See N.A. Dao, J.I. D´ıaz, and Q.-H. Nguyen [DDN18], D.S. McCormick, J.C. Robinson, and J.L. Rodrigo [McRR13], H. Kozono and H. Wadade [KoW08], H. Wadade [Wad10], and the references therein, for example. Gagliardo–Nirenberg type inequalities involving homogeneous Besov spaces of negative order have been studied, among others, by M. Ledoux [Led03], V.I. Kolyada and F.J. P´erez L´azaro [KoPL14], H. Bahouri and A. Cohen [BaC11] using entirely different techniques like semigroup and rearrangement methods. Multiplicative inequalities of the type studied here are of great importance for the theory of nonlinear partial differential equations. For example, they allow to reduce the problem of global existence in evolution equations to find a priori bounds in relatively weak norms. See, for instance, [Ama85]. That paper seems to

VII.5 Triebel–Lizorkin Spaces

221

contain the first (non-optimal) extension of Gagliardo–Nirenberg inequalities to fractional order spaces. Another important line of applications rests on the fact that such inequalities can be used to estimate seminorms of intermediate strength by arbitrarily small contributions of higher order seminorms. This is due to the following well-known simple observation: Suppose x, y, z > 0 and 0 < θ < 1. Then there exists a constant c0 ≥ 1 such that x ≤ c0 y 1−θ z θ

(5.8.1)

iff there exists a constant c1 ≥ 1 such that x ≤ εz + c1 ε−θ/(1−θ) y , Proof

ε>0.

(5.8.2)

(1) Let (5.8.1) be satisfied. Write ¡ 1/(1−θ) −θ/(1−θ) ¢1−θ c0 y 1−θ z θ = (εz)θ c0 ε y

and apply Young’s inequality ξ 1−θ η θ ≤ (1 − θ)ξ + θη < ξ + η for ξ, η > 0. (2) Assume (5.8.2) applies. Then (5.8.1) is obtained by minimizing the function f (t) := tz + c1 t−θ/(1−θ) y, t > 0. ¥ It is well-known (see [Tri95]) that . s Fp,2 (Rd ) = Hps (Rd ) ,

s∈R,

1 − , q p q |ω| + ¡ s/ν ¢ then ∂ α ∈ L Bq,r (X, E), Lp (X, E) . The equality sign can be admitted if either s − α q ω > |ω|/q or r = 1. (ii) Assume K is closed and s − t ≥ α q ω. If 1 1 ³1 s − t − α q ω ´ ≥ ≥ − , q p q |ω| + then ¡ s/ν ¢ ¡ ¢ t/ν ∂ α ∈ L Bq,r (X, E), Bp,r (X, E) ∩ L Bqs/ν (X, E), Bpt/ν (X, E) . Proof

We omit (X, E). It follows from Theorems 2.2.2 and 2.8.3 that 1 1 ³1 s − σ ´ s/ν σ/ν Bq,r ,→ Bp,r if ≥ ≥ − q p q |ω| +

(6.1.1)

for 0 ≤ σ ≤ s. (1) Using Theorem 2.6.5, σ/ν Bp,r ,→ Wpα

p ω/ν

∂α

−→ Lp

if either σ > α q ω or r = 1. This and (6.1.1) imply (i). (2) If K is closed, then ¡ σ/ν σ−α p ω ¢ ∂ α ∈ L Bp,r , Bp,r by Theorem 2.8.6. From this and (6.1.1) we get the first part of (ii) by setting σ = t + α q ω. For the second part we have to invoke Corollary 5.6.4. ¥ 6.1.2 Corollary Suppose m ∈ νN. Then ¡ ¢ ∂ α ∈ L Wqm/ν (X, E), Lp (X, E) , provided

1 1 ³1 m − α q ω ´ ≥ ≥ − q p q |ω| +

and either 13 x

+

α qω ≤ m ,

:= max{x, 0} and x− := min{x, 0} for x ∈ R.

224

VII Function Spaces

(i) the second equality is strict. or (ii) m − α q ω > |ω|/q. or (iii) p, q ∈ (1, ∞) and α q ω ∈ νN. Proof

m/ν

Use either Wq

m/ν

,→ Bq,∞ and the theorem, or Theorem 5.6.5.

¥

Point-Wise Products We denote by L(E0 , E1 ; E2 ) the Banach space of all maps β : E0 × E1 → E2 which are bilinear and bounded, endowed with the norm © ª β 7→ |β| := sup |β(e0 , e1 )|E2 ; |e0 |E0 ≤ 1, |e1 |E1 ≤ 1 . With β ∈ L(E0 , E1 ; E2 ) we associate the continuous linear map Bβ : E0 7→ L(E1 , E2 ) , e0 7→ β(e0 , ·) . ¡ ¢ Then |Bβ | ≤ |β|. Conversely, given B ∈ L E0 , L(E1 , E2 ) , we put βB : E0 × E1 7→ E2 ,

(e0 , e1 ) 7→ (Be0 )e1 .

It follows βB ∈ L(E0 , E1 ; E2 ) and |βB | ≤ |B|. Note that βBβ = β and BβB = B. Hence ¡ ¢ L(E0 , E1 ; E2 ) → L E0 , L(E1 , E2 ) , β → 7 Bβ (6.1.2) is an isometric isomorphism by which we often identify these two Banach spaces. If β ∈ L(E0 , E1 ; E2 ), then its point-wise extension, mβ , over X is the bilinear map BC(X, E0 ) × L1,loc (X, E1 ) → L1,loc (X, E2 ) , defined by

¡ ¢ mβ (u0 , u1 )(x) := β u0 (x), u1 (x) ,

(u0 , u1 ) 7→ mβ (u0 , u1 ) a.a. x ∈ X .

We often write β(u0 , u1 ) for mβ without fearing confusion. Throughout the rest of this section •

β ∈ L(E0 , E1 ; E2 ) ,

unless explicitly stated otherwise. Furthermore, we frequently simply write • mβ : F0 (X, E0 ) × F1 (X, E1 ) ,→ F2 (X, E2 ) for

¡ ¢ mβ ∈ L F0 (X, E0 ), F1 (X, E1 ); F2 (X, E2 ) ,

where Fj (X, Ej ) are suitable subspaces of S 0 (X, Ej ).

VII.6 Point-Wise Multiplications

6.2

225

Multiplications in Classical Spaces

First we prove point-wise multiplier theorems in spaces of bounded and continuous functions and Sobolev spaces. Spaces of Bounded Continuous Functions The following elementary point-wise multiplier theorem is basically a consequence of Leibniz’ rule. 6.2.1 Theorem Suppose m ∈ νN. (i) If K is closed and F ∈ {BC, BUC}, then ¡ ¢ mβ ∈ L Fm/ν (X, E0 ), Fm/ν (X, E1 ); Fm/ν (X, E2 ) . ¡ ¢ m/ν m/ν (ii) mβ ∈ L BC m/ν (X, E0 ), Wq (X, E1 ); Wq (X, E2 ) . Proof

(1) Let α ∈ Nd and uj ∈ C |α| (X, Ej ) for j = 0, 1. Then, by Leibniz’ rule, ¡ ¢ X³ α ´ ∂ α β(u0 , u1 ) = β(∂ α−γ u0 , ∂ γ u1 ) . (6.2.1) γ γ≤α

Moreover, kβ(u0 , u1 )kr ≤ |β| ku0 k∞ ku1 kr . From this we get |||β(u0 , u1 )|||m/ν,r ≤ c(m) |β| |||u0 |||m/ν,∞ |||u1 |||m/ν,r .

(6.2.2)

This implies (i) if F = BC. (2) We infer from β(u0 , u1 )(x) − β(u0 , u1 )(y) ¡ ¢ ¡ ¢ = β u0 (x) − u0 (y), u1 (x) + β u0 (y), u1 (x) − u1 (y) that β(u0 , u1 ) ∈ BUC(X, E2 ) if uj ∈ BUC(X, Ej ). Now (1) and the definition of BUC m/ν imply (i) for F = BUC. (3) Suppose u0 ∈ BC ∞ (X, E0 ) and u1 ∈ S(X, E1 ). It follows from (VI.1.1.2), (1.1.9), and (6.2.1) that ¡ ¢ ∂ α β(u0 , u1 ) |∂j K = 0 , α ∈ Nd , j ∈ JK∗ , if X = K. From this and (6.2.2) we deduce β(u0 , u1 ) ∈ S(X, E2 ) and that (6.2.2) m/ν applies. Since S(X, E1 ) is dense in Wq (X, E1 ), estimate (6.2.2) holds equally

226

VII Function Spaces m/ν

well for u0 in BC ∞ (X, E0 ) and u1 in Wq to arrive at assertion (ii). ¥

(X, E1 ). Now we apply Theorem 1.6.4 m/ν

6.2.2 Remarks (a) BC m/ν (X, E0 ) is a multiplier space for Wq pendently of q. Thus it is universal.

(X, E1 ), inde-

(b) The assumption that u0 belongs to BC m/ν (K, E) means that u0 does not need to vanish on ∂j K for j ∈ J ∗ if K is not closed. (c) Let K be closed. In agreement with Theorem 1.4.1, we set m/ν W∞ (X, E) :=

©

u ∈ S 0 (˚ X, E) ; ∂ α u ∈ L∞ (X, E) for α q ω ≤ m

ª

.

It is a Banach space with the norm |||·|||m/ν,∞ which contains BC m/ν (X, E) as a closed linear subspace. Clearly, Theorem 6.2.1(ii) remains valid if K is closed and BC m/ν (X, E0 ) is m/ν replaced by W∞ (X, E0 ). (d) In Theorem 6.2.1 the map β 7→ mβ is linear and continuous. This is also true in all other point-wise multiplier theorems of this section. ¥ Sobolev Spaces The following theorem shows, in particular, that we can replace BC m/ν (X, E0 ) by m/ν Wq0 (X, E0 ) if m > |ω|/q0 . 6.2.3 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. Suppose m0 , m ∈ νN with m ≤ m0 and 1 ≤ q0 < ∞ are such that m0 > |ω|/q0 . Then 0 /ν mβ : Wqm (X, E0 ) × Wqm/ν (X, E1 ) ,→ Wqm/ν (X, E2 ) , 0

1≤q 1 if m = m0 . Proof

Assume α 0 + α1 = α ,

α qω ≤ m .

(6.2.3)

Then, by H¨older’s inequality, kβ(∂ α0 u0 , ∂ α1 u1 )kq ≤ |β| k∂ α0 u0 kp0 k∂ α1 u1 )kp1 ,

(6.2.4)

with p0 , p1 ∈ [1, ∞] satisfying 1 1 1 + = . p0 p1 q

(6.2.5)

VII.6 Point-Wise Multiplications

227

We set m1 := m and q1 := q. Suppose we can choose pi satisfying (6.2.5) such that ³1 1 1 mi − αi q ω ´ ≥ ≥ − , qi pi qi |ω| +

(6.2.6)

where the second inequality is strict unless either mi − αi q ω > |ω|/qi or αi q ωi belongs to νN and qi > 1. Then it follows from Corollary 6.1.2 that kβ(∂ α0 u0 , ∂ α1 u1 )kp ≤ c |β| |||u0 |||m0 /ν,q0 |||u1 |||m1 /ν,q1 .

(6.2.7)

If α0 = 0 and α q ω = m1 , then we can choose p0 = ∞ and p1 = q1 . Let either α0 > 0 and α q ω = m1 , or α q ω < m1 . Then α0 q ω ≤ m1 and q α1 ω < m1 . Hence, if either α0 q ω < m1 or m0 > m1 , 1 m0 − α0 q ω 1 − < q0 |ω| q0 and 1/q1 − (m1 − α1 q ω)/|ω| < 1/q1 . Consequently, by (6.2.3), ³1 m0 − α0 q ω ´ ³ 1 m1 − α1 q ω ´ 1 m0 1 1 1 1 − + − < − + < < + . q0 |ω| q1 |ω| q0 |ω| q1 q1 q0 q1 This shows that we can fix pi satisfying (6.2.6) with a strict inequality sign in the second place such that (6.2.5) holds true. Lastly, if α0 q ω = m1 = m0 , then the choice p0 = q0 = q1 and p1 = ∞ is possible. These considerations show that (6.2.7) is true whenever (6.2.3) is satisfied, provided q0 = q if m0 = m. Now the assertion is a consequence of (6.2.1). ¥ 6.2.4 Corollary Suppose m ∈ νN satisfies m > |ω|/q with 1 < q < ∞. If E is a m/ν continuous multiplication algebra, then Wq (X, E) is one as well. A slight modification of the preceding proof leads to the following point-wise multiplier theorem for Sobolev spaces of low regularity. 6.2.5 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. Suppose m0 , m1 , and m2 , belonging to νN, and q0 , q1 , q2 ∈ [1, ∞) satisfy mi ≤ |ω|/qi for i = 0, 1 , and

Then

m 2 ≤ m1 ≤ m0 ,

1/q2 ≤ 1/q0 + 1/q1 ,

1 m2 1 m0 1 m1 − > − + − . q2 |ω| q0 |ω| q1 |ω| 0 /ν 1 /ν 2 /ν mβ : Wqm (X, E0 ) × Wqm (X, E1 ) ,→ Wqm (X, E2 ) . 0 1 2

228

VII Function Spaces

Proof

Let α0 + α1 = α with α q ω ≤ m2 . Then

³1 m0 − α0 q ω ´ ³ 1 m1 − α1 q ω ´ − + − q0 |ω| q1 |ω| 1 m0 1 m1 α qω 1 m0 1 m1 m2 1 = − + − + ≤ − + − + < . q0 |ω| q1 |ω| |ω| q0 |ω| q1 |ω| |ω| q2 This shows that we can choose pi such that (6.2.6) holds with strict inequalities and 1/p0 + 1/p1 = 1/q2 . Hence (6.2.7) applies with p = q2 . Now the claim follows once more from (6.2.1). ¥ Spaces of Negative Order Let E1 and E2 be reflexive, p1 , p2 ∈ (1, ∞), and denote by |||·|||−m/ν,pi the norm −m/ν

of Wpi

(X, Ei ). Assume |||β(u0 , u1 )|||−m2 /ν,p2 ≤ c |β| |||u0 |||m0 /ν,q0 |||u1 |||−m1 /ν,p1 m /ν

for u0 ∈ Wq0 0 (X, E0 ) if 1 ≤ q0 < ∞, and u0 ∈ BC m0 /ν (X, E0 ) for q0 = ∞, and −m /ν for u1 ∈ S(X, E1 ). Then, by the density of S(X, E1 ) in Wq1 1 (X, E1 ), there exists ¯ β such that a unique m 0 /ν 1 /ν1 2 /ν ¯ β : Wqm m (X, E0 ) × Wp−m (X, E1 ) ,→ Wp−m (X, E2 ) 0 1 2

(6.2.8)

if 1 ≤ q0 < ∞, and 1 /ν1 2 /ν ¯ β : BC m0 /ν (X, E0 ) × Wp−m m (X, E1 ) ,→ Wp−m (X, E2 ) 1 2

(6.2.9)

¯ β (·, u1 ) = mβ (·, u1 ) for u1 ∈ S(X, E1 ). Thus it is feasible to if q0 = ∞, such that m ¯ β. write again mβ for m The following theorems extend point-wise multiplication to Sobolev spaces of negative order. 6.2.6 Theorem Let K be a corner in Rd and X ∈ {Rd , K}. Suppose E1 and E2 are reflexive, p0 , p ∈ (1, ∞), and m0 , m ∈ νN. Then −m/ν

(i) mβ : BC m/ν (X, E0 ) × Wp

−m/ν

(X, E1 ) ,→ Wp

(X, E2 ).

(ii) If m0 > |ω|/p0 and 0 ≤ m < m0 , then 0 /ν mβ : Wpm (X, E0 ) × Wp−m/ν (X∗ , E1 ) ,→ Wp−m/ν (X∗ , E2 ) . 0

VII.6 Point-Wise Multiplications

Proof

229

(1) We define a bilinear map β 0 : E0 × E20 → E10 ,

­ ® (e0 , e02 ) 7→ e02 , β(e0 , ·) E2 .

Then ¯­ ¯­ ® ¯ ® ¯ |β 0 (e0 , e02 )|E10 = sup ¯ β 0 (e0 , e02 ), e1 E1 ¯ = sup ¯ e02 , β(e0 , e1 ) E2 ¯ |e1 |≤1

|e1 |≤1

≤ |β| |e0 |E0 |e02 |E20 . This shows β 0 ∈ L(E0 , E20 ; E10 ) ,

|β 0 | ≤ |β| .

(6.2.10)

(2) It follows from Theorem 6.2.1(ii) and cl(K∗ ) = K that ¡ ¢ m/ν m/ν mβ 0 ∈ L BC m/ν (X, E0 ), Wp0 (X∗ , E20 ); Wp0 (X∗ , E10 )

(6.2.11)

and kmβ 0 k ≤ c |β|. Assume u1 ∈ S(X, E1 ). Then the definition of β 0 implies ­ ® ­ ® mβ 0 (u0 , u02 ), u1 = u02 , mβ (u0 , u1 ) m/ν

for u0 ∈ BC m/ν (X, E0 ) and u02 ∈ Wp0 (X∗ , E20 ). Hence we infer from (6.2.11), Theorem 1.5.1, and definition (1.2.5) ¯­ 0 ° ®¯ ° ¯ u2 , mβ (u0 , u1 ) ¯ ≤ °mβ 0 (u0 , u02 )° m/ν ∗ 0 ku1 k m/ν ∗ 0 0 (W (X ,E )) W (X ,E ) p0

≤

p0

1

c |β| |||u0 |||m/ν,∞ ku02 kW m/ν (X∗ ,E 0 ) 0 2 p

1

|||u1 |||−m/ν,p .

Consequently, using Theorem 1.5.1 once more, we see that (6.2.9) is satisfied with m1 = m2 = m and p1 = p2 = p. This proves the first claim. (3) By Theorem 6.2.3, m/ν

0 /ν mβ 0 : Wpm (X, E0 ) × Wp0 0

2

m/ν

(X, E20 ) ,→ Wp0

1

(X, E10 )

and kmβ 0 k ≤ c |β|. Now an obvious modification of step (2) proves statement (ii). ¥ Clearly, ‘dualizing’ Theorem 6.2.5, we can obtain a multiplication theorem for Sobolev spaces of negative order in which the multipliers possess low regularity.

6.3

Multiplications in Besov Spaces of Positive Order

First we prove a general multiplier theorem for positive order Besov spaces. It has a number of important consequences which we derive afterwards.

230

VII Function Spaces

6.3.1 Theorem Let K be a closed corner in Rd and X ∈ {Rd , K}. Suppose 0 < s2 ≤ s1 ≤ s0 ,

1 ≤ p0 , p1 , p2 ≤ ∞ with 1/p2 ≤ 1/p0 + 1/p1

(6.3.1)

sk 6= |ω|/pk ,

(6.3.2)

satisfy k = 0, 1 ,

and ³ ³1 1 s0 ´ s1 ´   − + − ,  p |ω| + p1 |ω| + 1 s2 0 − ≥ n1 sk o p2 |ω|    max − k=0,1 pk |ω|

n1 sk o − >0, k=0,1 pk |ω|

if max

(6.3.3)

otherwise ,

where s2 < s 1

n1 sk o − >0, k=0,1 pk |ω|

if

min

(6.3.4)

and (6.3.3) is a strict inequality if at least one of the s0 , s1 , and s2 is an integer. Assume, moreover, that either 1 ≤ r0 ≤ r1 ≤ r2 ≤ ∞ ,

(6.3.5)

or (6.3.3) is a strict inequality and 1 ≤ r0 ≤ ∞ ,

1 ≤ r1 ≤ r2 ≤ ∞ .

(6.3.6)

Then /ν /ν /ν mβ : Bps00 ,r (X, E0 ) × Bps11 ,r (X, E1 ) ,→ Bps22 ,r (X, E2 ) . 0 1 2

(6.3.7)

Proof It suffices to prove the theorem for r0 = r1 = r2 = r ∈ [1, ∞]. Then, applying this result with r = r1 and using (2.2.1), we obtain (6.3.7) under the assumption (6.3.5). (1) First we suppose s0 , s 1 , s 2 ∈ /N.

(6.3.8)

Theorem 6.1.1(i) implies k∂ α0 u0 kπ0 ≤ c ku0 kB s0 /ν p0 ,r

if α0 q ω < s0 and ³1 1 1 s 0 − α0 q ω ´ ≥ > − , p0 π0 p0 |ω| +

(6.3.9)

VII.6 Point-Wise Multiplications

231

where the equality sign is permitted if |ω|/p0 < s0 − α0 q ω. Similarly, k∂ α1 u1 kπ1 ≤ c ku1 kB s1 /ν p1 ,r

if α1 q ω < s1 and ³1 1 1 s 1 − α1 q ω ´ ≥ > − , p1 π1 p1 |ω| +

(6.3.10)

the equality being admissible if |ω|/p1 < s1 − α1 q ω. Thus, if these condition are satisfied and α0 + α1 = α with α q ω < s2 , then kβ(∂ α0 u0 , ∂ α1 u1 )kp2 ≤ c |β| ku0 kB s0 /ν ku1 kB s1 /ν ,

(6.3.11)

1/p2 = 1/π0 + 1/π1 .

(6.3.12)

p0 ,r

p1 ,r

provided

We observe that, for α q ω < s2 , 1 s0 − α0 q ω 1 s1 − α 1 q ω 1 s0 1 s1 α qω − + − = − + − + p0 |ω| p1 |ω| p0 |ω| p1 |ω| |ω| 1 s0 1 s1 s2 < − + − + . p0 |ω| p1 |ω| |ω|

(6.3.13)

If the first alternative of (6.3.3) applies, then the last expression is majorized by ³1 ³1 s0 ´ s1 ´ s2 1 − + − + ≤ . p0 |ω| + p1 |ω| + |ω| p2 This shows that, in this case, we can choose π0 and π1 such that (6.3.9), (6.3.10), and (6.3.12) are satisfied. If the second alternative of (6.3.3) is active, then the last expression of (6.3.13) is bounded from above by n1 sk o s2 1 max − + ≤ . k=0,1 pk |ω| |ω| p2 Hence in this case we can also find π0 and π1 satisfying (6.3.9), (6.3.10), and (6.3.12). Define mi ∈ νi N by mi < s2 < mi + νi for 1 ≤ i ≤ `. Then the above observations and Leibniz’ rule (6.2.1) imply ` m i /νi X X ° j¡ ¢° °∇x β(u0 , u1 ) ° ≤ c |β| ku0 k s0 /ν ku1 k s1 /ν . i p2 B B i=1 j=0

p0 ,r

p1 ,r

(6.3.14)

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VII Function Spaces

(2) Note that ¡ ¢ 4y β(u0 , u1 ) = β 4y u0 , u1 (· + y) + β(u0 , 4y u1 ) ¡ ¢ for y ∈ Xd . Thus k4y β(u0 , u1 )kπ2 ≤ |β| k4y u0 kπ0 ku1 kπ1 + ku0 kπ0 ku1 kπ1 , if 1 1 1 1 1 = + = + . π2 π0 π1 π e0 π e1 Consequently, given θ ∈ (0, 1), £ ¤ ¡ ¢ β(u0 , u1 ) θ,π2 ,r ≤ |β| [u0 ]θ,π0 ,r ku1 kπ1 + ku0 kπ0 [u1 ]θ,π1 ,r .

(6.3.15)

(3) Fix i ∈ {1, . . . , `} and suppose α0 + α1 = α ∈ Ndi and α q ω = |α| νi = mi . By Theorem 6.1.1(ii), [[∂xαi0 u0 ]](s2 −mi )/νi ,π0 ,r;i ≤ c ku0 kB s0 /ν p0 ,r

(recall Theorem 3.5.2), provided ³1 1 1 s0 − s2 + mi − |α0 | νi ´ ≥ ≥ − . p0 π0 p0 |ω| +

(6.3.16)

Similarly, [[∂xαi1 u1 ]](s2 −mi )/νi ,π1 ,r;i ≤ c ku1 kB s1 /ν , p1 ,r

if ³1 1 1 s1 − s2 + mi − |α1 | νi ´ ≥ ≥ − . p1 π e1 p1 |ω| +

(6.3.17)

Hence, if we can find π0 and π1 satisfying (6.3.12), (6.3.16), and (6.3.10), as well as π e0 and π e1 with 1/p2 = 1/e π0 + 1/e π1 such that (6.3.9) and (6.3.17) are valid, then it follows from (6.3.15) that ££ ¤¤ β(∂xαi0 u0 , ∂xαi1 u1 ) (s −m )/ν ,p ,r;i ≤ c |β| ku0 kB s0 /ν ku1 kB s1 /ν . (6.3.18) 2

i

i

2

p0 ,r

p1 ,r

We take the sum of the two expressions in ( · · · )+ in (6.3.16) and (6.3.10) to get

1 s0 s2 mi − |α0 | νi 1 s1 |α1 | νi − + − + − + p0 |ω| |ω| |ω| p1 |ω| |ω| 1 s0 1 s1 s2 = − + − + =: a . p0 |ω| p1 |ω| |ω|

Analogously, by adding the corresponding terms of (6.3.9) and (6.3.17), we arrive at 1 s0 − |α0 | νi 1 s1 s2 mi − |α1 | νi − + − + − =a. p0 |ω| p1 |ω| |ω| |ω|

VII.6 Point-Wise Multiplications

233

If the first alternative of (6.3.3) is in force, then a≤

³1 ³1 s0 ´ s1 ´ s2 1 − + − + ≤ . p0 |ω| + p1 |ω| + |ω| p2

Otherwise,

n1 sk o s2 1 − + ≤ . k=0,1 pk |ω| |ω| p2

a ≤ max

In each case we can thus choose π0 , π1 , and π e0 , π e1 , as desired. From this, (6.3.18) and (6.2.1) we obtain ££

m /νi ¡

∇xi i

β(u0 , u1 )

¢¤¤ (s2 −mi )/νi ,p2 ,r;i

≤ c |β| ku0 kB s0 /ν ku1 kB s1 /ν . p0 ,r

p1 ,r

This is true for each i ∈ {1, . . . , `}. Hence we see from it, (6.3.14), and Theorem 3.5.2 that the assertion is valid in this case. (4) We now drop assumption (6.3.8). Since (6.3.3) is a strict inequality, we can choose σj < sj for j = 0, 1, and σ2 > s2 such that (6.3.2), (6.3.3), and (6.3.4) hold with {s0 , s1 , s2 } replaced by {σ0 , σ1 , σ2 }. Then, by what we have already proved, /ν /ν /ν mβ : Bpσ00,r (X, E0 ) × Bpσ11,r (X, E1 ) ,→ Bpσ22,r (X, E2 ) . 0 1 2 s /ν

σ /ν

σ /ν

s /ν

Since Bpjj ,rj ,→ Bpjj,rj for j = 0, 1 and Bp00,r0 ,→ Bp00 ,r0 , the assertion follows. (5) Lastly, suppose (6.3.3) is a strict inequality and (6.3.6) applies. Then we fix σ0 < s0 such that we can apply the foregoing results to get σ /ν

/ν /ν mβ : Bp00,1 (X, E0 ) × Bps11 ,r (X, E1 ) ,→ Bps22 ,r (X, E2 ) . 1 2 s /ν

σ /ν

Now we get the claim from Bp00 ,r0 ,→ Bp00,1 . The theorem is proved.

¥

6.3.2 Remarks Let (6.3.1) be satisfied. (a) Assume s0 > |ω|/p0 and s0 − |ω|/p0 ≥ s1 − |ω|/p1 ,

s1 6= |ω|/p1 .

(6.3.19)

Then conditions (6.3.2) and (6.3.3) reduce to s2 − |ω|/p2 ≤ s1 − |ω|/p1 .

(6.3.20)

(b) Suppose s0 < |ω|/p0 and (6.3.19) is satisfied. Then (6.3.2) and (6.3.3) boil down to s2 − |ω|/p2 ≤ s0 − |ω|/p0 + (s1 − |ω|/p1 )− . Proof

This is obvious in both cases.

¥

234

VII Function Spaces

These general results have major implications some of which we collect in the next theorem. 6.3.3 Theorem Let K be a closed corner in Rd , X ∈ {Rd , K}, and 1 ≤ p0 , p ≤ ∞. (i) If s0 > |ω|/p0 and 0 < s ≤ s0 are such that s0 − |ω|/p0 ≥ s − |ω|/p ,

(6.3.21)

then /ν s/ν s/ν mβ : Bps00 ,r (X, E0 ) × Bp,r (X, E1 ) ,→ Bp,r (X, E2 ) , 0

0 < s ≤ s0 ,

where r0 , r ∈ [1, ∞] satisfy r0 ≤ r, unless (6.3.21) is a strict inequality, and s0 6= |ω|/p if s = s0 . (ii) For 0 < s < s0 , s0 /ν s/ν s/ν mβ : B ∞ (X, E0 ) × Bp,r (X, E1 ) ,→ Bp,r (X, E2 ) , 1 2

1 ≤ r1 ≤ r2 ≤ ∞ .

If r1 = r2 = ∞, then this applies to s = s0 also. s /ν

s/ν

s/ν

(iii) mβ : b∞0 (X, E0 ) × b∞ (X, E1 ) ,→ b∞ (X, E2 ), 0 < s ≤ s0 . (iv) Suppose s0 < |ω|/p and 0 < s1 ≤ s0 . Then s0 /ν s1 /ν s0 +s1 −|ω|/p mβ : Bp,r (X, E0 ) × Bp,r (X, E1 ) ,→ Bp,r (X, E2 ) 0 1 2

with 1 ≤ r0 ≤ r1 ≤ r2 ≤ ∞. Proof (1) Statement (i) follows from Remark 6.3.2(a) and Theorem 6.3.1, provided s 6= |ω|/p. Assume s = |ω|/p. Then s < s0 . Thus we can choose 0 < t0 < s < t1 < s0 s0 /ν such that tj 6= |ω|/p. Then, given u0 ∈ Bp,r 0 , ¡ tj /ν tj /ν ¢ β(u0 , ·) ∈ L Bp,r , Bp,r ,

kβ(u0 , ·)k ≤ c |β| ku0 kB s0 /ν . p,r0

(6.3.22)

Set θ := (s − t0 )/(t1 − t0 ). Then we obtain from (6.3.22) and Theorems 2.7.1(i) and 2.8.3, by interpolating with the functor (·, ·)θ,r , ¡ s/ν s/ν ¢ β(u0 , ·) ∈ L Bp,r , Bp,r ,

kβ(u0 , ·)k ≤ c |β| ku0 kB s0 /ν . p,r0

This implies (i). (2) Claim (ii) is a special case of (i). (3) Claim (iii) is an easy consequence of (ii) (with r1 = r2 = ∞), definition (2.2.7), and Theorem 2.8.3. (4) The last assertion is clear from Remark 6.3.2(b) and Theorem 6.3.1.

¥

VII.6 Point-Wise Multiplications

235

6.3.4 Corollary Suppose 1 ≤ p < ∞. (i) Let s > |ω|/p0 > 0. Then mβ : Wps/ν (X, E0 ) × Wpt/ν (X, E1 ) ,→ Wpt/ν (X, E2 ) 0 if either s > t and s − |ω|/p0 > t − |ω|/p or s = t and p0 = p. t/ν

(ii) mβ : BUC s/ν (X, E0 ) × Wp s/ν

t/ν

(X, E1 ) ,→ Wp

(X, E2 ), 0 ≤ t < s.

t/ν

(iii) mβ : BUC (X, E0 ) × BUC (X, E1 ) ,→ BUC t/ν (X, E2 ) and mβ : bucs/ν (X, E0 ) × buct/ν (X, E1 ) ,→ buct/ν (X, E2 ) for 0 ≤ t ≤ s. r/ν

Proof

Recall that Wp

r/ν

= Bp

r/ν

and BUC r/ν = B∞ for r ∈ R+ \νN.

Suppose s, t ∈ / νN. Then the claims follow from the theorem. If s, t ∈ νN, then they are contained in Theorems 6.2.1 and 6.2.3. If s ∈ / νN and t ∈ νN, or s ∈ νN and t ∈ / νN, then we use –– in an obvious manner –– the embeddings Fs/ν ,→ Ft/ν for s > t and F ∈ {Wq , BUC, buc}. ¥

6.4

Multiplications in Besov Spaces of Negative Order

Now we consider multiplications in Besov spaces with one factor being a singular distribution. The Reflexive Case First we restrict ourselves to the reflexive case and employ the duality argument introduced in Subsection 6.2. 6.4.1 Theorem Let K be a closed corner in Rd , X ∈ {Rd , K}, and E1 , E2 reflexive. Assume −s0 ≤ s2 ≤ s1 < 0

and

p0 ∈ [1, ∞], p1 , p2 ∈ (1, ∞) with 1/p2 ≤ 1/p0 + 1/p1

satisfy s0 6= |ω|/p0 ,

s2 6= −|ω| (1 − 1/p2 ) .

(6.4.1)

Also assume r0 ∈ [1, ∞] and r1 , r2 ∈ (1, ∞) are such that r0 ≤ r20 ≤ r10 . Then /ν /ν /ν mβ : Bps00 ,r (X, E0 ) × Bps11 ,r (X∗ , E1 ) ,→ Bps22 ,r (X∗ , E2 ) , 0 1 2

(6.4.2)

236

VII Function Spaces

provided s0 + s1 ≥ |ω|

³1 ´ 1 + −1 p0 p2

(6.4.3)

and one of the following conditions is satisfied: (i) s0 > |ω|/p0 and s2 − |ω|/p2 ≤ s1 − |ω|/p1 . (ii) s0 < |ω|/p0 and s0 + s1 − |ω|

³1 ´ ³ ³ 1 1 ´´ + − 1 ≥ s2 + |ω| 1 − . p0 p1 p2 +

where s2 < s1 if s2 + |ω| (1 − 1/p2 ) > 0. If the second inequality in (i), resp. (ii), is strict, then the restriction r0 ≤ r20 can be omitted. Proof We set σ0 := s0 , σ1 := −s2 , σ2 := −s1 and π0 := p0 , π1 := p02 , π2 := p01 , as well as ρ0 := r0 , ρ1 := r20 , ρ2 := r10 . Then σj and πj satisfy conditions (6.3.1) and (6.3.2). It is also verified that assumptions (i) and (ii) guarantee that (6.3.3) holds (with sj , pj , rj replaced by σj , πj , ρj ). Thus, since ρ0 ≤ ρ1 ≤ ρ2 , it follows from Theorem 6.3.1 that −s /ν

−s /ν

/ν mβ 0 : Bps00 ,r (X, E0 ) × Bp0 ,r2 0 (X, E20 ) ,→ Bp0 ,r1 0 (X, E10 ) . 0 2

2

1

1

From this we obtain the assertion by Theorem 2.8.4 and the duality argument employed in the proof of Theorem 6.2.6. The last assertion is a consequence of Remarks 6.3.2(a) and (b). ¥ 6.4.2 Corollary Let p0 ∈ [1, ∞) and p ∈ (1, ∞). Assume 0 < s ≤ s0 with s < s0 if s0 ∈ νN. (i) If s0 > |ω|/p0 , then mβ : Wps00 /ν (X, E0 ) × Bp−s/ν (X∗ , E1 ) ,→ Bp−s/ν (X∗ , E2 ) , provided s0 − s ≥ |ω| −s/ν

(ii) mβ : BUC s0 (X, E0 ) × Bp

³1 ´ 1 + −1 ≥0 . p0 p −s/ν

(X∗ , E1 ) ,→ Bp

(6.4.4)

(X∗ , E2 ) if 0 < s < s0 .

(iii) Suppose 0 < s0 < |ω|/p0 and (6.4.4) is satisfied. Then mβ : Wps00 /ν (X, E0 ) × Bp−s/ν (X∗ , E1 ) ,→ Bp(s0 −s−|ω|/p0 )/ν (X∗ , E2 ) .

VII.6 Point-Wise Multiplications

237

Proof (1) If s0 ∈ / νN and s 6= |ω|/p0 , then the assertion is immediate by the theorem, due to the fact that the second inequality in (6.4.4) implies r0 = p0 ≤ p0 = r20 . The case where s = |ω|/p0 is then handled by interpolation. If s0 ∈ νN, we fix σ with s < σ < s0 and σ ∈ / νN. Then we obtain the assertion from what has just s0 /ν σ/ν been shown and Wp0 ,→ Bp0 . (2) Claims (ii) and (iii) follow analogously.

¥

The Non-Reflexive Case It is the purpose of the following considerations to prove that BUC s0 /ν is a pointwise multiplier space for negative order Besov spaces even if E1 and E2 are not reflexive. For this we need some preparation. We use the notations and conventions of Section VI.3. First we study the case X = Rd . In this situation we employ the following stipulations: if F(Rd , E) is a linear subspace of D0 (Rd , E), then we denote it simply by F(E), and F := F(C). ¡We fix a ¢ν-quasinorm Q on Rd , set Ω := [Q < 1]. Let ψ be Ω-adapted. Then (Ωk )(ψk ) is the ν-dyadic partition of unity ine where duced by (Ω, ψ). Thus, see Example VI.3.6.1, ψ0 := ψ, ψj := σ2−j ψ, ψe := ψ − σ2 ψ, Ω0 = [Q < 2], and Ωk = [2k−1 < Q < 2k+1 ] for k ≥ 1. Since F −1 ψk ∈ S, it follows (see (III.4.2.14) or Proposition 1.2.7 of the Appendix) that ψk (D)u = F −1 ψk ∗ u ∈ OM (E) ,

u ∈ S 0 (E) .

(6.4.5)

Hence S n u :=

n X

ψk (D)u ∈ OM (E) ,

n∈N,

u ∈ S 0 (E) .

(6.4.6)

k=0

We get from (6.4.6) and Theorem 1.6.4 of the Appendix that β(S n u0 , S n u1 ) ∈ OM (E2 ) ,

uj ∈ S 0 (Ej ) ,

j = 0, 1 .

(6.4.7)

Suppose s > −t > 0. It is the purpose of the following considerations to show that π(u0 , u1 ) := lim β(S n u0 , S n u1 ) n→∞

t/ν

s/ν

t/ν

exists in Bq,r (E2 ) for (u0 , u1 ) ∈ B∞ (E0 ) × Bq,r (E1 ), that π is a multiplication, s0 /ν t/ν and π(u0 , u1 ) = β(u0 , u1 ) if u1 ∈ Bq,r (E1 ) ,→ Bq,r (E1 ) for some s0 > 0. In this t/ν s/ν t/ν sense, π is the point-wise product in Bq,r (E2 ) of (u0 , u1 ) ∈ B∞ (E0 ) × Bq,r (E1 ).

238

VII Function Spaces

For abbreviation, Sj := ψj (D). Then we set, for n ≥ 3, π1n (u0 , u1 ) :=

n X

β(Sj u0 , S j−3 u1 ) ,

j=3

π2n (u0 , u1 ) :=

n X 2 X

β(Sj u0 , Sj+i u1 ) − β(Sn−1 u0 , Sn+1 u1 )

j=0 i=−2

π3n (u0 , u1 ) :=

n X

¡ ¢ − β Sn u0 , (Sn+1 + Sn+2 )u1 ,

β(S j−3 u0 , Sj u1 ) ,

j=3

where Sj := 0 for j < 0. Then π n :=

n X

β(Sj u0 , Sj u1 ) = π1n + π2n + π3n .

(6.4.8)

j=0

The importance of this decomposition stems from the following support properties. 6.4.3 Lemma Let (u0 , u1 ) ∈ S 0 (E0 ) × S 0 (E1 ). (i) If j ≥ 3, then F β(Sj u0 , S j−3 u1 ) and F β(S j−3 u0 , Sj u1 ) are supported in [2j−2 < Q < 2j+2 ]. ¡ ¡ ¢¢ Pj+2 (ii) If j ∈ N, then supp F β Sj u0 , k=j−2 Sk u1 ⊂ [Q < 2j+4 ]. Proof (1) Using the notations of the Appendix, we set v0 q v1 := β(v0 , v1 ) for (v0 , v1 ) ∈ S 0 (E0 ) × OM (E1 ). By replacing β by |β|−1 β, we can assume that q is a multiplication in the sense of the Appendix, that is, it has norm at most one. Hence the convolution theorem for vector-valued distributions (Theorem 1.9.10 of the Appendix) guarantees (v0 , v1 ) ∈ S 0 (E0 ) × S(E1 ) ,

where ∗q denotes the convolution induced by β. Theorem 1.6.4 of the Appendix and the continuity of b b

b

β(v0 , v1 ) = β(v 0 , v 1 ) ,

b

(v0 ∗ q v1 )b = vb0 q vb1 ,

on S and S 0 imply

(v0 , v1 ) ∈ S 0 (E0 ) × S(E1 ) .

b

Recall F −1 = (2π)−d F (see (III.4.2.2)). From this and (6.4.9) we deduce F −1 (v0 ∗ q v1 ) = (2π)d F −1 v0 q F −1 v1 and thus, setting wj := F −1 vj , F β(w0 , w1 ) = (2π)−d w c0 ∗ q w b1 ,

(6.4.9)

(w0 , w1 ) ∈ S 0 (E0 ) × S(E1 ) .

VII.6 Point-Wise Multiplications

239

Now we get from Remark 1.9.6(f) of the Appendix that ¡ ¢ supp F β(w0 , w1 ) ⊂ supp(w b0 ) + supp(w b1 ) ,

(6.4.10)

whenever w b0 and w b1 are compactly supported. (2) Suppose w0 ∈ OM (E0 ) and w b0 is compactly supported. Let (w1,j ) be a sequence in S(E1 ) converging in S 0 (E1 ) towards w1 such that supp(w b1,j ) ⊂ K for some K b Rd . Then, by (6.4.10), ¡ ¢ supp F β(w0 , w1,j ) ⊂ supp(w b0 ) + K , j∈N. (6.4.11) It follows from Theorem 1.6.4 of the Appendix that β(w0 , ·) is a continuous linear map from S 0 (E1 ) into S 0 (E2 ). Thus F β(w0 , ·) has this property also. From this and (6.4.11) we deduce that ¡ ¢ supp F β(w0 , w1 ) ⊂ supp(w b0 ) + K . (6.4.12) ¡ ¢ Assume u1 ∈ S 0 (E1 ). Since Sk = ψk (D) ∈ L S 0 (E1 ) by Lemma VI.3.6.3(ii), and S(E1 ) is dense in S 0 (E1 ), we can find a sequence (u1,j ) in S(E1 ) such that w1,j := Sk u1,j → Sk u1 in S 0 (E1 ). Note that supp(w b1,j ) = supp(ψk u b1,j ) ⊂ Ωk , Hence it follows from (6.4.12) that ¡ ¢ supp F β(w0 , Sk u1 ) ⊂ supp(w b0 ) + Ωk ,

j∈N.

k∈N,

u1 ∈ S 0 (E1 ) ,

(6.4.13)

provided w0 belongs to OM (E0 ) and w b0 is is compactly supported. Similarly, given w1 ∈ OM (E1 ) with w b1 being compactly supported, ¡ ¢ supp F β(Sj u0 , w1 ) ⊂ Ωj + supp(w b1 ) (6.4.14) for j ∈ N and u0 ∈ S 0 (E0 ).

Pm mu = [ (3) Let m ∈ N. Then S b for u ∈ S 0 (E). Hence (see (VI.3.6.1) k=0 ψk u and (VI.3.6.3)) ¡ ¢ m u ⊂ 2m+1 q Ω = [Q < 2m+1 ] . [ supp S Consequently, if k ≥ 3, we obtain from (6.4.13) ¡ ¢ supp F β(S k−3 u0 , Sk u1 ) ⊂ [Q < 2k−2 ] + [2k−1 < Q < 2k+1 ] . Thus |Q(ξ) − Q(η)| ≤ Q(ξ + η) ≤ Q(ξ) + Q(η) implies ¡ ¢ supp F β(S k−3 u0 , Sk u1 ) ⊂ [2k−1 − 2k−2 < Q < 2k+1 + 2k−2 ] ⊂ [2k−2 < Q < 2k+2 ] for (u0 , u1 ) ∈ S 0 (E0 ) × S 0 (E1 ), due to (6.4.7). This proves one half of (i). The second half follows analogously.

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VII Function Spaces

Pj+2 (4) Since k=j−2 ψk equals σ2−j−2 ψ − σ2−j+1 ψ if j ≥ 3, respectively σ2−j−2 ψ if 0 ≤ j ≤ 2, we find that its support is contained in [2j−3 < Q < 2j+3 ] if j ≥ 3, and in [Q < 2j+3 ] otherwise. We deduce from [2j−1 < Q < 2j+1 ] + [2j−3 < Q < 2j+3 ] ⊂ [Q < 2j+4 ] that, in either case, j+2 ³ ´ X supp F β(Sj u0 , Sk u1 ) ⊂ [Q < 2j+4 ] k=j−2

for (u0 , u1 ) ∈ S 0 (E0 ) × S 0 (E1 ). This proves (ii).

¥

Henceforth, setting π0n := π n , we define πi (u0 , u1 ) := lim πin (u0 , u1 ) , n→∞

i = 0, . . . , 3 ,

for those (u0 , u1 ) ∈ S 0 (E0 ) × S 0 (E1 ) for which this limit exists in S 0 (E2 ). 6.4.4 Lemma Assume s0 > −s1 > 0. Then π is a continuous bilinear map from s /ν s1 /ν s1 /ν B∞0 (E0 ) × Bq,r (E1 ) into Bq,r (E2 ). Its norm is bounded by c |β|. Proof (1) It suffices to prove the assertion for r = 1. Indeed, then we apply this fact with si replaced by σi , where s0 > −σ0 > −s1 > −σ1 > 0. Thus, given s /ν u0 ∈ B∞0 (E0 ), ¡ σi /ν ¢ σi /ν π(u0 , ·) ∈ L Bq,1 (E1 ), Bq,1 (E2 ) , kπ(u0 , ·)k ≤ c |β| ku0 kB s0 /ν ∞

for i = 0, 1. Hence interpolation with the real interpolation functor (·, ·)θ,r , where θ = (s1 − σ0 )/(σ1 − σ0 ), gives ¡ s1 /ν ¢ s1 /ν π(u0 , ·) ∈ L Bq,r (E1 ), Bq,r (E2 ) , kπ(u0 , ·)k ≤ c |β| ku0 kB s0 /ν , ∞

s /ν B∞0 (E0 ),

due to Theorem 2.7.1(i). This being true for all u0 ∈ the assertion follows. (2) It suffices to prove the claim with π replaced by πi , 1 ≤ i ≤ 3. Clearly, this is a consequence of (6.4.8). (3) We infer from supp(ψk ) ⊂ Ωk and Lemma 6.4.3(i) that ψk (D)β(Sj u0 , S j−3 u1 ) = 0 ,

|j − k| ≥ 3 .

Hence ψk (D)π1n (u0 , u1 ) =

k+2 X

¡ ¢ ψk (D)β ψj (D)u0 , S j−3 u1 ,

j=k−2

and ψk (D)π1n (u0 , u1 ) = 0 for k > n + 2.

k ≤n+2 ,

VII.6 Point-Wise Multiplications

241

Let 3 ≤ m < n. Then we find ψk (D)(π1n − π1m )(u0 , u1 ) =

k+2 X

¡ ¢ ψk (D)β ψj (D)u0 , S j−3 u1

j=k−2

if m − 1 ≤ k ≤ n + 2, and ψk (D)(π1n − π1m )(u0 , u1 ) = 0 otherwise. Thus we infer from (2.1.1), since s1 < 0 and s0 + s1 > 0, that ∞ X

2ks1 kψk (D)(π1n − π1m )(u0 , u1 )kq

k=0 n+2 X

≤ c |β|

2 X

X

k+i−3 X

³ X

kψj (D)u1 kq

j=0

2ks1 2−(k+i)s0 2−(k+i−3)s1 ku0 kB s0 /ν ku1 kB s1 /ν ∞

k≥m−1 i=−2

≤ c |β|

kψk+i (D)u0 k∞

i=−2

k=m−1

≤ c |β|

2 X

2ks1

q,1

´

2ks1 ku0 kB s0 /ν ku1 kB s1 /ν . ∞

k≥m−1

q,1

s /ν

1 From this we deduce that (π1n ) is a Cauchy sequence in Bq,1 (E2 ). Thus it converges therein towards π1 (u0 , u1 ). Obvious modifications of the preceding estimates yield

kπ1 (u0 , u1 )kB s1 /ν = lim kπ1n (u0 , u1 )kB s1 /ν ≤ c |β| ku0 kB s0 /ν ku1 kB s1 /ν . q,1

n→∞

∞

q,1

q1 ,1

(4) Analogously to the above considerations we find, for τ > 0, ∞ X

2ks1 kψk (D)(π3n − π3m )(u0 , u1 )kq

k=0

≤ c |β|

X

kψj (D)u0 k∞ kψk+i (D)u1 kq

i=−2 j=0

k≥m−1

≤ c |β|

2 k+i−3 X X

2ks1

³ X

2ks1 kψk (D)u1 kq

k≥m−1

= c |β|

X ³

2ks1 kψk (D)u1 kq

∞ ´X

´

2jτ kψk (D)u0 k∞

j=0

ku0 kB τ /ν . ∞,1

k≥m−1

s /ν

1 (E1 ), we Since the sum over k ≥ m − 1 converges to 0 for m → ∞ if u1 ∈ Bq,1

s /ν

1 see that π3 (u0 , u1 ) exists in Bq,1 (E2 ) and satisfies

kπ3 (u0 , u1 )kB s1 /ν ≤ c(τ ) |β| ku0 kB τ /ν ku1 kB s1 /ν q,1

τ /ν

s /ν

1 for (u0 , u1 ) ∈ B∞,1 (E0 ) × Bq,1 (E1 ).

∞,1

q,1

(6.4.15)

242

VII Function Spaces

Finally, the assertion for π3 is obtained by choosing τ such that s0 > τ > −s1 s /ν τ /ν and recalling B∞0 (E0 ) ,→ B∞,1 (E0 ). (5) The definition of π2n and Lemma 6.4.3(ii) imply that the support of F π2n is contained in [Q < 2n+4 ]. Hence ψk (D)π2n = 0 for k ≥ n + 5. Thus, given n > m, ψk (D)(π2n − π2m )(u0 , u1 ) n 2 ³ X X = ψk (D) β(Sj u0 , Sj+i u1 ) j=m+1 i=−2

¡ ¢ − β(Sn−1 u0 , Sn+1 u1 ) − β Sn u0 , (Sn+1 + Sn+2 )u1 ¡ ¢´ + β(Sm−1 u0 , Sm+1 u1 ) + β Sm u0 , (Sm+1 + Sm+2 )u1 vanishes if k ≥ n + 5. From this and (2.1.1) we infer that ∞ X

2ks1 kψk (D)(π2n − π2m )(u0 , u1 )kq

k=0

≤c

n+4 X

2ks1

kβ(Sj u0 , Sj+i u1 )kq

(6.4.16)

j=m i=−2

k=0

≤ c |β|

n X 2 X

∞ ³X k=0

2ks1

∞ X 2 ´X j=m i=−2

−m(s0 +s1 )

≤ c |β| 2

2−(js0 +(j+i)s1 ) k2js0 ψj (D)u0 k∞ k2(j+i)s1 ψj+i (D)u1 kq

°¡ js ¢° ° k2 0 ψj (D)u0 k∞ °

`∞

°¡ js ¢° ° k2 1 ψj (D)u1 kq °

`1

= c |β| 2−m(s0 +s1 ) ku0 kB s0 /ν ku1 kB s1 /ν . ∞ ¡ q,1 ¢ s1 /ν Since s0 + s1 > 0, we see that π2n (u0 , u1 ) is a Cauchy sequence in Bq,1 (E2 ). From this and estimate (6.4.16) we deduce that π2 is a continuous bilinear map s /ν s1 /ν s1 /ν from B∞0 (E0 ) × Bq,1 (E1 ) into Bq,1 (E2 ), whose norm is bounded by c |β|. This proves the lemma. ¥ The next lemma shows that π is an extension of the point-wise multiplication operator mβ studied in the preceding subsection. s /ν

6.4.5 Lemma Suppose s0 > s > 0. Then π(u0 , u1 ) = mβ (u0 , u1 ) for u0 ∈ B∞0 (E0 ) s/ν and u1 ∈ Bq,r (E1 ). t/ν

Proof (1) Assume t ∈ R and u ∈ Bp,r (E), where 1 ≤ p ≤ ∞. We claim that t/ν S n u → u in Bp,r (E). To see this, let n > m > 0. Then ψk (D)(S n − S m )u =

n X j=m+1

ψk (D)ψj (D)u =

1 X i=−1

ψk (D)ψk+i (D)u

VII.6 Point-Wise Multiplications

243

for m ≤ k ≤ n + 1, and ψk (D)(S n − S m )u = 0 otherwise. Hence, using once more (2.1.1), ∞ X X 2kt kψk (D)(S n − S m )ukp ≤ c 2kt kψk (D)ukp . k≥0

k=m−1 t/ν

This shows that (S n u) is a Cauchy sequence in Bp,1 (E). Therefore S n u converges t/ν

t/ν

t/ν

t/ν

in Bp,1 (E), consequently, due to Bp,1 (E) ,→ Bp,r (E) ,→ S 0 (E), in Bp,r (E) and in S 0 (E) towards some v. From Lemma VI.3.6.2 we know that S n u → u in S 0 (E). Hence v = u. This proves the claim. (2) Let (u0 , u1 ) ∈ S 0 (E0 ) × S 0 (E1 ). It is obvious from (6.4.7) that π n (u0 , u1 ) = β(S n u0 , S n u1 ) = mβ (S n u0 , S n u1 ) , s /ν

n∈N.

(6.4.17)

s/ν

Suppose u0 ∈ B∞0 (E0 ) and u1 ∈ Bq,r (E1 ). Step (1) and Theorem 6.3.3(ii) imply s/ν mβ (S n u0 , S n u1 ) → mβ (u0 , u1 ) in Bq,r (E2 ) . s/ν

−s/ν

−s/ν

Hence, by Bq,r (E2 ) ,→ Bq,r (E2 ), in Bq,r that π(u0 , u1 ) = mβ (u0 , u1 ). ¥

(E2 ). Now we deduce from (6.4.17)

s /ν

Let K be closed, 0 < s < s0 , and r < ∞. We set, for u0 ∈ B∞0 (K, E0 ) and −s/ν u1 ∈ Bq,r (K, E1 ), πK (u0 , u1 ) := RK π(EK u0 , EK u1 ) . It follows from Lemma 6.4.4 and Theorem 2.8.2 that πK is a continuous bilins /ν −s/ν −s/ν ear map from B∞0 (K, E0 ) × Bq,r (K, E1 ) into Bq,r (K, E2 ), whose norm is bounded by c |β|. s /ν

Assume u1 ∈ S(K, E1 ). Then (EK u0 , EK u1 ) ∈ B∞0 (E0 ) × S(E1 ) by Theorems 2.8.2 and VI.1.2.3. Hence, by Lemma 6.4.5, πK (u0 , u1 ) = RK mβ (EK u0 , EK u1 ) = mβ,K (u0 , u1 ) , where, for clarity, we write mβ,K for the mβ of Subsection 6.3 if X = K. Indeed, this is obvious by the point-wise definition of mβ,K and the fact that RK is the −s/ν point-wise restriction. Since S(K, E1 ) is dense in Bq,r (K, E1 ), this implies that, s0 /ν given u0 ∈ B∞ (K, E0 ), ¡ −s/ν ¢ −s/ν πK (u0 , ·) ∈ L Bq,r (K, E1 ), Bq,r (K, E2 ) is the unique continuous extension of mβ,K (u0 , ·). Thus it is feasible to write again mβ,K for πK . Moreover, as before, we drop the index K and write simply mβ , even if X = K. Thus, if K = K, r < ∞, and s0 > s > 0, ¡ s0 /ν ¢ −s/ν −s/ν mβ ∈ L B ∞ (K, E0 ), Bq,r (K, E1 ); Bq,r (K, E2 ) . (6.4.18) From this we get, by obvious real interpolation with the functor (·, ·)θ,∞ , that we can admit r = ∞ also. Furthermore, mβ is universal, that is, independent of the particular choice of the parameters s0 , s, q, and r.

244

VII Function Spaces

Now we can easily prove the following general point-wise multiplier result. 6.4.6 Theorem Suppose K is a closed corner in Rd , X ∈ {Rd , K}, and 1≤q s > −s0 .

Then s0 /ν s/ν s/ν m β : B∞ (X, E0 ) × Bq,r (X, E1 ) ,→ Bq,r (X, E2 )

(6.4.19)

s0 /ν s/ν m β : B∞ (X, E0 ) × bs/ν q,∞ (X, E1 ) ,→ bq,∞ (X, E2 ) .

(6.4.20)

and

Proof Due to Theorem 6.3.3 and (6.4.18) it suffices to prove (6.4.19) for s = 0 and assertion (6.4.20). The first case is settled by interpolation with the func0 tor (·, ·)θ,∞ , and the second one by employing (6.4.19) and (·, ·)θ,∞ . ¥

6.5

Multiplications in Bessel Potential Spaces

Now it is not difficult to prove that Besov–H¨older spaces are point-wise multipliers for Bessel potential spaces as well. 6.5.1 Theorem Suppose K is a corner in Rd and X ∈ {Rd , K}. Let E1 and E2 be ν-admissible and 1 < p < ∞. Then s0 /ν B∞ (X, E0 ) × Hps/ν (X, E1 ) ,→ Hps/ν (X, E2 )

for s0 > s > −s0 . Proof (1) Suppose s0 > t > s > 0. We can assume that ε := t − s < ν. We fix m in νN with m > s and set θ := s/m. It follows from Theorems 4.3.2 and 4.5.1 that ¤ . £ Hps/ν (X, Ej ) = Lp (X, Ej ), Wpm/ν (X, Ej ) θ (6.5.1) for j = 1, 2. Theorems 2.7.1(v) and 2.8.3 guarantee, due to (1 − θ)ε + θ(m + ε) = s + ε = t , that ¤ . £ ε/ν (m+ε)/ν bt/ν (X, E0 ) θ . ∞ (X, E0 ) = b∞ (X, E0 ), b∞ (j+ε)/ν

(6.5.2)

It is a consequence of b∞ (X, E0 ) ,→ BUC j/ν (X, E0 ) for j ∈ {0, m} and Theorem 6.2.1 that ¡ ¢ mβ ∈ L b(j+ε)/ν (X, E), Wpj/ν (X, E1 ); Wpj/ν (X, E2 ) (6.5.3) ∞

VII.6 Point-Wise Multiplications

245

for j ∈ {0, m}. Now we use properties of A.P. Calder´on’s second complex interpoθ lation functor [·, ·] , 0 < θ < 1. Namely, suppose (F0 , F1 ) is an interpolation pair of Banach spaces. Then [F0 , F1 ]θ ,→ [F0 , F1 ]θ

(6.5.4)

and [F0 , F1 ]θ = [F0 , F1 ]θ

if

F0 is reflexive

(6.5.5)

(see [Cal64, Sections 6–9] or [BeL76, Theorems 4.1.4 and 4.3.1]). Hence, by (6.5.1) and (6.5.5), £ ¤θ Hps/ν (X, Ej ) = Lp (X, Ej ), Wpm (X, Ej ) , j = 1, 2 . (6.5.6) It follows from (6.5.2) and (6.5.4) that £ ε/ν ¤θ (m+ε)/ν bt/ν (X, E0 ) =: X θ . ∞ (X, E0 ) ,→ b∞ (X, E0 ), b∞

(6.5.7)

Using (6.5.6) and Calder´on’s multilinear interpolation theorem (cf. [Cal64, Section 10.1] or [BeL76, Theorems 4.4.1]), we obtain from (6.5.3) that ¡ ¢ mβ ∈ L X θ , Hps/ν (X, E1 ); Hps/ν (X, E0 ) . s /ν

Thus (6.5.7) and B∞0

t/ν

,→ b∞ imply

¡ s0 /ν ¢ mβ ∈ L B ∞ (X, E0 ), Hps/ν (X, E1 ); Hps/ν (X, E0 ) .

(2) Suppose −s0 < s < 0. Then the claim follows from (1) by duality, see the proof of Theorem 6.2.6, due to that theorem and Theorem 4.4.2. (3) The case s = 0 is already contained in (6.5.3). ¥

6.6

Space-Dependent Bilinear Maps

For certain applications, in particular in situations involving Riemannian metrics on manifolds, it is necessary to consider point-wise multiplications which depend on the underlying space variables also. Such situations are considered in the present subsection. More precisely, suppose ¡ ¢ β ∈ BC X, L(E0 , E1 ; E2 ) (6.6.1) and write mβ for the point-wise extension, BC(X, E0 ) × L1,loc (X, E1 ) → L1,loc (X, E2 ) ,

(u0 , u1 ) 7→ mβ (u0 , u1 ) ,

now defined by

¡ ¢ mβ (u0 , u1 )(x) := β(x) u0 (x), u1 (x) ,

a.a. x ∈ X .

Observe that the point-wise multiplier theorems proved so far are universal in the sense that they depend neither on the particular structure of β nor on the Banach

246

VII Function Spaces

spaces E0 , E1 , and E2 , except for occasional reflexivity assumptions. In other words, suppose Fj (X, Ej ) ,→ S 0 (X, Ej ) and mβ : F0 (X, E0 ) × F1 (X, E1 ) ,→ F2 (X, E2 ) . Then, given Banach spaces F0 , F1 , F2 , and γ ∈ L(F0 , F1 ; F2 ), it follows mγ : F0 (X, F0 ) × F1 (X, F1 ) ,→ F2 (X, F2 ) . To express this fact we use the somewhat unprecise but intuitive notation F0 × F1 ,→ F2 . Based on this observation, we can employ a simple argument to carry over many of the preceding results to space-dependent multiplications. 6.6.1 Lemma Suppose Fj = Fj (X, Ej ) ,→ S 0 (X, Ej ), j = 0, 1, 2, are such that F0 × F0 ,→ F0 Then

Proof

and

F0 × F1 ,→ F2 .

(6.6.2)

¡ ¢ mβ : L F0 (X, E0 ), F1 (X, E1 ); F2 (X, E2 ) . Consider the multiplication β0 : L(E0 , E1 ; E2 ) × E0 → L(E1 , E2 ) ,

(β, e0 ) 7→ β(e0 , ·)

and let m0 be its point-wise extension. Then the first part of (6.6.2) implies ¡ ¡ ¢ ¢ m0 ∈ L F0 X, L(E0 , E1 ; E2 ) , F0 (X, E0 ); F0 (X, E2 ) . (6.6.3) Next we introduce the multiplication L(E1 , E2 ) × E1 → E2 ,

(A, e) 7→ Ae

and denote its point-wise extension by m1 . Then, by the second part of (6.6.2), ¡ ¡ ¢ ¢ m1 ∈ L F0 X, L(E1 , E0 ) , F1 (X, E1 ); F2 (X, E2 ) . (6.6.4) Note that

¡ ¢ mβ (u0 , u1 ) = m1 m0 (β, u0 ), u1 ,

This proves the lemma.

(u0 , u1 ) ∈ E0X × E1X .

¥

6.6.2 Examples Assume K is closed. (a) Let one of the following conditions be satisfied: (α) m ∈ νN, F0 ∈ {BC m/ν , BUC m/ν }. (β) s0 > 0, F0 ∈ {BUC s0 /ν , bucs0 /ν }. s /ν

(γ) s0 > |ω|/p0 > 0, F0 ∈ {Wp00 Then F0 × F0 ,→ F0 .

s /ν

, Bp00 ,r }.

VII.6 Point-Wise Multiplications

247

Proof Theorem 6.2.1 implies (α). From (α) and Theorem 6.3.3 we get (β). Lastly, (γ) follows from Corollary 6.3.4(i). ¥ (b) Theorems 6.2.1, 6.2.3, 6.2.6, 6.3.3(i)–(iii), Corollary 6.3.4, Theorem 6.4.1(i) and parts (i) and (ii) of its corollary, and Theorems 6.4.6 and 6.5.1 apply to spacedependent multiplications as well. Proof

6.7

Clear by (a).

¥

Notes

A detailed study of point-wise multiplication in scalar-valued Besov (and Triebel– Lizorkin) spaces on Rd can be found in Th. Runst and W. Sickel [RS96] for isotropic spaces, and in J. Johnson [Joh95] for the anisotropic case. It follows from those publications that the multiplier theorems of this section are optimal, except for a few limit cases. The proofs given here in the case of positive order spaces and –– in the reflexive setting –– for negative order spaces are essentially the same as the ones in our paper [Ama91]. There we formulated the results for general Banachspace-valued functions. However, the proofs were restricted to finite-dimensional settings. This is now rectified by the reasoning in this section. In fact, the theorems presented here are even sharper than the ones in our earlier paper. It should be noted that [Ama91] is the first paper in which the optimal conditions for the validity of the point-wise multiplier theorems for positive order Sobolev and Besov spaces are given. In earlier work (see [Han85], [Val85], [Val88], [Zol78]) additional restrictions had been imposed. The reader should consult [RS96] for references to earlier work as well as the book by V.G. Maz’ya and T.O. Shaposhnikova [MaS85]. In [Ama91] and [RS96] there are also considered multiplications with more than two factors. It is clear that this could be done here also. The proofs of Subsection 6.4 in the non-reflexive setting follow the work of W. Sickel and J. Johnson inasmuch as we use paramultiplication (also see [Yam86]). For the proof of the support properties of Lemma 6.4.3 we have to employ the full strength of the theory of vector-valued distributions, namely the Schwartz kernel theorem, presented in the Appendix. Given this lemma, our proof of Lemma 6.4.5 is simpler and more direct than the demonstrations of the other authors since we do not rely on convergence theorems of M. Yamazaki. In [Ama01] we have given a (somewhat sketchy) proof of Theorem 6.4.6 by following closely the arguments of J. Johnson [Joh95] and Th. Runst and W. Sickel [RS96]. M. Meyries and M. Veraar [MeV15] extend many of the results known for point-wise multipliers on scalar-valued Bessel potential spaces to the vector-valued setting. More precisely, they consider weighted Bessel potential spaces s Hp,w (Rd , E) := J −s Lp (Rd , w, E) ,

248

VII Function Spaces

where w is a power weight acting on one variable only, and E is a UMD space. In this setting, they obtain the analogue of Theorem 6.5.1. ¡ dBy R-boundedness ¢ s techniques, ‘irregular’ multiplier spaces of the form Bq,∞ R , w, L(E1 , E2 ) are considered also. The idea of using multilinear interpolation in the proof of Theorem 6.5.1 is taken from [MeV15]. M. K¨ohne and J. Saal [K¨oS17] present a study of point-wise multiplication in anisotropic vector-valued Sobolev, Besov, and Bessel potential spaces. They consider the case of possibly more than two factors and positive order spaces. These authors base their proofs on the results of [Ama09]. Thus they have to assume that all occurring target spaces are ν-admissible. We refer to the book [RS96] for rather detailed references to earlier work on point-wise multiplication in scalar Besov (and Triebel–Lizorkin) spaces.

VII.7 Compactness

7

249

Compactness

This section is concerned with compact embeddings of Sobolev, Besov, and Bessel potential spaces. Such results are fundamental for many investigations of qualitative properties of evolution and nonlinear partial differential equations. The main result of the first subsection is an extension to the vector-valued setting of the sufficiency part of the Fr´echet–Kolmogorov theorem characterizing compact subsets of Lebesgue spaces. This and the Arzela–Ascoli theorem provide the basis for compact embedding theorems involving Besov and Bessel potential spaces. They are derived in Subsection 7.2. In the rest of this section we consider vector-valued function spaces on compact intervals. In Subsection 7.3 it is shown how these spaces relate to the general theory established in the earlier sections. First we define E-valued Sobolev– Slobodeckii spaces on compact intervals by means of the classical seminorms. Then we prove a retraction-coretraction theorem. On its basis, we can transpose embedding and interpolation theorems, proved earlier for function spaces on corners, to the present setting. Vector-valued function spaces on intervals play a fundamental role in the theory of evolution equations, which is seen, for example, by the theory developed in Volume I. Here they are introduced to establish, in the last subsection, generalizations of compact embedding theorems of Aubin-Lions type. Throughout this section, • K is a closed corner in Rd . • X ∈ {Rd , K}. • K is a compact subset of X with nonempty interior. Given an LCS F(X, E) ,→ D0 (X, E), we set © ª FK (X, E) := u ∈ F(X, E) ; supp(u) ⊂ K .

7.1

(7.0.1)

(7.0.2)

Equicontinuity

In the scalar-valued isotropic case, the fundamental Rellich–Kondrachov compact embedding theorem is based on the Arzela–Ascoli and the Fr´echet–Kolmogorov theorem. Motivated by this, we give in this subsection sufficient conditions guaranteeing compactness in the spaces BUC(X, E) and Lq (X, E). Compact Sets in BUC Our first result is a simple application of the Arzela–Ascoli theorem. Recall that b and ,− ,→ stand for compact embeddings.

250

VII Function Spaces

7.1.1 Theorem Let (7.0.1) be satisfied, E1 ,− ,→ E0 , and K ⊂ BUCK (X, E1 ). Assume: (i) K is bounded in BUC(X, E1 ). (ii) K is equicontinuous in C(X, E0 ), that is, for each x ∈ X lim |u(x + y) − u(x)|E0 = 0 ,

y→0 y∈X

uniformly with respect to u ∈ K. Then K b BUC(X, E0 ). © ª Proof Since E1 ,− ,→ E0 , it follows from (i) that K(x) := u(x) ; u ∈ K is for each x ∈ K relatively compact in E0 . Due to (ii) and Lemma 1.1.1, the Arzela– Ascoli theorem (e.g., [DuS57, Theorem IV.6.7]) implies the assertion. ¥ Compact Sets in Lq The next theorem is a vector-valued variant of the sufficiency part of the Fr´echet– Kolmogorov theorem (e.g., [DuS57, Theorem IV.8.2]). 7.1.2 Theorem Let (7.0.1) be satisfied, E1 ,− ,→ E0 , and 1 ≤ q < ∞. Suppose: (i) K is bounded in Lq (X, E1 ). (ii) K is equicontinuous in Lq (X, E0 ), that is, Z lim |u(x + y) − u(x)|qE0 dx = 0 , y→0 y∈X

X

uniformly with respect to u ∈ K. (iii) For each ε > 0,

Z lim

R→∞

X∩[|x|≥R]

|u(x)|qE0 dx = 0 ,

uniformly for u ∈ K. Then K b Lq (X, E0 ). Proof

(1) We set Bt := { x ∈ X ; |x| ≤ t }, (Ty u)(x) := u(x + y), and Z 1 Mt u := Ty u dy , t>0. |Bt | Bt

Then, by H¨older’s inequality (if q > 1) and the Fubini–Tonelli theorem, we get from Z ¡ ¢ 1 (Mt u − u)(x) = u(x + y) − u(x) dy , a.a. x ∈ X , |Bt | Bt

VII.7 Compactness

251

the estimate ³Z ³ 1 Z ´q ´1/q |u(x + y) − u(x)|E0 dy dx X |Bt | Bt Z Z ´1/q 0 1 ³ ≤ |u(x + y) − u(x)|qE0 dy |Bt |q/q dx |Bt | X Bt ³ 1 Z Z ´1/q = |u(x + y) − u(x)|qE0 dx dy |Bt | Bt X

kMt u − ukLq (X,E0 ) ≤

≤ sup kTy u − ukLq (X,E0 ) . |y|≤t y∈X

Let ε > 0. Then this estimate and (ii) imply the existence of τ > 0 such that kMτ u − ukLq (X,E0 ) ≤ ε/6 ,

u∈K.

(7.1.1)

(2) Similarly as above, Z ³ 1 Z ´1/q 1 |u(x + y)|E1 dy ≤ |u(x + y)|qE1 dy |Bτ | Bτ |Bτ | Bτ Z ³ ´ 1/q 1 1 ≤ |u(x + y)|qE1 dy ≤ kukLq (X,E1 ) . |Bτ |1/q X |Bτ |1/q

|Mτ u(x)|E1 ≤

By (i), |Mτ u(x)|E1 ≤ c ,

a.a. x ∈ X ,

u∈K.

(7.1.2)

By the same arguments we find Z 1 |u(x + y + z) − u(x + z)|E0 dz |Bτ | Bτ ³ 1 Z ´1/q ≤ |u(x + y + z) − u(x + z)|qE0 dz |Bτ | Bτ Z ´1/q 1 ³ ≤ |u(x + y) − u(x)|qE0 dx . 1/q |Bτ | X

|Mτ u(x + y) − Mτ u(x)|E0 ≤

Thus we infer from E1 ,→ E0 , (7.1.2), and (ii) that { Mτ u ; u ∈ K } is equicontinuous in BUC(X, E0 ). (3) Using the arguments of step (1) once more, we find Z X\BR

|Mτ u|qE0 dx ≤

1 |Bτ |

Z

Z Bτ

Z X\BR

|u(x + y)|qE0 dx dy ≤

X\BR

|u(x)|qE0 dx .

252

VII Function Spaces

Hence it follows from (iii) that we can fix R > 0 such that, setting K := BR , ³Z X\K

|Mτ u|qE0 dx

´1/q

≤ ε/6 ,

u∈K.

(4) Since K is compact, we deduce from step (2) and Theorem 7.1.1 that { Mτ u | K ; u ∈ K } is relatively compact in BUC(K, E0 ), hence totally bounded. Let B(v, r) be the open ball in BUC(K, E0 ) with center at v and radius r, and Bq (v, r) the one in Lq (K, E0 ). There are v0 , . . . , vn ∈ { Mτ u ; u ∈ K } such that n [ ¡ ¢ { Mτ u | K ; u ∈ K } ⊂ B vj | K, ε/3 |K| . j=0

Since

³Z kMτ u − vj kLq (K,E0 ) =

K

|Mτ u(x) − vj (x)|qE0 dx

´1/q

for Mτ u ∈ B(vj , ε/3 |K|), we see that { Mτ u | K ; u ∈ K } ⊂

≤ ε/3

Sn j=0

Bq (vj , ε/3).

(5) Assume u ∈ Lq (X, E0 ). By step (4) we can find uj ∈ K such that Z K

|Mτ u − Mτ uj |qE0 dx ≤ (ε/3)q .

(7.1.3)

Note that (7.1.1) implies ku − uj kLq (X,E0 ) ≤ ku − Mτ ukLq (X,E0 ) + kMτ u − Mτ uj kLq (X,E0 ) + kMτ uj − uj kLq (X,E0 )

(7.1.4)

≤ ε/3 + kMτ u − Mτ uj kLq (X,E0 ) . From (7.1.3) and step (3) we infer kMτ u − Mτ uj kLq (X,E0 )

³Z

≤ kMτ u − Mτ uj kLq (K,E0 ) + ³Z ≤ kMτ u − Mτ uj kLq (K,E0 ) +

X\K

X\K

|Mτ u − Mτ uj |qE0 dx |Mτ u|qE0 dx

´1/q

´1/q

³Z + X\K

|Mτ uj |qE0 dx

´1/q

≤ 2ε/3 . Thus it follows from (7.1.4) that ku − uj kLq (X,E0 ) ≤ ε. This shows that K is totally bounded in Lq (X, E0 ), hence relatively compact. ¥

VII.7 Compactness

7.2

253

Compact Embeddings

For preparation, we include some simple observations. 7.2.1 Lemma L1,loc,K (X, E) is a closed linear subspace of L1,loc (X, E). Proof Let (uj ) be a sequence in L1,loc,K (X, E) converging in L1,loc (X, E) towards u. Given ϕ ∈ D(X, E) with supp(ϕ) ⊂ X\K, it follows that Z Z 0= ϕuj dx → ϕu dx . X

Hence supp(u) ⊂ K.

X

¥

7.2.2 Corollary Let F and F1 be LCSs satisfying F1 ,→ F ,→ L1,loc (X, E). Then FK is a closed linear subspace of F and F1,K ,→ FK . Let X, X0 , and X1 be Banach spaces. Recall that K(X1 , X0 ) is the set of all compact T ∈ L(X1 , X0 ). Below we shall repeatedly use the following well-known and easy to prove fact: Let T ∈ K(X1 , X0 ), S0 ∈ L(X0 , X), and S1 ∈ L(X, X1 ). Then S0 T ∈ K(X1 , X) and T S1 ∈ K(X, X0 ).

(7.2.1)

Compact Embeddings of Besov Spaces First we consider general Besov spaces. 7.2.3 Theorem Let (7.0.1) be true. Assume E1 ,− ,→ E0 , 1 ≤ q < ∞, 1 ≤ r ≤ ∞, and s0 < s1 with s1 > 0. Then s /ν

(7.2.2)

s /ν

(7.2.3)

1 s0 /ν Bq,r,K (X, E1 ) ,− ,→ Bq,r (X, E0 )

and 1 s0 /ν B∞,K (X, E1 ) ,− ,→ B∞ (X, E0 ) .

Proof

We use the notations of Subsection 3.5. s /ν

1 (1) Let K be a bounded subset of Bq,r,K (X, E1 ). Then

K is bounded in Lq,K (X, E1 ).

(7.2.4)

We fix s ∈ (0, s1 ) with s < νi for 1 ≤ i ≤ `. It follows from Theorems 2.2.2, 2.2.5, s/ν and 2.8.3 and Corollary 7.2.2 that K is a bounded subset of Bq,∞ (X, E0 ). Now

254

VII Function Spaces

Proposition 3.5.4 implies that Ki :=

©

xi 7→ u(xi ; ·) ; u ∈ K

ª

(7.2.5)

¡ ¢ is equicontinuous in Lq Xi , Lq (Xˆı , E0 ) . Given x, y ∈ X, u(x + y) − u(x) =

` ³ X

u(x1 , . . . , xi−1 , xi + yi , . . . , x` + y` )

i=1

´ − u(x1 , . . . , xi , xi+1 + yi+1 , . . . , x` + y` ) .

From this we infer ku(x + y) − u(x)kLq (X,E0 ) Z ¯ ` ³Z X ¯ ≤ ··· ¯u(x1 , . . . , xi−1 , xi + yi , . . . , x` + y` ) i=1

X1

X`

¯q ´1/q ¯ − u(x1 , . . . , xi , xi+1 + yi+1 , . . . , x` + y` )¯ dx` · · · dx1 E0

≤

` ³Z X i=1

° ° °u(xi + yi ; ·) − u(xi ; ·)°q

Lq (Xı ,E0 )

Xi

ˆ

dxi

´1/q

,

where the last inequality follows by the variable substitutions xj + yj → ξj for i + 1 ≤ j ≤ `. Thus (7.2.5) shows that K is equicontinuous in Lq (X, E0 ). From this, (7.2.4), and the fact that condition (iii) of Theorem 7.1.2 is trivially satisfied we deduce by that theorem that K b Lq (X, E0 ). This proves s /ν

1 Bq,r,K (X, E1 ) ,− ,→ Lq (X, E0 ) .

s /ν

1 (2) Suppose s0 > 0. Let K be a bounded sequence in Bq,r,K (X, E1 ). By (1) there exists a subsequence (uj ) which is a Cauchy sequence in Lq (X, E0 ). It follows from Theorems 2.7.1(i) and 2.8.3 and embedding (2.6.6) with m = 0 that ¢ . ¡ s0 /ν s1 /ν Bq,r (X, E0 ) = Lq (X, E0 ), Bq,r (X, E0 ) s /s ,r . 0

1

This and Proposition I.2.2.1 give s /s

1 0 /s1 kuj − uk ks0 /ν,q,r ≤ c kuj − uk k1−s kuj − uk ks01 /ν,q,r q 0 /s1 ≤ c kuj − uk k1−s q

for j, k ∈ N, where the norms belong to E0 -valued spaces. The last inequality s1 /ν holds since K is bounded in Bq,r (X, E0 ). Hence (uj ) is a Cauchy sequence in

VII.7 Compactness

255

s /ν

0 Bq,r (X, E0 ). This implies that K is sequentially compact, hence compact, in s0 /ν Bq,r (X, E0 ). Consequently, (7.2.2) applies if 0 < s0 < s1 . (3) Suppose s0 ≤ 0. Choose s ∈ (0, s1 ). Then, by (2),

s /ν

1 s/ν s0 /ν Bq,r,K (X, E1 ) ,− ,→ Bq,r (X, E0 ) ,→ Bq,r (X, E0 ) .

Now (7.2.2) follows from (7.2.1). (4) The proof of (7.2.3) is essentially the same, except that we have to apply Theorem 7.1.1. ¥ Compact Embeddings of H¨older, Sobolev–Slobodeckii, and Bessel Potential Spaces On the basis of the preceding results it is not difficult to prove compact embedding theorems for the spaces listed in this subtitle. The reader should recall the definitions of the Sobolev–Slobodeckii, H¨older, and little H¨older scales in Subsection 3.6. 7.2.4 Theorem Let (7.0.1) be satisfied. Suppose E1 ,− ,→ E0 and 0 ≤ s0 < s1 . Then s /ν

(X, E1 ) ,− ,→ BUC s0 /ν (X, E0 )

s /ν

(X, E1 ) ,− ,→ bucs0 /ν (X, E0 ) .

BUCK1 and

bucK1 Proof

We fix t0 , t1 ∈ / νN with s0 < t0 < t1 < s1 . Then s /ν

bucK1

s /ν

(X, E1 ) ,→ BUCK1

t /ν

1 (X, E1 ) ,→ B∞,K (X, E1 )

t0 /ν ,− ,→ B∞ (X, E0 ) ,→ bucs0 /ν (X, E0 ) ,→ BUC s0 /ν (X, E0 ) .

Thus the assertion follows from (7.2.1).

¥

As an application of these results we can easily prove the following farreaching generalizations of the classical Rellich–Kondrachov embedding theorem. 7.2.5 Theorem Let assumption (7.0.1) be satisfied and suppose E1 ,− ,→ E0 . (i) Assume q0 , q1 ∈ [1, ∞), 0 ≤ s0 < s1 , and s1 − |ω|/q1 > s0 − |ω|/q0 . Then s /ν

Wq11,K (X, E1 ) ,− ,→ Wqs00 /ν (X, E0 )

and

s /ν

Hq11,K (X, E1 ) ,− ,→ Hqs00 /ν (X, E0 ) .

(ii) If 1 ≤ q < ∞ and s − |ω|/q > t ≥ 0, then s/ν

Wq,K (X, E1 ) ,− ,→ buct/ν (X, E0 )

and

s/ν

Hq,K (X, E1 ) ,− ,→ buct/ν (X, E0 ) .

256

VII Function Spaces

Proof (1) We fix s, t, t0 with s0 < s < t < t1 < s1 and t − |ω|/q1 ≥ s − |ω|/q0 . Then we get from Theorems 2.2.2, 2.6.5, and 2.8.3, Corollary 7.2.2 and Theorem 7.2.3 s /ν t /ν Wq11,K (X, E1 ) ,→ Bq11 ,K (X, E1 ) ,− ,→ Bqt/ν (X, E0 ) 1 ,→ Bqs/ν (X, E0 ) ,→ Wqs00 /ν (X, E0 ) . 0 ,q1 This implies the first part of (i). (2) Let 0 ≤ t < s < t0 < t1 < s1 be such that t1 − |ω|/q > t0 . Then, similarly as in the preceding step, s /ν

t /ν

1 1 t0 /ν Wq,K (X, E1 ) ,→ Bq,K (X, E1 ) ,→ B∞ (X, E1 )

s/ν ,− ,→ B∞ (X, E0 ) ,→ bucs0 /ν (X, E0 ) ,

where the last embedding is implied by (2.2.10). Hence the first part of (ii) is proved. (3) The assertions about the Bessel potential spaces are now obvious consequences of Theorem 4.1.3(i), taking Theorems 2.2.4(i) and 2.8.3 into account. ¥

7.3

Function Spaces on Intervals

It is the purpose of this subsection to introduce the basic spaces of distributions on compact intervals and to put them in relation to general results obtained in the preceding sections. It is no loss of generality to assume throughout • 0 ν1 (m + 1/q). Then we know from Lemma 1.2.7 t/ν that the series in (1.2.15) converges in Bq,1 . From this, step (1), and (1.2.23) we infer trk (Km v) =

∞ X

∞ X ¡ q q ¢ q q trk 2−jν1 F1−1 ηjm ⊗ (ψj (D )v) = δk,m ψj (D )v = δk,m v

j=0

j=0

for 0 ≤ k ≤ m, where the last equality is a consequence of Lemma VI.3.6.2. This and the last part of Lemma 1.2.7 show that trcm := Km is a continuous right inverse p (s−ν1 (m+1/q))/ν on Bq,r for trm if s > ν1 (m + 1/q). Thus the theorem is proved in the s/ν s/ν case of Besov spaces. Lemma 1.2.8 and the embedding Fp,1 ,→ Fp,r guarantee p (s−ν (m+1/p))/ν s/ν that Km maps Bp 1 continuously into Fp,r . Thus the assertion holds for Triebel–Lizorkin spaces also. ¥ By means of the ‘sandwich theorem’ of Subsection VII.5.5 we can now show that the r-c pair (trm , trcm ) of Theorem 1.2.1 is an r-c pair for Sobolev and Bessel potential spaces too. Recall the definition of Slobodeckii spaces (VII.3.6.3). 1.2.9 Theorem Suppose 1 < p < ∞ and m ∈ N. (i) If k ∈ νN satisfies k > ν1 (m + 1/p), then (trm , trcm ) is an r-c pair for ¡ k/ν d ¢ p Wp (R , E), Bp(k−ν1 (m+1/p))/ν (Rd−1 , E) . (1.2.24) (ii) Let s > ν1 (m + 1/p). Then (trm , trcm ) is an r-c pair for ¡ s/ν d ¢ p Hp (R , E), Bp(s−ν1 (m+1/p))/ν (Rd−1 , E) . Furthermore, ¡ ¢ p trcm ∈ L Bp(t−ν1 (m+1/p))/ν (Rd−1 , E), Hpt/ν (Rd , E) for t ∈ R. Proof

This is immediate from Theorems VII.5.5.2 and 1.2.1.

¥

VIII.1 Traces

299

1.2.10 Corollary Let k ∈ νN satisfy k > ν1 (m + 1/p) and suppose that either q q k − ν1 (m + 1/p) ∈ / ν N or E is ν -admissible. Then (1.2.24) can be replaced by ¡ k/ν d ¢ p Wp (R , E), Wp(k−ν1 (m+1/p))/ν (Rd−1 , E) . p p q t/ν . t/ν Proof If t := k − ν1 (m + 1/p) ∈ / ν N, then Bp = Wp by (VII.3.6.3). Otherwise, this equivalence is implied by Theorem VII.4.3.2. ¥

1.3

Traces on Half-Spaces

We consider first the most important spaces, namely the scales of Bessel potential, Sobolev–Slobodeckii, H¨older, and little H¨older spaces. For the sake of a uniform s/ν notation, we leave out Besov spaces Bq,r with q 6= r and Triebel–Lizorkin spaces. Afterwards, general Besov spaces are easily included by interpolation. We write Ft/ν q , t ∈ R, to denote either one of the spaces •

Bqt/ν , 1 ≤ q ≤ ∞ ,

•

bt/ν q , 1≤q ≤∞ ,

•

Wqt/ν , 1 < q < ∞ , t ∈ νN ,

•

Hqt/ν , 1 < q < ∞ ,

where we recall that We denote by

(1.3.1)

bt/ν = Bqt/ν , q

1≤q s0 > ν1 (j + 1/q). Set • X := Lq if q < ∞

and

X := BUC if q = ∞ .

It follows from Theorems VII.2.6.5, VII.2.8.3, and VII.3.6.2 that ¡ ¢ ¡ ¢ Bqs0 /ν (H, E) ,→ Bqs0 /ν1 R+ , X (∂H, E) ,→ C j R+ , X (∂H, E) . s/ν

Furthermore, by the definition of bq VII.4.1.3, Corollary VII.4.1.2,

(1.3.2)

(1.3.3)

and Theorems VII.2.2.2, VII.2.6.5, VII.2.8.3,

s0 /ν Fs/ν (H, E) . q (H, E) ,→ Bq

300

VIII Traces and Boundary Operators

j j From this we deduce that the trace operator of order j on ∂H, γ j = γ∂H = ∂n (in s/ν the direction of the interior normal) is defined for u ∈ Fq (H, E) by

γ j u := ∂ j u|x1 =0 = ∂ j u(0, ·) ∈ X (∂H, E) . We set 0 γ = γ∂H := γ 0 = ∂n

(1.3.4)

and call it trace operator on ∂H. Note γ j = γ ◦ ∂1j ,

(1.3.5)

s/ν Fq (H, E)

and γ j maps continuously into X (∂H, E). The following theorem exhibits its precise image. For abbreviation, we set t(j, q) := t − ν1 (j + 1/q) and

(

p t/ν t/ν B∞ , if q = ∞ and Ft/ν ∞ = B∞ , := (1.3.6) p bt/ν in all other cases , q q q for t ∈ R. Moreover, ω = (ω1 , ω ) = (ν1 , ω ). Using these notations, we can prove the following fundamental boundary retraction theorem for half-spaces. ∂Ft/ν q

p

s/ν

1.3.1 Theorem Suppose j ∈ N and s > ν1 (j + 1/q). Let Fq satisfy (1.3.1). Then γ j is a universal retraction from p s(j,q)/ν Fs/ν (∂H, E) . q (H, E) onto ∂Fq There exists a universal map

¡ ¢ p (γ j )c ∈ L ∂Ft(j,q)/ν (∂H, E), Ft/ν q q (H, E)

(1.3.7)

for t ∈ R. It is a coretraction for γ j if t > ν1 (j + 1/q) and satisfies γ i ◦ (γ j )c = 0 for 0 ≤ i < j. It commutes with ‘tangential derivatives’, that is, given t ∈ R and β ∈ Nd−1 , the diagram t(j,q)/ν

∂Fq

p

(γ j )c (∂H, E)

∂xβ p

? p p (t(j,q)−β p ω )/ν

∂Fq

-

t/ν

Fq (H, E) ∂xβ p

j c

(γ ) (∂H, E)

-

p p

(t−β ω )/ν

Fq

is commutative, provided E is ν-admissible if 6 F = H. 6 Recall

t/ν

that then Wp

?

. t/ν = Hp for t ∈ νN by Theorem VII.4.3.2.

(H, E)

(1.3.8)

VIII.1 Traces

301

Proof We know from Theorems VII.1.3.1, VII.2.8.2 and VII.4.1.1 that (RH , EH ) ¡ s/ν ¢ s/ν is a universal r-e pair for Fq (Rd , E), Fq (H, E) . From this and Theorems 1.2.1 and 1.2.9 we deduce that ¡ ¢ p s(j,q)/ν trj ◦ EH ∈ L Fs/ν (∂H, E) . q (H, E), ∂Fq s/ν

s/ν

Suppose u ∈ Fq (H, E). Then v := EH u ∈ Fq (Rd , E). Since the derivative of a continuously differentiable function equals its right derivative, we find trj ◦ EH u = trj v = ∂1j v|x1 =0 = ∂1j (RH v)|x1 =0 = ∂1j (RH EH u)|x1 =0 = ∂1j u|x1 =0 = γ j u ∈ X (∂H, E) . Thus

(1.3.9)

¡ ¢ p s(j,q)/ν γ j ∈ L Fs/ν (∂H, E) . q (H, E), ∂Fq

We define (γ j )c := RH ◦ trcj . Theorems 1.2.1 and 1.2.9 imply (1.3.7). If 0 ≤ i ≤ j, then it follows from (1.3.9) and Theorems 1.2.1 and 1.2.9 that γ i ◦ (γ j )c g = ∂1i (RH ◦ trcj g)|x1 =0 = tri (trcj g) = δij g s(j,q)/ν

for g ∈ ∂Fq

p

(∂H, E).

It is clear that ∂xβ p commutes with RH . Hence it suffices to show that ∂xβ p commutes with (trj )c = Kj . It follows from Lemma 1.2.6 that ¡ q q ¢ q q ∂xβ p 2−i ν1 F1−1 ηij ⊗ ψi (D )v = 2−i ν1 F1−1 ηij ⊗ ∂xβ p ψi (D )v . q q q Since ∂xβ p = (iD )β commutes with ψi (D ), assertion (1.3.8) follows from (1.2.15) and Lemmas 1.2.7 and 1.2.8, by also using Theorems VII.2.8.6 and VII.4.3.1(i). ¥ Now it is not difficult to deduce from this the following general boundary retraction theorem, the main result of this subsection. s/ν

1.3.2 Theorem Suppose k ∈ N and s > ν1 (k + 1/q). Let Fq

satisfy (1.3.1). Then

0 k ~γ k := (γ 0 , . . . , γ k ) = (∂n , . . . , ∂n )

is a universal retraction from Fs/ν q (H, E)

onto

k Y p p ~ k,q)/ν ∂Fs( (∂H, E) := Fs(j,q)/ν (∂H, E) . q q

(1.3.10)

j=0

It possesses a universal coretraction (~γ k )c . q If β ∈ Nd−1 and s > ν1 (k + 1/q) + β q ω , then (~γ k )c commutes with ∂xβ p , provided E is ν-admissible if F = H.

302

Proof

VIII Traces and Boundary Operators

For abbreviation, we set X s := Fs/ν q (H, E) , ¡

Then ~γ k ∈ L X s ,

Qk

p Yjs := ∂Fs(j,q)/ν (∂H, E) . q

(1.3.11)

¢ s

Yj by the preceding theorem. Qk Suppose (v 0 , . . . , v k ) ∈ i=0 Yis . Set u0 := (γ 0 )c v 0 ∈ X s . Let 1 ≤ j ≤ k and assume u0 , . . . , uj−1 are already defined. Put j=0

uj := uj−1 + (γ j )c (v j − γ j uj−1 ) = pj uj−1 + (γ j )c v j , where pj := 1 − (γ j )c γ j ∈ L(X s ). Then uk = pk · · · p1 (γ 0 )c v 0 + pk · · · p2 (γ 1 )c v 1 + · · · + (γ k )c v k .

(1.3.12)

Note that γ j pj = 0 and γ i pj = γ i for 0 ≤ i < j. From this we infer γ j uk = v j for 0 ≤ j ≤ k. Hence, setting (~γ k )c (v 0 , . . . , v k ) := uk , we see that (~γ k )c is a universal coretraction for ~γ k . q q Let s > ν1 (k + 1/q) + β q ω so that s > ν1 (j + 1/q) + β q ω for 0 ≤ j ≤ k. p p Since ∂xβ p ∈ L(X s , X s−β ω ) and γ j commutes with ∂xβ p , we get from Theorem 1.3.1 that p p ∂xβ p γ j = γ j ∂xβ p ∈ L(X s , Y s−β ω ) . Hence

(γ j )c γ j ∂xβ p ∈ L(X s , X s−β

pωp

),

This and the commutativity of (γ j )c and ∂xβ p imply p p ∂xβ p pj = pj ∂xβ p ∈ L(X s , X s−β ω ) ,

0≤j≤k .

0≤j≤k .

Now the second part of the assertion is a consequence of the representation (1.3.12) of (~γ k )c . ¥ Parameter-Dependence

¡ ¢ The following technical considerations elucidate the behavior of ~γ k , (~γ k )c on parameter-dependent spaces. For this we make use of (VII.8.1.5) and definitions (VII.8.2.1) and (VII.8.2.2). ¡ ¢ p q We denote by σt = σtν the action of the multiplicative group (0, ∞), q asq sociated with the dilation t 7→ t νqp ξ on ∂H (see (VI.3.1.6)). 1.3.3 Theorem Suppose k ∈ N and s > ν1 (k + 1/q). Then ~γ k is an η-uniform unip s/ν s(~ k,q)/ν versal retraction from Fq;η (H, E) onto ∂Fq;η (∂H, E). Let (~γ k )c be a universal k coretraction for ~γ for the case η = 1 and set ¡ ¢ q (~γ k )cη (v 0 , . . . , v k ) := σ[η] (~γ k )c σ1/[η] (v 0 , [η]−ν1 v 1 , . . . , [η]−kν1 v k ) . (1.3.13)

VIII.1 Traces

Then

303

¡ ¢ p ~ k,q)/ν (~γ k )cη ∈ L ∂Fs( (∂H, E), Fs/ν q;η q;η (H, E)

η-uniformly ,

s/ν

and it is a universal coretraction for ~γ k on Fq;η (H, E). q p q Proof Set T sq,[η] := [η]−s+|ω |/q σ[η] and use the notation (1.3.11) also for the parameter-dependent case. s (1) Let 0 ≤ j ≤ k. We define a surjection γηj ∈ L(Xηs , Yj;η ) by the commutativity of the diagram s Tq,1/[η]

Xηs γηj

∼ =

?

s Yj;η

-

(1.3.14)

γj

s(j,q)

¾

Xs

?

Tq,[η]

Yjs

∼ =

Then, for v ∈ Xηs , ¢ p q ¡ γηj v = [η]−s(j,q)+|ω |/q σ[η] γ j ([η]s−|ω|/q σ[1/η] v) ¢ q ¡ = [η]ν1 j σ[η] γ j (σ[1/η] v) = γ j v ,

(1.3.15)

q due to γ j (σ[1/η] v) = [η]−jν1 σ1/[η] γ j v. From this and the fact that the horizontal arrows represent isometricQisomorphisms we see that ~γ k is an η-uniform continuous k s surjection from Xηs onto j=0 Yj;η . (2) We define (~γ k )cη by the commutativity of the diagram q s(j,q)

k j=0

(~γ k )cη

diag T q,1/[η]

s Yj;η

?

Xηs

∼ =

-

k j=0

Yjs (~γ k )c

¾

s Tq,[η]

∼ =

? Xs

Qk s Given (v 0 , . . . , v k ) ∈ j=0 Yj;η , it follows that (~γ k )cη is given by the right side of (1.3.13). Now the assertion follows from the result of step (1), (1.3.14), and (1.3.15). ¥ General Besov spaces Finally, we indicate the easy extension of the findings of this section to general Besov spaces. For this we can be rather brief.

304

VIII Traces and Boundary Operators s/ν

s/ν

s/ν

1.3.4 Theorem Assume 1 ≤ q, r ≤ ∞. Let Fq,r ∈ {Bq,r , bq,r } and set ( t/ν p t/ν Bq,r , if Ft/ν p q,r = Bq,r , t/ν ∂Fq,r := p t/ν bt/ν if Ft/ν q,r , q,r = bq,r . p s/ν t/ν Then all theorems of this subsection remain valid if (Fq , ∂Fq ) is replaced by p s/ν t/ν (Fq,r , ∂Fq,r ) with the appropriate choices of t. Proof

Fix s1 > s > s0 > ν1 (j + 1/q). Then s0 /ν Fsq1 /ν ,→ Fs/ν , q,r ,→ Fq

p p p ∂Fsq1 (j,q)/ν ,→ ∂Fs(j,q)/ν ,→ ∂Fsq0 (j,q)/ν q,r

by the embedding theorems of Section VII.2. From this, Theorem 1.3.1, and the interpolation Theorems VII.2.7.1 and VII.2.8.3 we get the analogue of Theorem 1.3.1 in the present setting. Now the counterparts of Theorems 1.3.2 and 1.3.4 follow by using the parameter-dependent version of Theorem 1.3.1 in the respective proofs. ¥

1.4

Spaces of Vanishing Traces

On the basis of the preceding results we can now give alternative characterizations ˚ of (most of the) Sobolev–Slobodeckii, Bessel potential, and H¨older spaces on H. Recalling convention (1.3.1), we put for k ∈ N and s > ν1 (k + 1/q) © ª s/ν Fq,~γk (H, E) := u ∈ Fs/ν γku = 0 . (1.4.1) q (H, E) ; ~ Furthermore, we define for q < ∞: s/ν ˚ ˚ Fs/ν q (H, E) is the closure of D(H, E) in Fq (H, E) .

(1.4.2)

s/ν

˚ E). First we conWe investigate the relations between these spaces and Fq (H, centrate on the Sobolev–Slobodeckii space scale. Sobolev–Slobodeckii Spaces Henceforth, unless stated otherwise, •

s ∈ R+ and 1 ≤ q < ∞ with q > 1 if s ∈ νN . s/ν

Recall that the Slobodeckii spaces are defined by Wq

s/ν

:= Bq

(1.4.3) s/ν

= bq

if s ∈ / νN.

1.4.1 Theorem The following dense embeddings prevail: d

d

d

˚ E) ,→ S(H, ˚ E) ,→ W s/ν (H, ˚ E) ,→ W ˚qs/ν (H, E) . D(H, q s/ν ˚ s/ν Proof Theorem VII.3.7.7 guarantees that Wq (H, E) ,→ Wq (H, E), provided s∈ / νN. If s ∈ νN, then we get this from definition (VII.1.2.2).

VIII.1 Traces

305 s/ν

It is a consequence of (VII.2.2.8) that S(Rd , E) is dense in Bq (Rd , E). ˚ E) is Hence Lemma VI.1.1.2 and Theorems VI.1.2.3 and 2.8.3 imply that S(H, s/ν ˚ dense in Bq (H, E). Thus, recalling definition (VII.1.2.2) once more, we see that the middle dense embedding prevails. The left one follows from Lemma VI.1.1.3(ii). Now the statement on the right is implied by definition (1.4.2). ¥ 1.4.2 Theorem Let (1.4.3) be satisfied. (i) If k ∈ N and ν1 (k + 1/q) < s < ν1 (k + 1 + 1/q) , then

(1.4.4)

s/ν

˚qs/ν (H, E) . Wq,~γ k (H, E) = W

(ii) If 0 < s < ν1 /q, then ˚qs/ν (H, E) . Wqs/ν (H, E) = W Proof We omit E. s/ν s/ν (1) Since Wq,~γ k (H) is the kernel of the continuous linear map ~γ k on Wq (H), s/ν ˚ ⊂ W s/νk (H) implies it is a closed linear subspace of Wq (H). Hence D(H) q,~ γ

˚ s/ν (H) ,→ W s/νk (H) . W q q,~ γ Consequently, the assertion follows, provided we show that d

s/ν

˚ ⊂ W k (H) . D(H) q,~ γ

(1.4.5)

Similarly, in order to prove assertion (ii), we have to verify that d

˚ ⊂ W s/ν (H) , D(H) q

0 < s < ν1 /q .

(1.4.6)

s/ν

(2) Let s ∈ R+ \ν1 (N + 1/q), u ∈ Wq (H), and ε > 0. Since S(H) is dense in this space, we can find v ∈ S(H) satisfying ku − vkW s/ν (H) < ε . q

(1.4.7)

By Theorem 1.3.2 we can choose a coretraction (~γ k )c for ~γ k such that ¡ ¢ (~γ k )c ◦ ~γ k ∈ L Wqt/ν (H) , t > ν1 (k + 1/q) . (1.4.8) From this, (VII.2.2.1), (VII.2.6.7), and Theorem VII.2.8.3 it follows that w := v − (~γ k )c~γ k v ∈ C ∞ (H) .

306

VIII Traces and Boundary Operators

Hence, if s > ν1 (k + 1/q), then w ∈ C ∞ ∩ Wqs/ν (H) ,

~γ k w = 0 .

Furthermore, (1.4.8) and ~γ k u = 0 imply, due to (1.4.7), ku − wkW s/ν (H) ≤ ku − vkW s/ν (H) + k(~γ k )c~γ k (u − v)kW s/ν (H) ≤ c ε , q

q

q

where c is independent of ε > 0. This shows that we can –– and do –– assume that u ∈ C ∞ ∩ Wqs/ν (H)

where ~γ k u = 0 if s > ν1 (k + 1/q) .

(1.4.9)

¡ ¢ (3) We fix ϕ ∈ D Rd , [0, 1] satisfying ϕ(x) = 1 for x ∈ H with |x| ≤ 1, and set ϕt (x) := ϕ(tx) for t > 0. Then ϕt (x) = 1 for x ∈ H with |x| ≤ 1/t and ∂ α ϕt = t|α| (∂ α ϕ)t ,

α ∈ Nd .

(1.4.10)

Hence ϕt ∈ D(H) and kϕt kBC n/ν ≤ c(n) ,

00. (1.4.14) 0

Note that ¡ ¢ ∆h (1 − ψ1/t )v (y) = −∆h ψ1/t (y)v(y + h) + (1 − ψ1/t )∆h v(y) .

(1.4.15)

Moreover, ∆h ψ(y) = 0 for y ≥ 1 ,

|∆h ψ(y)| ≤ c min{h, 1} ,

and ∆h ψ1/t = (∆h/t ψ)1/t for h, t > 0. Let σ ∈ (0, 1/q). Then it follows that Z ∞ Z ° ° −σq °(∆h ψ1/t )(y)v(y + h)°q dy dh h X h + 0 Y Z ∞ Z dh = t−σq (h/t)−σq |∆h/t ψ(y/t)|q kv(y + h)kqX dy h + 0 Y Z ∞ Z t ≤ ct−σq τ −σq−1 min{τ, 1}q dτ kv(y + tτ )kqX dy ≤ ct1−σq 0

0

for t > 0. Similarly, using |∆h v(y)| ≤ c min{h, 1} for y ∈ Y + , Z ∞ Z ° ° °(1 − ψ1/t )(y)∆h v(y)°q dy dh h−σq X h 0 Y+ Z ∞ Z t ≤c h−σq−1 min{h, 1}q dh dy ≤ ct 0

0

for t > 0. From these estimates and (1.4.15) we obtain £ ¤X (1 − ψ1/t )v σ,q ≤ ct−σ+1/q , 0 ν1 (k + 1/q). Hence ~γ k u = 0. Thus Taylor’s formula implies for 0 ≤ j ≤ k m X

∂ j v(y) =

i=k−j+1

1 i+j ∂ v(0)y i + Rm (∂ j v)(y)y m i!

for m ≥ k − j + 1, where Z

1

Rm (w)(y) := 0

¢ (1 − t)m−1 ¡ m ∂ w(ty) − ∂ m w(0) dt . (m + 1)!

Since y ≤ c on supp(v), it thus follows that k∂ j v(y)kX ≤ c y k−j+1 ,

0≤j ≤k+1 .

(1.4.18)

Note that ∂ i ψ1/t = t−i (∂ i ψ)1/t

(1.4.19)

implies |∂ i ψ1/t | ≤ ct−i . Moreover, ∂ i ψ1/t (y) = 0 for y ≥ t > 0. From this and j

¡

¢

j

∂ (1 − ψ1/t )v = (1 − ψ1/t )∂ v −

j ³ ´ X j i=1

i

∂ i ψ1/t ∂ j−i v

we infer j ³ ´ X ° j¡ ¢° −i k−j+i+1 °∂ (1 − ψ1/t )v ° ≤ c k(1 − ψ1/t )y k−j+1 kq + t ky k L (0,t) q q i=1

for 0 ≤ j ≤ k + 1. The first norm on the right is majorized by ³Z t ´1/q y (k−j+1)q dy ≤ ctk−j+1+1/q , t>0 , 0≤j ≤k+1 . 0

The sum too can be majorized by ctk−j+1+1/q . Hence ° j¡ ¢° °∂ (1 − ψ1/t )v ° ≤ ctk−j+1+1/q , t>0, Lq (Y + ,X )

0≤j ≤k+1 ,

and thus k(1 − ψ1/t )vkWqk+1 (Y + ,X ) ≤ ct1/q ,

00. y 0 t y Then u = v − w. Indeed, v(y) → 0 and w(y) → 0 as y → ∞. Hence the claim follows from ∂u = ∂v − ∂w, which is immediate.

VIII.1 Traces

315

q We put σ := s/ν1 . Then (1.5.11) will follow from the density of D(R+ , X ) in Wpσ (R+ , X ), provided we show that kvkLp,−σp (R+ ,X ) ≤ c kukWpσ (R+ ,X )

(1.5.12)

and kwkL

p+

≤ c kukWpσ (R+ ,X ) . ,X ) ¡ ¢ H¨older’s inequality, applied with 1 q u(x) − u(t) , yields Z 1 y |v(y)|pX ≤ |u(y) − u(t)|pX dt . y 0 Hence Z ∞

Z y

0

−σp

|v(y)|pX

dy ≤

Z

∞

y

−σp−1

y Z0 ∞ Z

∞

= Z0 ∞ Zt ∞ = Z0 ∞ ≤

(1.5.13)

p,−σp (R

0

τ

0

y −σp−1 |u(y) − u(t)|pX dy dt (t + τ )−σp−1 |u(t + τ ) − u(t)|pX dτ dt

−σp

0

|u(y) − u(t)|pX dt dy

(1.5.14)

dτ = [u]pσ,p τ ≤ c kukpW σ (˚ . H)

k∆τ ukpLp (R+ ,X )

≤ kukpW σ (R+ ,X ) p

p

This implies (1.5.12). Inequality (1.5.13) is a consequence of (1.5.12) and kwkL

p

+ p,−σp (R ,X )

≤ c kvkL

p

+ p,−σp (R ,X )

.

The latter follows from (1.5.9) since s < ν1 /p implies −σp > −1. This proves (1.5.11), hence the theorem. ¥ Next we extend tis result to the case where s > ν1 /p. 1.5.6 Theorem Suppose 1 < p < ∞ and s ∈ R+ \ν1 (N + 1/p). Then . ˚ E) . ˚ps/ν (H, E) = W Wps/ν (H, E) ∩ Lp,−sp/ν1 (H,

(1.5.15)

˚ps/ν (H, E) iff u ∈ Wps/ν (H, E) and y s/ν1 u ∈ Lp (H, E). Thus u ∈ W Proof We omit E and use the notations of the preceding proof. Due to Theorem 1.5.5, we can assume that ν1 (k + 1/p) < s < ν1 (k + 1 + 1/p) for some k ∈ N.

(1.5.16)

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VIII Traces and Boundary Operators

(1) Assume s 6= ν1 (k + 1). It follows from Theorem VII.3.4.1(i) that (1.5.14) still holds. We set Z y v(t) w1 (y) := dt , y>0. t 0 q By u(0) = v(0) = w1 (0) for u ∈ D(R+ , X ) and ∂u = ∂v + ∂w1 , we get u = v + w1 . From (1.5.10) we obtain kw1 kL

p

+ p,−σp (R ,X )

≤ c kvkL

p

+ p,−σp (R ,X )

,

since s > ν1 /p implies ρ := −σp < −1. From this we obtain, similarly as in step (2) of the proof of the preceding theorem, that ˚ , ˚ps/ν (H) ,→ Lp,−sp/ν (H) W 1

(1.5.17)

q ˚ σ (R+ , X ). due to the density of D(R+ , X ) in W p q (2) Suppose s = ν1 (k + 1). For u ∈ D(R+ , X ) set Z 1 t w(t) := u(τ ) dτ , t>0. t 0 Repeated partial integration gives Z Z (−1)k t k k (−1)k tk 1 k k w(t) = τ ∂ u(τ ) dτ = s ∂ u(st) ds . tk! tk! 0 0 Hence, given ρ ∈ R, ³Z ∞ ´1/p ³Z tρ |w(t)|pX dt =c 0

0 1

Z ≤c Z

0

∞

Z0 ∞ 0

¯p ´1/p ¯ tk+ρ/p sk ∂ k u(st) ds¯ dt X

ktk+ρ/p sk ∂ k u(st)kLp (R+ ,dt;X ) ds s

=c

1 0

1

=c

¯Z ¯ ¯

k

³Z

∞ 0

tkp+ρ |∂ k u(st)|pX dt

´1/p

ds

s−(ρ+1)/p ds ky p+ρ/p ∂ k ukLp (R+ ,dy;X ) ,

where the last equality results from the substitution y = st. Now we set ρ := −kp to obtain ³Z ∞ ´1/p t−kp |w(t)|pX dt ≤ c k∂ k ukLp (R+ ,X ) (1.5.18) 0

with c :=

¡R 1 0

sk−1/p ds

¢1/p

< ∞.

VIII.1 Traces

317

Observe that

Z

t

u(t) =

∂u(τ ) dτ = t 0

1 t

Z

t

∂u(τ ) dτ . 0

By replacing u in the preceding considerations by ∂u, we obtain from (1.5.18) ³Z ∞ ´1/p t−(k+1)p |u(t)|pX dt ≤ c k∂ k+1 ukLp (R+ ,X ) . (1.5.19) 0

(k+1)/ν

Definition (VII.1.2.2) of the norm of Wp

(H) implies

k∂ k+1 ukLp (R+ ,X ) ≤ |||u|||(k+1)/ν,p . (k+1)/ν

˚ in W ˚p From this, (1.5.19), and the density of D(H) (H) we infer that (1.5.17) holds for s = ν1 (k + 1) also. ˚ps/ν (H). Then ~γ k v 6= 0 by Theorem 1.4.2. (3) Assume v ∈ S(H), but u ∈ /W j + Let j be the least integer for which γ j v 6= 0. ¡ Since S(H)¢ ,→ C (R , X ), we deduce from Taylor’s theorem that v(y, ·) = y j γ j v + r(y, ·) with |r(y, ·)|X → 0 as y → 0. Hence we can choose κ, ε > 0 such that |v(y, ·)|X ≥ κy j for 0 ≤ y ≤ ε. Consequently, Z ∞ Z ε y −sp/ν1 |v(y, ·)|pX dy ≥ κ y −(s−ν1 j)p/ν1 dy = ∞ , 0

0

˚ From this and the since (1.5.16) implies (s − ν1 j)p/ν1 > 1. Thus v ∈ / Lp,−sp/ν1 (H). s/ν s/ν ˚ belongs to density of S(H) in Wp (H) we infer that u ∈ Wp (H) ∩ Lp,−sp/ν (H) 1

˚ps/ν (H). Now we get from the validity of (1.5.17) for each admissible s that W ˚ ⊂W ˚ s/ν (H) ,→ W s/ν (H) ∩ Lp,−sp/ν (H) ˚ s/ν (H) , W p p p 1 ˚ps/ν (H) is a closed linear subspace of Wps/ν (H). This thanks to the fact that W proves the theorem. ¥

1.6

Further Characterizations of Spaces with Vanishing Traces

˚ps/ν (H, E), With the aid of Theorem 1.5.6 we can establish a further property of W which turns out to have important consequences. 1.6.1 Theorem Suppose 1 < p < ∞ and s ≥ 0 with s ∈ / ν1 (N + 1/p). Then . ˚ E) . ˚ s/ν (H, E) = W s/ν (H, W p p Proof Henceforth, we omit E and assume without loss of generality that s > 0. ˚ ˚ ˚ (1) We write (R, E) for the r-e pair (R˚ H , E˚ H ), so that E is the trivial extension s/ν s/ν operator (VI.1.1.12). Recall that Wp = Bp if s ∈ / νN.

318

VIII Traces and Boundary Operators

Suppose we show that ¡ s/ν ¢ ˚ ˚p (H), Wps/ν (Rd ) . E∈L W

(1.6.1)

¡ ¢ ˚ ∈ L Wps/ν (Rd ), Wps/ν (H) ˚ . Hence we Theorems VII.1.3.1 and VII.2.8.2 imply R ˚˚ get from (1.6.1) and u ∈ R Eu that ˚ . ˚ps/ν (H) ,→ Wps/ν (H) W Then the assertion follows from Theorem 1.4.1. Thus we have to establish the validity of (1.6.1). (2) Let s ∈ νN. It is obvious that k∂ α˚ EukLp (Rd ) = k˚ E∂ α ukLp (Rd ) = k∂ α ukLp (H) ˚ From this and the density of D(H) ˚ in W ˚ps/ν (H, E) for a q ω ≤ s/ν and u ∈ D(H). property (1.6.1) follows in this case. (3) Assume that s ∈ / νN. Let X ∈ {Rd , H}, X1 ∈ {R, R+ }. Then X = X1 × ∂H d−1 and ∂H = b R . Corollary VII.3.6.3 guarantees that ¡ ¢ ¢ p . Wps/ν (X) = Wps/ν1 X1 , Lp (H) ∩ Lp (X, Bps/ν (∂H) . (1.6.2) s/ν

For abbreviation, we set X := Lp (∂H) and Y := Bp u e := ˚ Eu. Then, obviously, ˚ , u ∈ D(H)

ke ukLp (R,Z) = kukLp (R+ ,Z) ,

p

(∂H). We also write

Z ∈ {X , Y} .

(1.6.3)

˚ in W ˚ps/ν (H) we infer that it suffices to From this, (1.6.2), and the denisty of D(H) show that ke ukW s/ν1 (R,X ) ≤ c kukW s/ν (R+ ,X ) , p

p

˚ . u ∈ D(H)

(1.6.4)

˚ (4) First we establish an auxiliary estimate. Suppose r > 0 and u ∈ D(H). Then, given 0 < ϑ < ∞, Z ∞Z 0 Z ∞ Z rh dh dh Ir (u) := |u(x + rh)|pX dx 1+ϑp = |u(y)|pX dy 1+ϑp h h 0 −rh 0 0 Z ∞Z ∞ Z ∞ dh = |u(y)|pX dy = ϑp rϑp |y −ϑ u(y)|pX dy . 1+ϑp 0 y/r h 0 Thus Theorem 1.5.6 guarantees Ir (u) ≤ c(r, ϑ) kukpW ϑ (R+ ,X ) , p

provided ϑ ∈ / N + 1/p.

˚ , u ∈ D(H)

(1.6.5)

VIII.1 Traces

319

q (5) Suppose ν1 (k + 1/p) < s < ν1 (k + 1 + 1/p) for some k ∈ N, and s ∈ / ν1 N. Set ϑ := s/ν1 . By Theorem VII.3.4.1(i), v 7→ kvkLp (X1 ,X ) + [v]ϑ,p is a norm for

Wpϑ (X1 , X ),

where Z

[v]pϑ,p

∞

Z

:= 0

X1

p k4k+1 h vkX dh . hϑp h

Hence we infer from (1.6.3) that ¡ ¢ ke ukWpϑ (R,X ) ≤ c kukLp (R+ ,X ) + [e u]ϑ,p ,

˚ . u ∈ D(H)

(1.6.6)

Note that k+1 X

4k+1 h v(x) =

(−1)k+1−j

j=0

³

´ k+1 v(x + jh) j

(1.6.7)

(e.g., [AmE05, p. 124]). Moreover, [e u]pϑ,p = [u]pϑ,p + I , where

Z

∞

Z

0

I := 0

−∞

k4k+1 ekpX dh h u dx . hϑp h

It follows from (1.6.7) that k+1 ³ ´ X [e u]pϑ,p ≤ c [u]pϑ,p + Ij (u) ,

˚ . u ∈ D(H)

j=0

By applying (1.6.5) to each element of the last sum, we see that ¡ ¢ ˚ . [e u]pϑ,p ≤ c [u]pϑ,p + kukpW ϑ (R+ ,X ) ≤ c kukpW ϑ (R+ ,X ) , u ∈ D(H) p

p

Now we get (1.6.4) from (1.6.6). The theorem is proved.

¥

˚ps/ν in Because of their importance, we collect the properties of the spaces W the following theorem. 1.6.2 Theorem Suppose 1 < p < ∞. (i) If k ∈ N and ν1 (k + 1/p) < s < ν1 (k + 1 + 1/p), then . . s/ν ˚ E) = ˚ps/ν (H, E) = W Wps/ν (H, Wp,~γ k (H, E) . (ii) If 0 < s < ν1 /p, then . . ˚ E) = ˚ps/ν (H, E) = W Wps/ν (H, Wps/ν (H, E) . Proof

Theorems 1.4.2 and 1.6.1.

¥

320

VIII Traces and Boundary Operators

General Besov Spaces By interpolation we can now extend the preceding results to general Besov spaces. For this we make use of the J method of real interpolation, which we recall briefly. 1.6.3 Remark Let E0 and E1 be Banach spaces with E1 ,→ E0 , 0 < θ < 1, and 1 ≤ q ≤ ∞. For t ∈ R we set J(t, e) := max{kekE0 , t kekE1 } , e ∈ E1 . ¡ ¢ Then e ∈ (E0 , E1 )θ,q iff there exists f ∈ C (0, ∞), E1 satisfying Z ∞ J −θ kukθ,q := kt J(t, e)kLq ((0,∞),dt/t) < ∞ , e= f (t) dt/t ,

(1.6.8)

0

J

where the integral converges in E0 , and k·kθ,q is a norm for (E0 , E1 )θ,q (e.g., Chapter 3 of [BeL76] or Section 1.6 in [Tri95]). ¥ 1.6.4 Theorem Suppose 1 < p < ∞ and 1 ≤ r ≤ ∞. (i) If k ∈ N and ν1 (k + 1/p) < s < ν1 (k + 1 + 1/p), then s/ν

s/ν ˚ Bp,r (H, E) = Bp,r,~γ k (H, E) .

(ii) If 0 < s < ν1 /p, then . s/ν s/ν ˚ Bp,r (H, E) = Bp,r (H, E) . Proof

We use the following abbreviations; X t := Wpt/ν (H, E) ,

as well as

t/ν Xρt := Bp,ρ (H, E) ,

˚ E) , X0t := Wpt/ν (H,

p Y t := bt/ν (∂H, E) , p

t t/ν ˚ Xρ,0 := Bp,ρ (H, E)

. for t ∈ R and 1 ≤ ρ ≤ ∞. Note that X t = Xpt . (1) Assume

ν1 (k + 1/p) < s0 < s < s1 < ν1 (k + 1 + 1/p) . Set θ := (s − s0 )/(s1 − s0 ) so that sθ = (1 − θ)s0 + θs1 = s. Put B := ~γ k . It follows from XBsi ,→ X si and Theorem 1.6.2(i) that . (X0s0 , X0s1 )θ,r = (XBs0 , XBs1 )θ,r ,→ (X s0 , X s1 )θ,r . (1.6.9) Theorems VII.2.7.1(i), VII.2.7.4, and VII.2.8.3 imply . s . (X0s0 , X0s1 )θ,r = Xr,0 , (X s0 , X s1 )θ,r = Xrs .

(1.6.10)

VIII.1 Traces

321

Hence s Xr,0 ,→ Xrs .

(2) Suppose

(1.6.11)

. s u ∈ (XBs0 , XBs1 )θ,r = X0,r .

Fix f such that (1.6.8) is satisfied with q = r, e = u, and (E0 , E1 ) = (XBs0 , XBs1 ). It is obvious that Z 1/ε Z 1/ε B f (t) dt/t = Bf (t) dt/t = 0 , 0 0 with s ∈ / ν1 (N + 1/p). Then . ˚s/ν s/ν ˚ Bp,r (H, E) = B p,r (H, E) . ˚ is dense in X s . ˚s1 = X s1 , that is, D(H) Proof We know from Theorem 1.6.2 that X 0 0 . s0 s1 Since r < ∞ and Xr,0 = (X0 , X0 )θ,r , interpolation theory guarantees that X0s1 is

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VIII Traces and Boundary Operators

. s ˚ is dense in X s . Since X s = dense in Xr,0 (see (I.2.5.2)). Hence D(H) Xr,B , it is r,0 r,0 s a closed linear subspace of Xr . Now the assertion is clear. ¥ Bessel Potential Spaces In order to prove partially analogous results for Bessel potential spaces we prepare a simple technical density theorem. 1.6.6 Lemma Let Xj and Yj be Banach spaces for j = 0, 1 such that X1 ,→ X0 and Y1 ,→ Y0 . Suppose (r, rc ) is an r-c pair for (Xj , Yj ), j = 0, 1. Set Xj,r := { x ∈ Xj ; rx = 0 } = ker(r | Xj ) . d

d

If X1 ,→ X0 , then X1,r ,→ X0,r . Proof Clearly, X1,r ,→ X0,r ,→ X0 . Let x ∈ X0,r . Since X1 is dense in X0 , there exists a sequence (xj ) in X1 with xj → x in X0 . Set yj := (1 − rc r)xj . Then yj ∈ X1,r and yj − x = (1 − rc r)(xj − x) → 0 in X0 . Hence X1,r is dense in X0,r . ¥ By means of this lemma we can prove an embedding theorem for Bessel potential and Besov spaces with vanishing traces. 1.6.7 Theorem Suppose k ∈ N, 1 < p < ∞, and ν1 (k + 1/p) < s < ν1 (k + 1 + 1/p) . Then s/ν

d

s/ν

s/ν

Bp,1,~γ k (H, E) ,→ Hp,~γ k (H, E) ,→ Bp,∞,~γ k (H, E) . Proof

Theorem VII.4.1.3(i) implies s/ν

d

s/ν Bp,1 (H, E) ,→ Hps/ν (H, E) ,→ Bp,∞ (H, E) .

(1.6.13)

Hence the claim follows ¡from Theorem 1.3.2 and the (proof of the) preceding ¢ lemma, setting (r, rc ) := ~γ k , (~γ k )c . ¥ It should¡be remarked ¢ that the proof of this theorem uses in an essential way the fact that ~γ k , (~γ k )c is simultaneously a universal r-c pair for Besov, Bessel potential, and Sobolev spaces. Now it is easy to derive an analogue of Theorem 1.6.1 for Bessel potential spaces.

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323

1.6.8 Theorem Suppose 1 < p < ∞. (i) If k ∈ N and ν1 (k + 1/p) < s < ν1 (k + 1 + 1/p) ,

(1.6.14)

. ˚s/ν s/ν Hp,~γ k (H, E) = H p (H, E) .

(1.6.15)

then

(ii) If 0 ≤ s < ν1 /p, then ˚ps/ν (H, E) . Hps/ν (H, E) = H Proof We omit E. (1) Let (1.6.14) be satisfied. Then we deduce from Theorem 1.6.4 and its corollary and from Theorem 1.6.7 that d d . s/ν s/ν ˚ ,→ ˚s/ν (H) = D(H) B Bp,1,~γ k (H) ,→ Hp,~γ k (H) . p,1 s/ν

s/ν

˚ is dense in H k (H), which is a closed linear subspace of Hp (H). Thus D(H) p,~ γ ˚ps/ν (H) we get (1.6.15). From this and the definition of H (2) Let 0 ≤ s < ν1 /p. The claim is obvious if s = 0. If s > 0, then we find, similarly as above, d d s/ν ˚ . s/ν ˚ ,→ D(H) Bp,1 (H) = Bp,1 (H) ,→ Hps/ν (H) ,

where we also made use of (1.6.13). Now the second claim is clear.

¥

H¨older Spaces If s is suitably restricted, then the following theorem gives a related characterizas/ν s/ν tion of the H¨older spaces BUC~γ k (H, E) and buc~γ k (H, E). 1.6.9 Theorem Suppose ν1 k < s < ν1 (k + 1) for some k ∈ N. Then s/ν ˚ E) BUC~γ k (H, E) = BUC s/ν (H,

and

s/ν ˚ E) . buc~γ k (H, E) = bucs/ν (H,

Proof We use the notations of Subsection VI.1.1 and omit E. ˚ H v. ˚ There exists v ∈ BUC s/ν (Rd ) with u = R (1) Suppose u ∈ BUC s/ν (H). q q Writing x = (y, x ) ∈ H with x ∈ ∂H, Z ∞ q u(x) = v(x) − h(t)v(−ty, x ) dt . 0

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VIII Traces and Boundary Operators

From this and Lemma VI.1.1.1 it follows that ~γ k u = 0. Hence s/ν

˚ ,→ BUC k (H) BUC s/ν (H) ~ γ ˚ ,→ bucs/ν and, similarly, bucs/ν (H) (H). ~ γk (2) Let Y be a Banach space and 0 < ϑ < 1. Suppose w ∈ BUC ϑ (R+ , Y) satisfies w(0) = 0. Denote by w e its trivial extension. Then w e ∈ BUC(R, Y) , and

kwk e ∞ = kwk∞ ,

|w(y) e − w(z)| e |w(0) − w(z)|Y Y ≤ , ϑ |y − z| |0 − z|ϑ

y ν1 (m + 1/q) for some m ∈ N. The trace operator of order m for ∂1 K is defined by q m γ1m u = ∂n u := ∂1m u|x1 =0 ∈ X (K , E) , 1

u ∈ Fs/ν q (K, E) ,

(recall (1.3.2) and (1.3.3)). Thus γ1m = γ m if K = H. The following result extends Theorem 1.3.1 to this situation. 1.8.1 Theorem Suppose s > ν1 (m + 1/q) with m ∈ N. Then ~γ1m := (γ10 , γ11 , . . . , γ1m ) s/ν is a universal retraction from Fq (K, E) onto m Y p p m,q)/ν ~ ∂Fs( (∂ K, E) := ∂Fs(j,q)/ν (∂1 K, E) . 1 q q j=0

It possesses a universal coretraction, (~γ m )c , which commutes with ∂2 , . . . , ∂d . Proof (1) In accordance with the definitions in Subsection VI.1.1, we can apply (RK p , EK p ) to functions defined on K by letting these maps operate on the variables q q x ∈ K . Then, using Theorems VII.1.3.1, VII.2.8.2, and VII.4.1.1, ¡ ¢ s/ν EK p ∈ L Fs/ν q (K), Fq (H) . Hence, by Theorem 1.3.1, ¡ ¢ p s(j,q)/ν γ j EK p ∈ L Fs/ν (∂H) . q (K), ∂Fq Consequently, ¡ ¢ p s(j,q)/ν RK p γ j EK p ∈ L Fs/ν (∂1 K) . q (K), ∂Fq

(1.8.1)

q Since γ j operates on the first variable of x = (y, x ) ∈ K, it commutes with RK p . Thus RK p γ j EK p u = ∂1j u(0, ·) = γ1j u , u ∈ Fs/ν q (K) , that is, γ1j is represented by (1.8.1). (2) We set (γ1j )c := RK p (γ j )c EK p . By Theorem 1.3.1, ¡ ¢ p q s/ν (γ1j )c ∈ L ∂Fs(j,q)/ν (K ), Fq (K) , q and

γ1i (γ1j )c v = RK p γ i (γ j )c EK p v = δ ij RK p EK p v = δ ij v p for v ∈ ∂1 Fs(j,q)/ν (∂1 K). Now the assertion follows by the arguments used in the proof of Theorem 1.3.2. ¥

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VIII Traces and Boundary Operators

1.8.2 Remark It is obvious from Theorem 1.3.4 that the same assertions apply to general Besov spaces. ¥ Vanishing Traces on Corners Now we extend Theorem 1.6.2 and Corollary 1.7.2 to corners, using obvious adaptions of the notation employed there. 1.8.3 Theorem Suppose either 1 < q < ∞ and F = W , or q = ∞ and F ∈ {B, b}. (i) Let m ∈ N and ν1 (m + 1/q) < s < ν1 (m + 1 + 1/q) . Then

(1.8.2)

¡ q ¢ s/ν Fq,~γ m (K, E) = Fs/ν (0, ∞) × K , E . q 1

(ii) Assume either 0 < s < ν1 /q, or E is reflexive Then

and

ν1 (−1 + 1/q) < s < ν1 /q .

¡ q ¢ Wqs/ν (K, E) = Wqs/ν (0, ∞) × K , E .

(iii) If q < ∞, then, in either case, ¡ ¡ q ¢ q ¢ Wqs/ν (0, ∞) × K , E is the closure of D (0, ∞) × K , E in Wqs/ν (K, E) . (1) Let (1.8.2) be satisfied. Suppose u ∈ Fq,~γ m (K). Then v := EK p u belongs 1 s/ν s/ν ˚ to Fq,~γ m (H). Hence v ∈ Fq (H) by Theorem 1.6.2(i). Thus 1 ¡ q¢ u = RK p v ∈ Fs/ν (0, ∞) × K . q s/ν

Proof

An analogous argument gives the converse inclusion. This proves (i). (2) Assertion (ii) follows similarly from Theorem 1.6.2(i) and Corollary 1.7.2. q¢ s/ν ¡ (3) Let q < ∞ and u ∈ Wq (0, ∞) × K . The theorems used in step (1) s/ν ˚ guarantee that v = EK p u ∈ Wq (H) can be approximated arbitrarily closely in s/ν ˚ The continuity of RK p , which is established in Wq (H) by elements w ∈ D(H). ¡ q¢ Theorem VI.1.2.3, implies that RK p w ∈ D (0, ∞) × K . From this we obtain assertion (iii). ¥ Faces of Higher Codimensions Now we write q ωi := (ω1 , . . . , ωbi , . . . , ωd ) ,

si (j, q) := s − ωi (j + 1/q)

for 1 ≤ i ≤ d. (Recall the definitions of Subsection VI.3.1.)

VIII.1 Traces

333

Suppose j ∈ J and s > ωj (mj + 1/q) with mj ∈ N. Then, given 0 ≤ k ≤ mj , k γjk u := ∂n u := ∂jk u|xj =0 , j

u ∈ Fs/ν q (K, E) ,

is the trace operator of order k for ∂j K. Using obvious notation, we deduce from m Theorem 1.8.1 by a permutation of coordinates that ~γj j is a retraction from p sj ( m ~ j ,q)/ω

s/ν

j Fq (K, E) onto ∂Fq (∂j K, E) and that it possesses a universal coretracmj c tion (~γj ) which commutes with ∂k for k 6= j. Generalizing the concept of (d − 1)-dimensional faces, that is, codim(1)-faces, we introduce faces of higher codimension. Set ` := card(J). Suppose 1 ≤ r ≤ ` and put © ª r J6= := (j) := (j1 , . . . , jr ) ∈ J r ; ji 6= jk for i 6= k

1 r so that J6= = J. Then, given (j) ∈ J6= , © ª r ∂(j) K = ∂(j) K := x ∈ K ; xi = 0, i ∈ {j1 , . . . , jr }

is ²a©codim(r)-face of K. It is (naturally a corner in Rd−r of type ª ¡ ` ¢ identified with) r ` J (j) . Note that card(J6= ) = r . In particular, ∂(j1 ,...,j` ) K is either an open q r corner in Rd−` , or it equals Rd−` . Also observe that, writing (j) = (j1 ,  ) ∈ J6= q r−1 with  ∈ J6= , r−1 r−1 r ∂(j) K = ∂j1 (∂( p ) K) = ∂( p ) ∂j1 K .

(1.8.3)

r ∂(j) K = ∂j1 ∂j2 · · · ∂jr K = ∂jσ(1) · · · ∂jσ(r) K ,

(1.8.4)

Hence

r where σ is any permutation of {1, . . . , r}. We also write ∂j1 ,...,jr K for ∂(j) K.

Of particular interest are corners of codim(2). In this case we put qq ωij := (ω1 , . . . , ωbi , . . . , ω cj , . . . , ωd ) , 1≤i ωi (mi + 1/q) + ωj (mj + 1/q) . Then

pp ¡ s/ν ¢ sij (mi ,mj ,q)/ωij mi mj ∂n ∂ ∈ L F (K, E), ∂F (∂ij K, E) q n j q i m

m

mi mi and ∂n ∂njj = ∂njj ∂n . i i

(1.8.5)

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VIII Traces and Boundary Operators

Proof Claim (1.8.5) is a consequence of Theorem 1.8.1 and the preceding considm erations. The commutativity assertion is clear if the partial derivatives ∂imi ∂j j u mj mi and ∂j ∂i u are continuous. This is the case if q = ∞. Otherwise, it is true if s/ν u ∈ S(K, E). Since this space is dense in Fq (K, E) with q < ∞, the assertion follows for q < ∞ also, due to (1.8.4). ¥ It is clear from (1.8.3) that this lemma can be iterated to obtain analogous results for more than two trace operators. Compatibility Conditions 2 2 Let I be a subset of J with card(I) ≥ 2, and I6= := I 2 ∩ J6= . We set [ dI K := ∂i K i∈I

and say that dK := dJ K is the essential (set) boundary of K. Note that dK is a ˚ the topological boundary, which is proper unless K is subset of bdry(K) = K\ K, closed. Suppose mi ∈ N and s > ωi (mi + 1/q) , i∈I . We set m ~ I := (mi )i∈I

~I and ~γIm := (~γimi )i∈I .

This is the trace operator of order m ~ I for dI K, resp. the essential trace operator if I = J. In particular, γdK := (γj )j∈J is the essential trace operator. It follows from Theorem 1.8.1 that ³ ´ p Q s (m ~ /q)ωi s/ν ~I ~γIm ∈ L Fq (K, E), i∈I ∂Fqi i (∂i K, E) . s/ν

In general, this map is not surjective. Indeed, given u ∈ Fq (K, E), Lemma 1.8.4 m implies that there may exist compatibility relations between ~γimi u and ~γj j u for 2 (i, j) ∈ I6= . Namely, suppose 2 (i, j) ∈ I6= and 0 ≤ µρ ≤ mρ , with µρ ∈ N for ρ ∈ {i, j},

satisfy ωi (µi + 1/q) + ωj (µj + 1/q) < s. µ

(1.8.6)

µ

Set gρ ρ := γρ ρ u. Then Lemma 1.8.4 implies µ

µ

γiµi gj j = γj j giµi . ~I We define the trace space for ~γIm , denoted by p m ~ I ,q)/ω ∂Fs( (dI K, E) , q

(1.8.7)

(1.8.8)

VIII.1 Traces

335

to be the closed linear subspace of Y s (m p ~ ,q)/ωi ∂Fqi i (∂i K, E) i∈I

consisting of all (gi )i∈I with gi = (gi0 , . . . , gimi ) satisfying the compatibility conditions (1.8.7) whenever the relations (1.8.6) apply. Clearly, ¡ ¢ p ~I s(m ~ I ,q)/ω ~γIm ∈ L Fs/ν (dI K, E) . (1.8.9) q (K, E), ∂Fq p ~ J ,q)/ω If I = J, then ∂Fs(m is called essential boundary trace space of order m ~ J. The Retraction Theorem for Corners Now we can prove the main result of this subsection, a retraction theorem for corners. It applies, in particular, to the essential boundary. 1.8.5 Theorem Suppose that either

1 < q < ∞ and F = W

or

q = ∞ and F ∈ {B, b} .

Recall that ∂F = B if F = B, and ∂F = b in all other cases. Let s > 0 and mi ∈ N satisfy ωi (mi + 1/q) < s < ωi (mi + 1 + 1/q) ,

i∈I .

(1.8.10)

2 Also assume that, given (i, j) ∈ I6= ,

s 6= ωi (µi + 1/q) + ωj (µj + 1/q)

(1.8.11)

~I for µρ ∈ N with µρ ≤ mρ , ρ ∈ {i, j}. Then ~γIm is a retraction

from

Fs/ν q (K, E)

onto

p m ~ I ,q)/ω ∂Fs( (dI K, E) . q

It has a coretraction which is universal subject to (1.8.10) and (1.8.11). Proof

Due to (1.8.9), it suffices to construct a universal coretraction.

We can (and do) assume that I = {0, . . . , k} so that K = (R+ )k × L, where L is a corner of type (J \I) in Rd−k . Then we proceed by induction on k. p s(m ~ ,q)/ω (1) Suppose k = 2. Let (g1 , g2 ) ∈ ∂Fq I (dI K) be given. Thus p

s (m ~ i ,q)/ωi

gi ∈ ∂Fqi

(∂i K) ,

i = 1, 2 ,

and (g1 , g2 ) satisfies the compatibility conditions (1.8.6) and (1.8.7).

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VIII Traces and Boundary Operators

By applying Theorem 1.8.1, we set u1 := (~γ1m1 )c g1 ∈ Fs/ν q (K) . Then

p

s (m ~ 2 ,q)/ω2

h2 := g2 − ~γ2m2 u1 ∈ ∂Fq2 +

(1.8.12)

(∂2 K) ,

+

where ∂2 K = R × {0} × L = b R × L. Hence p

s (i,q)/ω2

hi2 = g2i − γ2i u1 ∈ ∂Fq2

(R+ × L)

for 0 ≤ i ≤ m2 . Thus γ1j hi2 = γ1j g2i − γ1j γ2i (~γ1m1 )c g1 = γ1j g2i − γ2i γ1j (~γ1m1 )c g1 = γ1j g2i − γ2i g1j = 0

(1.8.13)

for 0 ≤ j ≤ m1 with ω1 (j + 1/q) < s2 (i, q) = s − ω2 (i + 1/q) . Here we used Lemma 1.8.4 and the compatibility conditions (1.8.6). It follows from assumption (1.8.11) that s2p(i, q) ∈ / ω1 (N + 1/q). Hence (1.8.13) and Theorem 1.8.3 ¢ s (i,q)/ω2 ¡ imply that hi2 ∈ ∂Fq2 (0, ∞) × L for 0 ≤ i ≤ m2 , that is, p

s (m ~ 2 ,q)/ω2 ¡

h2 ∈ ∂Fq2

¢ (0, ∞) × L . s/ν ¡

It is a consequence of (1.8.10) and Theorem 1.8.3 that Fq s/ν a closed linear subspace of Fq (K). In fact,

¢ (0, ∞) × R+ × L is

¡ ¢ s/ν Fs/ν (0, ∞) × R+ × L = Fq,~γ m1 (K) . q

(1.8.14)

¯ ¡ ¢ ~ m2 := ~γ m2 ¯ Fs/ν (0, ∞) × R+ × L , B q 2 2

(1.8.15)

1

Hence, setting

we deduce from Theorem 1.8.1 that ³ p ¡ ¢ ¢´ s2 ( m ~ 2 ,q)/ω2 ¡ + ~ m2 ∈ L Fs/ν B (0, ∞) × R × L , ∂F (0, ∞) × L q q 2 ~ m 2 )c . and that this map is a retraction possessing a universal coretraction ( B 2 Now we put ¡ ¢ ~ m2 )c h2 ∈ Fs/ν (0, ∞) × R+ × L . u2 := (B q 2 Then

u := u1 + u2 ∈ Fs/ν q (K) .

VIII.1 Traces

337

Moreover,

~γ1m1 u = ~γ1m1 u1 + ~γ1m1 u2 = g1

by (1.8.12) and (1.8.14). Furthermore, ~ m2 )c (g2 − ~γ m2 u1 ) = g2 , ~γ2m2 u = ~γ2m2 u1 + ~γ2m2 u2 = ~γ2m2 u1 + ~γ2m2 (B 2 2 due to (1.8.15). The map ¡ ¢ ~ m2 )c g2 − ~γ m2 (~γ m1 )c g1 (g1 , g2 ) 7→ u = (~γ1m1 )c g1 + (B 2 2 1 p (m ~ ,q)/ω s/ν is continuous and linear from ∂Fq I (dI K) into Fq (K), and it is a coretracm ~ tion for ~γdI K . It is obviously universal. This proves the theorem if k = 2. (2) Suppose k = 3. The following arguments are analogous to the ones of the preceding step. Thus we can be brief. We set ¯ ¡ ¢ ~ m3 := ~γ m3 ¯ Fs/ν B (0, ∞)2 × R+ × L . q 3 3 It is a retraction from ¡ ¢ © ª Fs/ν (0, ∞)2 × R+ × L = u ∈ Fs/ν γimi u = 0, i = 1, 2 q q (K) ; ~ onto

p

s (m ~ 3 ,q)/ω3 ¡

∂Fq3

¢ (0, ∞)2 × L .

Let u1 and u2 be as in step (1). Set p

s (m ~ 3 ,q)/ω3

h3 := g3 − ~γ3m3 (u1 + u2 ) ∈ ∂Fq3

(∂3 K) .

The compatibility conditions imply, similarly as in step (1), p ¢ s (m ~ ,q)/ω3 ¡ h3 ∈ ∂Fq3 3 (0, ∞)2 × L . ~ m3 )c a universal coretraction for B ~ m3 , we put Denoting by (B 3 3 ¡ ¢ ~ m3 )c h3 ∈ Fs/ν (0, ∞)2 × R+ × L . u3 := (B q 3 Hence ~γimi u3 = 0 for i = 1, 2. s/ν

We define u := u1 + u2 + u3 ∈ Fq (K). Then ~γimi u = ~γimi (u1 + u2 ) = gi , Moreover,

i = 1, 2 .

~ m3 )h3 ~γ3m3 u = ~γ3m3 (u1 + u2 ) + ~γ3m3 (B 3 = ~γ3m3 (u1 + u2 ) + h3 = g3 .

This proves the theorem if k = 3. The general case is now clear.

¥

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VIII Traces and Boundary Operators

1.8.6 Example (Parabolic weight systems) 2-corner K = Hm × R+ in Rm+1 , so that

q Let m ∈ N and consider the closed

∂1 K = {0} × Rm−1 × R+ = b Rm−1 × R+ and ∂m+1 K = Hm × {0} = b Hm . q q Thus dK = ∂1 K ∪ ∂m+1 K. We denote by x = (y, x ) with x ∈ Rm−1 the general m point of H and by (x, t) the one of K. £ ¤ Suppose ν ∈ N with ν ≥ 2 and ω := (1m , ν) so that [m, ν] = 2, (m, 1), (1, ν) is a reduced parabolic weight system on Rm+1 (cf. Example VI.3.1.1). Note that q ω1 = (1m−1 , ν) and

q ωm+1 = 1m .

In the following, we explicate the compatibility conditions (1.8.7) for small values of s. (a) (Sobolev–Slobodeckii spaces) s > ν/p

Let 1 < p < ∞ and with

s, s/ν ∈ / N + 1/p .

Set m1 := [s − 1/p] ,

m2 := [s/ν − 1] .

Assume g1i ∈ Wp(s−i−1/p)/(1,ν) (Rm−1 × R+ , E) , and

g2j ∈ Wps−j−1/p (Hm , E) ,

0 ≤ i ≤ m1 ,

0 ≤ j ≤ m2 .

Suppose CC(s, p, ν)

∂yi g2j (0, ·) = ∂tj g1j (·, 0) if

ν+1 0 with aa∗ ≥ α (2.1.12) of L(E, F ). (2) Let x, ξ ∈ ∂H. Then 4ξ (uv)(x) = 4ξ u(x)v(x + ξ) + u(x)4ξ v(x) implies 42ξ (uv)(x) = 42ξ u(x)v(x + 2ξ) + 24ξ u(x)4ξ v(x + ξ) + u(x)42ξ v(x) whenever u and on ∂H with values in some algebra. Similarly, we ¡ v are functions ¢ get from 4ξ u−1 (x)u(x) = 0 4ξ u−1 (x) = −u−1 (x)4ξ u(x)u−1 (x + ξ) and, consequently, 42ξ u−1 (x) = u−1 (x)4ξ u(x)u−1 (x + ξ)4ξ u(x + ξ)u−1 (x + 2ξ) − u−1 (x)42ξ u(x)u−1 (x + 2ξ) + u−1 (x)4ξ u(x)u−1 (x + ξ)4ξ u(x + ξ)u−1 (x + 2ξ) if u maps into the group of invertible elements. p ¢ ρ/ν ¡ (3) Let b ∈ B∞ ∂H, L(E, F ) satisfy (2.1.7). Set a := bb∗ and α := β 2 . Also put bc := b∗ a−1 . Then it follows from step (2) and the fact that ¡ ¢∗ b(x) 7→ b∗ (x) = b(x) are isometric maps for x ∈ ∂H that |4ξ bc (x)|L(F,E) ≤ c(α−1 , kbk∞ ) |4ξ b(x)|L(E,F ) ,

x, ξ ∈ ∂H .

This and (2.1.11) imply, in particular, ¡ ¢ bc ∈ BUC ∂H, L(F, E) , kbc k∞ ≤ c(α−1 , kbk∞ ) .

(2.1.13)

(2.1.14)

Furthermore, if 0 < θ < 1, we get (recall (VII.3.4.2)) [bc ]θ,∞;j ≤ c(α−1 , kbk∞ ) [b]θ,∞;j ,

2≤j≤d.

(2.1.15)

Similarly, k42ξ bc k∞ ≤ c(α−1 , kbk∞ ) (k42ξ bk∞ + k4ξ bk2∞ )

(2.1.16)

VIII.2 Boundary Operators

347

so that ¡ ¢ [bc ]1,∞;j ≤ c(α−1 , kbk∞ ) [b]1,∞;j + [b]21/2,∞;j ,

2≤j≤d.

(2.1.17)

¡ ¢ (4) Suppose 0 < ρ/ωj ≤ 1 and set X := BUC Rd−2 , L(E, F ) . Then we obtain from (2.1.14)–(2.1.17) that ¡ j ¢ ρ/ωj x 7→ bc (xj ; ·) ∈ B∞ (R, X ) (2.1.18) and the norm of this map is bounded by a constant depending on the norm of ¡ j ¢ ρ/ωj x 7→ b(xj ; ·) ∈ B∞ (R, X ) , (2.1.19) It is clear that (2.1.13) implies [bc ]δθ,∞;j ≤ c(α−1 , kbk∞ ) [b]δθ,∞;j ,

2≤j≤d,

Similarly, we get from (2.1.17) ¡ ¡ ¢2 ¢ [bc ]δ1,∞;j ≤ c(α−1 , kbk∞ ) [b]δ1,∞;j + [b]δ1/2,∞;j ,

δ>0.

2≤j≤d,

δ>0.

1/2

Since b1∞ (R, X ) ,→ b∞ (R, X ), it follows from Theorem VII.3.7.1 that [bc ]δ1,∞;j → 0 ρ/ω

j as δ → 0 if b ∈ b∞ too. 1 (R, X ). This shows that (2.1.18) holds for b∞ q ¡ j ¢ j k (5) Now we assume k ∈ N and x 7→ b(x ; ·) ∈ BUC (R, X ). Then it is a consequence of Leibniz’ rule and Lemma VI.3.3.7 that ¡ j ¢ x 7→ bc (xj ; ·) ∈ BUC k (R, X ) .

From this, step (4), and Theorems VII.3.6.2 and VII.3.7.1 it follows that p ¡ ¢ ¢ p¡ ρ/ν b ∈ b∞,surj ∂H, L(E, F ) =⇒ bc ∈ bρ/ν ∂H, L(F, E) ∞ for ρ > 0. (6) Given b0 satisfying (2.1.9), we see from (2.1.12) and steps (3) and (4) that p ¢ ρ/ν ¡ there exists an open neighborhood U of b0 in b∞ ∂H, L(E, F ) such that bc is p ¡ ¢ ρ/ν well-defined for b in U and bc belongs to b∞ ∂H, L(F, E) . Furthermore, the map (2.1.10) is continuously differentiable, hence analytic. This proves the lemma. ¥

2.2

Systems of Boundary Operators

In this subsection we consider systems of boundary operators of different order. We suppose •

0 ≤ m0 < m1 · · · < mk are integers for some k ∈ N.

•

F0 , . . . , Fk are Banach spaces.

• B mi is a normal boundary operator of type (s, mi , Fi , q).

348

VIII Traces and Boundary Operators

Then m := (m0 , . . . , mk ) is said to be an order sequence (of length k + 1), and F := (F0 , . . . , Fk ) is a sequence of boundary range spaces. Moreover, B := (B m0 , . . . , B mk ) is a normal boundary operator of type (s, m, F , q) on ∂H. Note that this implies s > ν1 (mk + 1/q) . If Fi = F for 0 ≤ i ≤ k, then we simply write F instead of F . For abbreviation, k Y p p i ,q)/ν ∂Fs(m,q)/ν (∂H, F ) := ∂Fs(m (∂H, Fi ) . q q i=0

The Boundary Operator Retraction Theorem 2.2.1 Theorem Let F satisfy (2.0.1). Suppose B is a normal boundary operator of type (s, m, F , q). Then B is a retraction from p s(m,q)/ν Fs/ν (∂H, F ) . q (H, E) onto ∂Fq There exists a coretraction B c satisfying j ∂n ◦ Bc = 0 ,

0 ≤ j < mk ,

j∈ / {m0 , . . . , mk−1 } .

It is universal with respect to s and q, subject to condition (2.1.1). Proof

Lemma 2.1.1 implies that B is a continuous linear map. We write B mi =

mi X

j bi,j ∂n ,

0≤i≤k ,

j=0

and fix a coretraction bci,mi satisfying (2.1.6) for bi,mi . Then, recall part (iii) of Corollary VII.6.3.4, C mi := −

m i −1 X

j bci,mi bi,j ∂n ,

C 0 := 0 ,

j=0

p s/ν s(m ,q)/ν is a continuous linear map from Fq (H, E) into ∂Fq i (∂H, E). We introduce mk ³ ´ Y p s(j,q)/ν C := (C 0 , . . . , C mk ) ∈ L Fs/ν (H, E), ∂F (∂H, E) q q j=0

by setting C j := 0 for 0 ≤ j < mk with j ∈ / {m0 , . . . , mk−1 }.

VIII.2 Boundary Operators

349 s(m,q)/ν

Suppose g = (g m0 , . . . , g mk ) ∈ ∂Fq

h = (h0 , . . . , hmk ) ∈

mk Y

p

(∂H, F ). Define

p ∂Fs(j,q)/ν (∂H, E) q

j=0

by hmi := bci,mi g mi for 0 ≤ i ≤ k, and hj := 0 otherwise. for

We deduce from Theorem 1.3.2 that there exists a universal coretraction γjc satisfying

j ∂n

` ∂n ◦ γjc = δj` id ,

j 6= ` .

(2.2.1)

s/ν

Let u0 := γ0c h0 ∈ Fq (H, E) and u−1 := 0. Suppose u0 , u1 , . . . , uj−1 have already been defined, where 1 ≤ j ≤ mk . Set j uj := uj−1 + γjc (hj + C j uj−1 − ∂n uj−1 ) .

(2.2.2)

s/ν

This results in u0 , u1 , . . . , umk ∈ Fq (H, E). It follows from (2.2.1) that j ∂n uj = hj + C j uj−1 ,

0 ≤ j ≤ mk ,

(2.2.3)

0 ≤ i < j ≤ mk .

(2.2.4)

and i i ∂n uj = ∂ n uj−1 ,

Suppose 0 ≤ ` ≤ j ≤ mk . Then we infer from (2.2.3) and (2.2.4) that ` ` ` ∂n u j = ∂n uj−1 = · · · = ∂n u` = h` + C ` u`−1 .

(2.2.5)

If, moreover, ` = mi ∈ {m0 , . . . , mk }, then (2.2.4) implies C ` u`−1 = −

`−1 X

j bci,mi bi,j ∂n u`−1

j=0

=−

`−1 X

j bci,mi bi,j ∂n u` = C ` u` = · · · = C ` uj .

j=0

Consequently, mi ∂n uj = hmi + C mi uj ,

mi ≤ j ≤ m k ,

1≤i≤k .

By multiplying these equations from the left by bi,mi , we obtain B m i uj = g m i ,

mi ≤ j ≤ mk .

(2.2.6)

350

VIII Traces and Boundary Operators

We define B mi ,c ∈ L

i ³Y

´ p s/ν j ,q)/ν ∂Fs(m (∂H, F ), F (H, E) j q q

j=0

by B

mi ,c

(g

m0

,...,g

mi

) := umi for 0 ≤ i ≤ k. Then, for 0 ≤ α ≤ i ≤ j ≤ k,

B mα B mi ,c (g m0 , . . . , g mi ) = B mα umi = B mα umj = B mα B mj ,c (g m0 , . . . , g mj ) = g mα , due to (2.2.6). Finally, we set B c := B mk ,c . Then B c is a coretraction for B. If `∈ / {m0 , . . . , mk }, then it follows from h` = 0, C ` = 0, and (2.2.5) that ` mk ,c m0 ` ∂n B (g , . . . , g mk ) = ∂n umk = 0 .

The asserted universality with respect to s, q, E, and F is clear from these arguments. This proves the theorem. ¥ s/ν

2.2.2 Remark An analogous result applies to the general Besov spaces Bq,r s/ν and bq,r as well as to Bessel potential spaces. ¥ Embeddings with Boundary Conditions

On the basis of the preceding results we can easily extend Theorem 1.6.7 to general boundary conditions. 2.2.3 Theorem Let 1 < p < ∞ and suppose B is a normal boundary operator of type (s, m, F , p). (i) Assume ν1 (mk + 1/p) < σ ≤ s. Then d

σ/ν

σ/ν

σ/ν

Bp,1,B (H, E) ,→ Hp,B (H, E) ,→ Bp,∞,B (H, E) . (i) If ν1 (mk + 1/p) < s0 < σ < s1 ≤ s, then s /ν

d

σ/ν

1 Bp,B (H, E) ,→ Bp,1,B (H, E)

and

d

σ/ν

s /ν

0 Bp,∞,B (H, E) ,→ Bp,B (H, E) .

Proof (1) Assertion (i) follows from(1.6.13), the preceding remark, and Lemma 1.6.6, setting (r, rc ) := (B, B c ). (2) We infer from Theorems VII.2.2.2, VII.2.2.4, and VII.2.8.3 that d

σ/ν

Bps1 /ν (H, E) ,→ Bp,1 (H, E) and

d

σ/ν Bp,∞ (H, E) ,→ Bps0 /ν (H, E) .

From these embeddings we get claim (ii) by again applying Lemma 1.6.6 with (r, rc ) = (B, B c ). ¥

VIII.2 Boundary Operators

351

It is an immediate consequence of this theorem that s /ν

d

d

σ/ν

s /ν

1 0 Bp,B (H, E) ,→ Hp,B (H, E) ,→ Bp,B (H, E) ,

provided ν1 (mk + 1/p) < s0 < σ < s1 ≤ s. Our next theorem generalizes these embeddings to cases in which s0 , σ, and s1 may be further apart. 2.2.4 Theorem Let B be a normal boundary operator of type (s, m, F , p), where 1 < p < ∞. Suppose 0 < s0 < σ < s1 < s with © ª s0 , s 1 , σ ∈ / ν1 (mi + 1/p) ; 0 ≤ i ≤ k . Then s /ν

d

d

σ/ν

s /ν

1 0 Bp,B (H, E) ,→ Hp,B (H, E) ,→ Bp,B (H, E) .

Proof We omit (H, E) and let F ∈ {B, H}. (1) It follows from Theorems VII.2.2.2, VII.2.8.3, and VII.4.1.3(i) that d

d

Bps1 /ν ,→ Hpσ/ν ,→ Bps0 /ν . t/ν

(2.2.7)

t/ν

Since Fp,B = Fp if t < ν1 (m0 + 1/p), the assertion is clear if s1 < ν1 (m0 + 1/p). (2) Suppose σ > ν1 (m0 + 1/p). Let `, resp. `1 , be the largest integer ≤ k such that ν1 (m` + 1/p) < σ, resp. ν1 (m`1 + 1/p) < s1 . Set n := (m0 , . . . , m` ) and n1 := (m0 , . . . , m`1 ) as well as G := (F0 , . . . , F` ) and G1 := (F0 , . . . , F`1 ). Then R := (B m0 , . . . , B m` ), resp. R1 := (B m0 , . . . , B m`1 ), is a normal boundary operator of type (s, n, G, p), resp. (s, n1 , G1 , p). Thus Theorem 2.2.1 (also recall Remark 2.2.2) guarantees that R1 is a retraction p from Fsp1 /ν onto ∂Fsp1 (n1 ,p)/ν (∂H, G1 ) possessing a universal coretraction Rc1 . Assume p g1 := (g 0 , . . . , g `1 ) ∈ ∂Fsp1 (n1 ,p)/ν (∂H, G1 ) . Then p p g := (g 0 , . . . , g ` ) ∈ ∂Fsp1 (n,p)/ν (∂H, G) ,→ ∂Fσ(n,p)/ν (∂H, G) . p

(2.2.8)

Moreover, B mi Rc1 (g 0 , . . . , g `1 ) = g mi ,

0≤i≤`.

(2.2.9)

Rc (g 0 , . . . , g ` ) := Rc1 (g 0 , . . . , g ` , 0, . . . , 0) .

(2.2.10)

Set

352

VIII Traces and Boundary Operators

Then the construction of Rc1 in the proof of Theorem 2.2.1 shows that ¡ ¢ p Rc ∈ L ∂Fσ(n,p)/ν (∂H, G), Fσ/ν . p p

(2.2.11)

c It follows from the definition of R, R p 1 , and ¢(2.2.8)–(2.2.11) that (R, R ) is a ¡ σ/ν σ(n,p)/ν universal r-c pair for Fp , ∂Fp (∂H, G) . Also note that s /ν

s /ν

1 1 Fp,B = Fp,R 1

and

σ/ν

σ/ν

Fp,B = Fp,R . σ/ν

σ/ν

(3) Let σ > ν1 (m0 + 1/p). Suppose u ∈ Hp,B ,→ Hp s /ν

that there exists a sequence (uj ) in Bp1 vj := (1 − Rc1 R1 )uj ,

(2.2.12) . We infer from (2.2.7)

σ/ν

converging in Hp

g1,j := R1 uj ,

towards u. Set

gj := Ruj .

Then, given 0 ≤ i ≤ `, it is a consequence of (2.2.9), (2.2.10), the definition of (R, Rc ), and (2.2.12) that B mi vj = B mi uj − B mi Rc1 g1,j = B mi uj − B mi Rc gj = B mi (1 − Rc R)uj = 0 s /ν

for j ∈ N. From this and Bp1

σ/ν

,→ Hp

σ/ν

we deduce that vj ∈ Hp,B . Hence

vj − u = (1 − Rc R)(uj − u) → 0

in Hpσ/ν .

This proves s /ν

d

σ/ν

1 Bp,B ,→ Hp,B .

(4) Suppose s0 > ν1 (m0 + 1/p). Then we obtain d

σ/ν

s /ν

0 Hp,B ,→ Bp,B

by interchanging the roles of B and H in step (3). (5) Assume σ > ν1 (m0 + 1/p) > s0 . Let i ≤ k be the largest integer such that ν1 (mi + 1/p) < σ and set ` := mi . Then σ/ν

σ/ν

Hp,~γ ` ,→ Hp,B ,→ Bps0 /ν .

(2.2.13)

σ/ν . ˚ σ/ν We know from Theorem 1.6.8(i) that Hp,~γ ` = H , and Theorem 1.6.2(ii) guaranp s0 /ν s0 /ν . ˚ s0 /ν σ/ν s /ν ˚ tees that Bp = Wp = Wp . Hence D(H, E) is dense in H ` and in Bp0 .

Thus, since

σ/ν Hp,~γ `

,→

s /ν Bp 0

duce from (2.2.13) that

by (2.2.13), we see that

σ/ν Hp,B

d

,→

σ/ν Hp,~γ `

d

,→

p,~ γ s0 /ν Bp .

Now we de-

s /ν Bp 0 . σ/ν

s /ν

(6) Finally, let s1 > ν1 (m0 + 1/p) > σ. By replacing (Hp,B , Bp0 preceding step by

s1 /ν σ/ν (Bp,B , Hp ), s /ν

) in the

we see that d

d

1 Bp,B ,→ Hpσ/ν ,→ Bps0 /ν ,

where the last dense embedding is obtained from (2.2.7). The theorem is proved.

¥

VIII.2 Boundary Operators

2.3

353

Transmission Operators

We denote by S the hyperplane {0} × Rd−1 in Rd , oriented by the positive normal nS = n := e1 = (1, 0, . . . , 0). If it is convenient and clear from the context, then we identify S naturally with Rd−1 . By ρS we mean the reflection on S, ρS x := (−x1 , x2 , . . . , xd ) ,

x ∈ Rd .

(2.3.1)

This defines an operation of the two element group {±1} on D0 (Rd , E) by setting ρS u := u ◦ ρS for u ∈ L1,loc (Rd , E), and hρS u, ϕiRd := hu, ρS ϕiRd ,

u ∈ D0 (Rd , E) ,

ϕ ∈ D(Rd ) .

The closed negative half-space H− of Rd is defined by H− := ρS H. Then H+ := H and H− are two adjacent closed half-spaces of Rd whose common boundary is the interface S = ∂H+ = ∂H− between H+ and H− . It is clear that the entire theory of function spaces on H, the trace theorems in particular, developed so far, can equally well be established by replacing H by H− . Then ¡ ¢ ρS ∈ Lis S 0 (H− , E), S 0 (H+ , E) , ρ−1 S = ρS . Furthermore, this operator restricts to an idempotent isometric isomorphism ρS : G(H− , E) → G(H+ , E) , if G is any one of the Banach subspaces of S 0 introduced in the preceding sections.

(2.3.2)

Unless stated otherwise, we suppose •

s > ν1 (m + 1/q) for some m ∈ N.

•

F is a Banach space. s/ν

Setting X := Lq (Rd−1 , E), we know that8 Fq (H± , E) ,→ C m (±R+ , X ). Suppose v± ∈ C m (±R+ , X ). Then v± (0) = lim v± (±t) = lim v± (t) t→0+

t→0±

is the trace of v+ , resp. v− , from the right, resp. left, at {0} = ∂R+ = ∂(−R+ ). We set γ± u± = lim u(y, ·) , u± ∈ Fs/ν q (H± , E) . y→0±

s/ν

Hence γ+ u+ = γu+ for u+ ∈ Fq (H+ , E). Note that γ− u− = lim u− (y, ·) = lim u− (−z, ·) = lim (ρS u− )(z, ·) . y→0−

8 Here

z→0+

z→0+

and in similar situations, everywhere either the upper or the lower sign has to be chosen.

354

VIII Traces and Boundary Operators

Consequently, γ− = γ ◦ ρS . On the basis of this observation we define the trace operator of order m on H− by m γ− := γ m ◦ ρS . Let X be open in Rd and F a Banach space. Given x ∈ X and e ∈ Rd , ∂v v(x + τ e) − v(x) (x) := lim ∈F τ →0+ ∂e τ is the directional derivative of v ∈ C 1 (X, F ) at x in the direction of e. q s/ν ˚− . Then Suppose m ≥ 1, u− ∈ Fq (H− , E), and x = (y, x ) ∈ H q q u− (y + h, x ) − u− (x) u− (y + τ, x ) − u− (x) = lim τ →0+ h→0 h τ u− (x + τ n) − u− (x) ∂u− = lim = (x) τ →0+ τ ∂n

∂1 u− (x) = lim

q q by the continuity of ∂1 u− (·, x ) (for a.a. x ∈ ∂H− ). Hence ∂1 u− (0, ·) = lim ∂1 u− (y, ·) = y→0−

By definition,

∂u− ¯¯ =: ∂n u− . ¯ ∂n y=0

(2.3.3)

¡ ¢ 1 γ− u− = γ 1 (ρS u− ) = lim ∂1 ρS u− (t, ·) t→0+

= − lim ∂1 u− (−t, ·) = −∂1 u− (0, ·) . t→0+

From this and (2.3.3) it follows that 1 γ− u− = −

∂u− ¯¯ = −∂n u− , ¯ ∂n x1 =0

(2.3.4)

that is, the trace operator of order 1 on H− is the negative of the derivative on ∂H− with respect to the exterior normal n of H− . In general, we see from m γ− u− = γ m ρS u− = ∂1m (ρS u− )|x1 =0 m = (−1)m ∂1m u− |x1 =0 = (−1)m ∂n u−

that m m γ− = (−1)m ∂n .

Let F satisfy (2.0.1). For abbreviation, we set s/ν s/ν Fs/ν q (H+ , H− ; E) := Fq (H+ , E) ⊕ Fq (H− , E)

(2.3.5)

VIII.2 Boundary Operators

355

and denote its general point by u = u+ ⊕ u− . We suppose that m C± :=

m X

j c± j γ

j=0

are boundary operators of type (s, m, F, q) on ∂H. Setting m m C m u := C+ u+ − C − ρS u − ,

u = u+ ⊕ u− ∈ Fs/ν q (H+ , H− ; F ) ,

we infer from Lemma 2.1.1 and (2.3.2) that ¡ ¢ p s(m,q)/ν C m ∈ L Fs/ν (S, F ) . q (H+ , H− ; E), ∂Fq We say that C m is a transmission operator across S (in the direction of n). It is of m m order m if C+ and C− are of order m. 0 2.3.1 Examples (a) (m = 0) Here C± = c± γ and C 0 u = c+ γ+ u+ − c− γ− u− . In particular, if F = E and c+ = c− = 1E , then ¡ ¢ C 0 u = [[u]] := γ+ u+ − γ− u− = lim u+ (t, ·) − u− (−t, ·) , t→0+

the jump of u across S (in the direction of n). (b) (m = 1)

Now − 1 + − 1 C 1 u = c+ 1 γ + u+ − c 1 γ − u− + c 0 γ + u+ − c 0 γ − u− .

± Suppose F = E, c± 1 = ±1E and c0 = 0. Then

C 1 u = ∂n u + − ∂n u − is the jump, [[∂n u]], of the ‘normal derivative’ of u across S.

¥

To C m we associate a boundary operator, B(C m ), on ∂H, acting on E 2 -valued functions, as follows. We define an isomorphism ¡ ¢ s/ν 2 T ∈ Lis Fs/ν (2.3.6) q (H+ , H− ; E), Fq (H, E ) by setting T (u+ ⊕ u− ) := (u+ , ρS u− ). Then B(C m ) := C m ◦ T −1 . Hence

¡ ¢ p 2 s(m,q)/ν B(C m ) ∈ L Fs/ν (∂H, F ) . q (H, E ), ∂Fq

Given9 u = (u1 , u2 ) = b 9 Whenever

hu i 1

u2

¡ ¢2 2 ∈ Fs/ν b Fs/ν , q (H, E ) = q (H, E)

convenient, we use standard matrix notation.

(2.3.7)

356

VIII Traces and Boundary Operators

we infer from (2.3.6) that m m B(C m )u = C+ u1 − C − u2 =

m X

− j j (c+ j ∂n u 1 − c j ∂ n u 2 )

j=0

=

m X

− j [c+ j , −cj ] ∂n

j=0

hu i 1

u2

=

m X

j bj (C m )∂n u,

j=0

− where bj (C m ) := [c+ j , −cj ].

2.3.2 Lemma B(C m ) is a boundary operator of type (s, m, F, q) on 10 ∂H. It is m m normal if C+ and C− are normal. Proof The first claim is obvious. q m m Suppose C+ and C− are normal. Thus, given x ∈ S, there exist continuq q q q c − ous linear right inverses bcm,1 (x ) for c+ m (x ) and bm,2 (x ) for −cm (x ). We de¡ ¢± q q q q fine bcm (x ) ∈ L(F, E 2 ) by bcm (x )f := bcm,1 (x )f, bcm,2 (x )f 2 for f ∈ F . Then q c q bm (x )bm (x )f = f . This implies the second claim. ¥ Henceforth, C m is said to be a normal transmission operator (across S) of type (s, m, F, q) if B(C m ) is a normal boundary operator on ∂H. Now we can prove a retraction theorem which corresponds to Theorem 2.2.1. For this we suppose • m = (m0 , . . . , mk ) is an order sequence. • F := (F0 , . . . , Fk ) is a sequence of Banach spaces. • s > ν1 (mk + 1/q). •

(2.3.8)

C mi is a normal transmission operator of type (s, mi , Fi , q).

Then C := (C m0 , . . . , C mk ) is said to be a normal transmission operator across S) of type (s, m, F , q). 2.3.3 Theorem Let F satisfy (2.0.1). Suppose C is a normal transmission ops/ν erator of type (s, m, F , q). Then C is a retraction from Fq (H+ , H− ; E) onto p s(m,q)/ν ∂Fq (S, F ). ¡ ¢ Proof The preceding lemma guarantees that B(C) := B(C m0 ), . . . , B(C mk ) is a boundary operator of type (s, m, F , q) on ∂H acting on E 2 -valued functions. By assumption, it is normal. Hence, by Theorem 2.2.1, there exists a continuous right inverse B c (C) for it. By (2.3.6), T −1 ◦ B c (C) is then a continuous right inverse for C. ¥ 10 Of

course, S is identified with ∂H.

VIII.2 Boundary Operators

357

Patching Together Half-Spaces ˚+, ˚ We know that (R+ , E+ ) := (RH , EH ) and (R E+ ) := (R˚ , E˚) are r-e pairs for ¡ s/ν d ¢ ¡ s/ν d ¢ H H s/ν s/ν ˚ Fq (R , E), Fq (H, E) and Fq (R , E), Fq (H, E) , respectively. By replac˚ 1 , and ˚ ing H = H+ by H− in the definition of R1 , E1 , R E1 in Subsection VI.1.1, ˚− , ˚ we obtain r-e pairs (R− , E− ) and (R E− ) for ¡ s/ν d ¢ ¡ ¢ d s/ν ˚ Fq (R , E), Fs/ν and Fs/ν q (H− , E) q (R , E), Fq (H− , E) , ˚+, ˚ possessing properties completely analogous to the ones of (R+ , E+ ) and (R E+ ), s/ν d respectively. These facts will now be used to characterize u ∈ Fq (R , E) by means of its restrictions R± u to H± . Generalizing the notation introduced in Examples 2.3.1, we define the jump, m [[∂n u]], of the m-th order normal derivative across S by m m m [[∂n u]] := ∂n u + − ∂n u− ,

u = u+ ⊕ u− ∈ Fs/ν q (H+ , H− ; E) ,

provided s > ν1 (m + 1/q). Then, given s > 0, we denote by ˚ Fs/ν q (H+ , H− ; E) s/ν

the set of all u = u+ ⊕ u− in Fq (H+ , H− ; E) satisfying j [[∂n u]] = 0 ,

j∈N,

ν1 (j + 1/q) < s .

(2.3.9)

s/ν

Clearly, it is a closed linear subspace of Fq (H+ , H− ; E), and s/ν ˚ Fs/ν q (H+ , H− ; E) = Fq (H+ , H− ; E) ,

s < ν1 /q ,

(2.3.10)

since condition (2.3.9) is then void. 2.3.4 Theorem Let F satisfy (2.0.1). Suppose s > 0 and s ∈ / ν1 (N + 1/q). Then d ˚s/ν Fs/ν q (R , E) → Fq (H+ , H− ; E) ,

u 7→ R+ u ⊕ R− u

(2.3.11)

is a toplinear isomorphism. Proof (1) It follows from the continuity properties of R± that u 7→ R+ u ⊕ R− u s/ν s/ν is a continuous linear map from Fq (Rd , E) into Fq (H+ , H− ; E). Suppose m ∈ N with ν1 (m + 1/q) < s < ν1 (m + 1 + 1/q) . s/ν

(2.3.12)

j Recall Fq (Rd , E) ,→ C m (R, X ), where X := Lq (Rd−1 , E). Clearly, [[∂n v]] = 0 for m 0 ≤ j ≤ m if v ∈ C (R, X ). This implies, together with (2.3.9), that (2.3.11) is well-defined and continuous.

358

VIII Traces and Boundary Operators

s/ν (2) Let (2.3.12) be satisfied. For v := v+ ⊕ v− in ˚ Fq (H+ , H− ; E) we put s/ν w := v+ − R+ E− v− . Then w ∈ Fq (H+ , E). Since R+ is the operator of pointwise restriction, j j j ∂n w = ∂n v+ − ∂n E− v − ,

0≤j≤m.

s/ν

From E− v− ∈ Fq (Rd , E) we infer j j ∂n E− v− = ∂n v− . s/ν

j ˚+ , E) by Theorems 1.6.2(i) and Hence ∂n w = 0 for 0 ≤ j ≤ m, that is, w ∈ Fq (H s/ν ˚ 1.6.9. Therefore u := E− v− + E+ w belongs to Fq (Rd , E). Note that

R+ u = R+ E− v− + R+˚ E+ w = v + and R− u = R− E− v− = v− . Consequently, the map v 7→ u is a continuous right inverse for (2.3.11). Since the latter is clearly injective, the assertion is proved in this case. . s/ν s/ν ˚ (3) Assume 0 < s < ν1 /q. Then Wq (H± , E) = Wq (H ± , E), due to Thes/ν orem 1.6.2(ii). Thus, given v = v+ ⊕ v− ∈ Fq (H+ , H− ; E), d u := ˚ E+ v + + ˚ E− v− ∈ Fs/ν q (R , E)

and R± u = v± . This proves that v 7→ u is a continuous right inverse for the injective map (2.3.11). ¥ 2.3.5 Corollary Suppose ν1 (m + 1/q) < s < ν1 (m + 1 + 1/q) . Then d u ∈ Fs/ν q (R , E)

⇐ ⇒

j R± u ∈ Fs/ν q (H± , E) and [[∂n u]] = 0 with 0 ≤ j ≤ m . s/ν

If 0 < s < ν1 /q, then u ∈ Wq

2.4

s/ν

(Rd , E) iff R± u ∈ Wq

(H± , E).

Interpolation With Boundary Conditions

In this subsection we introduce spaces with vanishing normal boundary conditions which generalize the spaces with vanishing traces studied in Subsection 1.4. Then we investigate the behavior of these spaces under real and complex interpolation. Preliminaries For the reader’s convenience, we review some facts of interpolation theory of which we make use in the proof of the main theorem. We set S := [0 < Rez < 1] ⊂ C and

VIII.2 Boundary Operators

359

Sx := [Rez = x] for 0 ≤ x ≤ 1. Let E0 and E1 be Banach spaces with E1 ,→ E0 . Recall from Example I.2.4.2 that F (E0 , E1 ) is the Banach space of bounded and . continuous functions from S into E0 + E1 = E0 , which are holomorphic in S and satisfy f | Sj ∈ C0 (Sj , Ej ) for j = 0, 1, endowed with the norm © ª f 7→ kf kF := max supt∈R kf (i t)kE0 , supt∈R kf (1 + i t)kE1 . Then [E0 , E1 ]θ is, for 0 < θ < 1, the image space of the evaluation map F (E0 , E1 ) → E0 ,

f 7→ f (θ) .

Assume f ∈ F (E0 , E1 ) and 0 < ϑ < 1. Set gt (z) := f (z + it) for z ∈ S and t ∈ R. Then gt ∈ F (E0 , E1 ) and gt (ϑ) := f (ϑ + i t). Hence, cf. Remark VI.2.2.1, kf (ϑ + i t)k[E0 ,E1 ]ϑ ≤ c kf kF ,

t∈R.

(2.4.1)

Below we need the following extension of the reiteration theorem (I.2.8.4) to the case where E1 is not necessarily dense in E0 . Recall that [E0 , E1 ][j] := Ej for j = 0, 1. 2.4.1 Lemma Suppose E1 ,→ E0 , θ0 ∈ [0, 1) and θ1 , η ∈ (0, 1). Then £ ¤ [E0 , E1 ]θ0 , [E0 , E1 ]θ1 η = [E0 , E1 ](1−η)θ0 +ηθ1 . ˚0 the closure Proof Let X0 and X1 be Banach spaces with X1 ,→ X0 . Denote by X of X1 in X0 . Then ˚0 , X1 ]ϑ , [X0 , X1 ]ϑ = [X

0 0, or E is reflexive and s0 > ν1 (−1 + 1/q), then £ s0 /ν ¤ . ¡ ¢ s1 /ν s1 /ν Bq (H, E), Bq,B (H, E) θ = Bqs0 /ν (H, E), Bq,B (H, E) θ,q . sθ /ν = Bq,B (H, E) . (ii) Let q = ∞ and s0 > 0. Then £ s0 /ν ¤ . ¡ ¢0 s1 /ν s1 /ν b∞ (H, E), b∞,B (H, E) θ = bs∞0 /ν (H, E), b∞,B (H, E) θ,∞ . sθ /ν = b∞,B (H, E) . Proof

For abbreviation, with F ∈ {B, b}, X t := Ft/ν q (H, E) ,

p Y t(j) := ∂Ft(j,p)/ν (∂H, Fj ) q

for t ∈ R and j ∈ N. First we consider complex interpolation.

VIII.2 Boundary Operators

361

(1) It follows from (2.4.3)(vi) and the continuity of B that XBs1 is a closed linear subspace of X s1 . Hence XBs1 ,→ X s1 implies . [X s0 , XBs1 ]θ ,→ [X s0 , X s1 ]θ = X sθ ,

(2.4.5)

where the second part is implied by Theorems VII.2.7.1(iii) and (v) and VII.2.8.3. (2) Suppose u ∈ [X s0 , XBs1 ]θ . Assume sθ > ν1 (m0 + 1/q) and let i be the largest integer ≤ k such that ν1 (mi + 1/q) < sθ . There exists f ∈ F (X s0 , XBs1 ) with f (θ) = u. We deduce from step (1) and (2.4.1) that the restriction of f to [θ ≤ Rez ≤ 1] is a bounded continuous function with values in X sθ , which is holomorphic on [θ ≤ Rez < 1]. Thus B mj f is for 0 ≤ j ≤ i a bounded continuous function from [θ ≤ Rez ≤ 1] into Y sθ (mj ) , which is holomorphic on [θ ≤ Rez < 1] and vanishes on S1 . Hence, by Lemma 2.4.2, B mj f vanishes identically. Consequently, B mj u = B mj f (θ) = 0 ,

0≤j≤i.

From this and (2.4.5) we get [X s0 , XBs1 ]θ ,→ XBsθ ,

(2.4.6)

provided sθ > ν1 (m0 + 1/q). If sθ < ν1 (m0 + 1/q), then XBsθ = X sθ and (2.4.6) follows from (2.4.5). (3) Let bi,mi be the top-order coefficient of B mi for 0 ≤ i ≤ k. We fix a coretraction bci,mi for bi,mi satisfying (2.1.6). Assume B mi u = 0. Then Bemi u := bci,mi Bu = 0 . Conversely, if Bemi u = 0, then B mi u = bi,mi Bemi u = 0. Hence ker(B mi ) = ker(Bemi ) , We write Bemi =

mi X

ebi,j ∂ j , n

0≤i≤k .

ebi,j := bc bi,j . i,mi

j=0

In particular, ¢ p¡ πmi := ebi,mi ∈ bσ/ν ∂H, L(E) ∞

(2.4.7)

q q and πmi ebi,j = ebi,j for 0 ≤ j ≤ mi . Thus πmi (x ) ∈ L(E) is a projection for x ∈ ∂H, and πmi Bemi = Bemi for mi ∈ M := {m0 , . . . , mk }.

(2.4.8)

362

VIII Traces and Boundary Operators

(4) Assume s1 > ν1 (mk + 1/q) ,

s1 ∈ / ν1 (N + 1/q) .

(2.4.9)

Let ` be the largest integer satisfying ν1 (` + 1/q) < s1 and set ` := (0, 1, . . . , `). Then we define a normal boundary operator of type (s1 , `, E, q), C = (C 0 , . . . , C ` ) : X s1 → Y s1 (`) :=

` Y

Y s1 (j) ,

j=0

by

( j

C :=

j ∂n ,

(1 −

if j ∈ /M , j πj )∂n

ej

⊕B ,

if j ∈ M .

Of course, now Y s(j) is a space of E-valued functions. Note that πj C j = πj Bej = Bej ,

j∈M .

(2.4.10)

Suppose Cv = 0. Let 0 ≤ r ≤ `. Then r C r v = ∂n v=0,

r∈ /M .

(2.4.11)

If r = mi ∈ M , then, by (2.4.7), (2.4.8), and (2.4.10), r

C v = (1 −

r πr )∂n v

r ⊕ πr Ber v = ∂n v+

r−1 X

ebi,j ∂ j v = 0 . n

j=0

From this and (2.4.11) we deduce successively mi C mi v = ∂n v=0,

i = 0, 1, . . . , k .

This and (2.4.11) imply that Cv = 0 iff ~γ ` v = 0. Hence . ˚ E) , XCs1 = X~γs`1 = X s1 (H,

(2.4.12)

where the equivalence is a consequence of (2.4.9), the choice of `, and Theorems 1.6.4 and 1.6.9. Note that Theorem 2.2.1 (and its proof) implies¡the existence ¤of a coretraction C c for C which is universal with respect to s1 in ν1 (` + 1/q), s . (5) Suppose u ∈ XBsθ = XBsθ . Let n be the largest integer such that ν1 (n + 1/q) is strictly smaller than sθ . Define ( j C u, if j ∈ / M, j ≤ n , gj := j (1 − πj )C u , if j ∈ M, j ≤ n .

VIII.2 Boundary Operators

Then

363

. gj ∈ Y sθ (j) = [Y s0 (j) , Y s1 (j) ]θ

by Lemma 2.1.1 and Theorems VII.2.7.1(iii) and (v) and VII.4.5.1. Hence there exists fej ∈ F (Y s0 (j) , Y s1 (j) ) such that fej (θ) = gj for j ≤ n. We put ( fej , if j ∈ /M , fj := (1 − πj )fej , if j ∈ M . Note that fj has the same properties as fej . Let F := C c (f0 , . . . , fn , 0, . . . , 0). Clearly, F ∈ F (X s0 , X s1 ) and v := F (θ) = C c (g0 , . . . , gn , 0, . . . , 0) . Observe that F | S1 ∈ C0 (S1 , X s1 ) satisfies Bej F1 = πj Bej F1 = πj C j C c (f0 , . . . , fn , 0, . . . , 0) | S1 = 0 ,

j∈M ,

due to (2.4.10) and πj fj = πj (1 − πj )fej = 0 for j ∈ M . Thus F ∈ F (X s0 , XBs1 ) and, consequently, v ∈ [X s0 , XBs1 ]θ .

(2.4.13)

If j ∈ M with j ≤ n, then C j (u − v) = C j u − C j C c (g0 , . . . , gn , 0, . . . , 0) = C j u − gj = 0 , since πj C j u = Bej u = 0 implies C j u = (1 − πi )C j u = gj . If j ∈ / M , then C j u = gj . sθ . sθ ˚ Hence u − v ∈ XC = X (H, E) (cf. step (4)). We know from Theorems VII.2.7.1 and VII.2.8.3 that ¤ . £ s0 ˚ ˚ E) = ˚ E) . X sθ (H, X (H, E), X s1 (H, θ It follows from Theorem VII.3.7.7 that ˚ E) ,→ X s0 . X s0 (H,

(2.4.14)

˚ E) ,→ X s1 we obtain From this and the obvious embedding X s1 (H, B ˚ E) ,→ [X s0 , X s1 ]θ . X sθ (H, B Hence u = v + (u − v) ∈ [X s0 , XBs1 ]θ , due to (2.4.13). This and (2.4.6) show that . XBsθ = [X s0 , XBs1 ]θ , provided (2.4.9) is satisfied and sθ > ν1 (m0 + 1/p). Let sθ < ν1 (m0 + 1/q). Then XBsθ = X sθ and it is clear from the above that . sθ X = XBsθ = [X s0 , XBs1 ]θ in this case too, provided (2.4.9) is satisfied.

364

VIII Traces and Boundary Operators

(6) Assume s1 > ν1 (mk + 1/q)

(2.4.15)

and there exists r ∈ N with s1 = ν1 (r + 1/q). Fix t ∈ (s1 , s] with t ∈ / ν1 (N + 1/q) . and set ϑ := (s1 − s0 )/(t − s0 ). Then, by the preceding step, [X s0 , XBt ]ϑ = XBs1 . Hence Lemma 2.4.1 implies ¤ . . £ . [X s0 , XBs1 ]θ = X s0 , [X s0 , XBt ]ϑ θ = [X s0 , XBt ]θϑ = XBsθ , using step (5) once more. This proves that the assertions concerning the complex interpolation functor are valid, provided (2.4.15) applies. ¡ ¤ (7) Now suppose s1 < ν1 (mk + 1/q). Fix t ∈ ν1 (mk + 1/q), s and define ϑ as in the preceding step. Then the arguments used there show that . [X s0 , XBs1 ]θ = XBsθ . Hence the assertions of the theorem involving the complex interpolation functor have been proved. (8) Let ν1 (mk + 1/q) < t0 < t1 ≤ s. Write (·, ·)θ := (·, ·)θ,q if q < ∞, so that t/ν

t/ν

t/ν

0

t/ν

Fq = Bq , and (·, ·)θ := (·, ·)θ,∞ if F∞ = b∞ . Then we know from the interpo. lation results of Section VII.2 that (X t0 , X t1 )θ = X tθ . . Assume u ∈ (X t0 , XBt1 )θ ,→ (X t0 , X t1 )θ = X tθ . Choose f such that (1.6.8) is t0 t1 satisfied with e = u and (E0 , E1 ) := (XB , XB ). Then Bf (t) = 0 implies11 Z ∞ Z ∞ Bu = B f (t) dt/t = Bf (t) dt/t = 0 , 0

that is, u ∈

XBtθ .

(Recall that

0

(E0 , E1 )0θ,∞

is the closure of E1 in (E0 , E1 )θ,∞ .)

(9) From Theorem 2.2.1 we know that B is a retraction from X t onto Y t(m) for ν1 (mk + 1/q) < t ≤ s . It possesses a coretraction B c which is universal with respect to these t. Hence P := 1X t − B c B is a projection from (X t0 , X t1 ) onto (XBt0 , XBt1 ) (cf. Subsection I.2.3). Suppose v ∈ X tθ satisfies Bv = 0. Choose g such that (1.6.8) holds with e = v, f = g, and (E0 , E1 ) = (X t0 , X t1 ). As above, since Bv = 0, Z ∞ v = Pv = Pg(t) dt/t

11 Cf.

¡

0

¢ in X , where Pg ∈ C (0, ∞), X . Thus, see Remark 1.6.3, v ∈ (XBt0 , XBt1 )θ . By combining this with the result of step (8), we obtain . (XBt0 , XBt1 )θ = XBtθ . (2.4.16) t0

the proof of Theorem 1.6.4.

t1

VIII.2 Boundary Operators

365

(10) Choose 0 < ε < min{θ, 1 − θ} such that ¡ ¢ sθ±ε ∈ ν1 (mi + 1/q), ν1 (mi+1 + 1/q) if 0 ≤ i ≤ k − 1 and sθ belongs to this interval. If sθ < ν1 (m0 + 1/q), assume that s0 < sθ±ε < ν1 (m0 + 1/q). Similarly, ν1 (mk + 1/q) < sθ±ε < s1

if

sθ > ν1 (mk + 1/q) .

Then, by the validity of the theorem for the complex interpolation functor, we ob. s tain [X s0 , XBs1 ]θ±ε = XBθ±ε . Thus the reiteration theorem for the real interpolation functor (e.g., [BeL76, Theorem 3.5.3] or [Tri95, Theorem 1.10.2]) implies ¢ . ¡ s s (X s0 , XBs1 )θ = [X s0 , XBs1 ]θ−ε , [XBs0 , XBs1 ]θ+ε 1/2 = (XBθ−ε , XBθ+ε )1/2 . = XBsθ . The last equivalence follows by applying the result of step (9) with mk replaced by mi and with the operator (B m0 , . . . , B mi ), provided s1 > ν1 (m0 + 1/q). Other. wise, (X s0 , XBs1 )θ = (X s0 , X s1 )θ = X sθ = XBsθ by (2.4.4). The theorem is proved. ¥ Generalizations By means of the reiteration theorems it is now not difficult to prove important extensions and complements of the main theorem. We start with complex interpolation results. 2.4.4 Theorem Let (2.4.3) apply. (i) Assume q < ∞. If either s0 > 0 or E is reflexive and s0 > −ν1 (1 − 1/q), then £ s0 /ν ¤ . sθ /ν s1 /ν Bq,B (H, E), Bq,B (H, E) θ = Bq,B (H, E) . (ii) Suppose q = ∞ and s0 > 0. Then £ s0 /ν ¤ . sθ /ν s1 /ν b∞,B (H, E), b∞,B (H, E) θ = b∞,B (H, E) . Proof Define X t as in the preceding proof. Fix σ0 < s0 with σ0 > 0 if s0 > 0, and σ0 > −ν1 (1 − 1/q) otherwise. Then, setting θ0 := (s0 − σ0 )/(s1 − σ0 ), it follows . from Theorem 2.4.3 that XBs0 = [X σ0 , XBs1 ]θ0 . Consequently, ¤ . . £ [XBs0 , XBs1 ]θ = [X σ0 , XBs1 ]θ0 , XBs1 θ = [X σ0 , XBs1 ](1−θ)θ0 +θ (2.4.17) ¡ ¢ by Lemma 2.4.1. Note that σ0 + (1 − θ)θ0 + θ (s1 − σ0 ) = sθ . Hence, by Theorem . 2.4.3, the third term of (2.4.17) equals XBsθ . Thus [XBs0 , XBs1 ]θ = XBsθ . This proves the theorem. ¥ Next we turn to real interpolation and consider general Besov spaces.

366

VIII Traces and Boundary Operators

2.4.5 Theorem Let (2.4.3) be satisfied with q < ∞. Assume that r, r0 , r1 ∈ [1, ∞] and either s0 > 0 or E is reflexive and s0 > −ν1 (1 − 1/q) . Then

¡ s0 /ν ¢ . sθ /ν s1 /ν Bq,r0 ,B (H, E), Bq,r (H, E) θ,r = Bq,r,B (H, E) 1 ,B

and

¡ s0 /ν ¢ . sθ /ν s1 /ν Hq,B (H, E), Hq,B (H, E) θ,r = Bq,r,B (H, E) .

Proof

t/ν

For 1 ≤ ρ ≤ ∞ we set Xρt := Bq,ρ (H, E).

. (1) Theorems VII.2.7.1(i) and VII.2.8.3 imply (Xrs00 , Xrs11 )θ,r = Xrsθ . Using this, the arguments of steps (8)–(10) of the proof of Theorem 2.4.3 show that . sθ (Xrs00 , Xrs11,B )θ,r = Xr,B . Now we follow the proof of the preceding theorem, with [·, ·]θ replaced by (·, ·)θ,r , to obtain the first assertion. (2) It is an obvious consequence of Theorems VII.4.1.3(i) that t/ν

t/ν

t/ν

Bq,1,B ,→ Hq,B ,→ Bq,∞,B . From this and (1), the second assertion is immediate.

(2.4.18) ¥

2.4.6 Corollary ¡

Proof

s /ν

s /ν

0 1 Bq,r (H, E), Hq,B (H, E) 0 ,B

¢ θ,r

¢ . ¡ s0 /ν s1 /ν = Hq,B (H, E), Bq,r (H, E) θ,r 1 ,B . sθ /ν = Bq,r,B (H, E) .

Using (2.4.18), we find s /ν

s /ν

s /ν

s /ν

s /ν

s /ν

0 1 0 1 0 1 (Bq,r , Bq,1,B )θ,r ,→ (Bq,r , Hq,B )θ,r ,→ (Bq,r , Bq,∞,B )θ,r . 0 ,B 0 ,B 0 ,B

Now the theorem implies . sθ /ν s0 /ν s1 /ν (Bq,r,B , Hq,B )θ,r = Bq,r,B . This proves the first half of the assertion. The second half follows analogously.

¥

We illustrate these results by an example which is motivated by boundary value problems for elliptic and parabolic equations.

VIII.2 Boundary Operators

367

2.4.7 Example Let (2.4.3) be satisfied with 1 < q < ∞ and s0 > −ν1 (1 − 1/q). Suppose E is ν-admissible. Then ¡ s0 /ν ¢ ¢ . ¡ s0 /ν s1 /ν s1 /ν Wq (H, E), Wq,B (H, E) θ,q = Wq,B (H, E), Wq,B (H, E) θ,q . sθ /ν = Bq,B (H, E) . Recall that

s /ν

s /ν

θ θ Wq,B (H, E) = Bq,B (H, E) if

s /ν

Proof Since Wq j results above. ¥

sθ ∈ / ν1 N .

. s /ν = Hq j if sj ∈ ν1 N, the assertions are immediate from the

Complex Interpolation of Bessel Potential Spaces Real interpolation of Bessel potential spaces with vanishing boundary conditions is covered by Theorem 2.4.5. Now we turn to the complex interpolation of these spaces. Here we restrict ourselves to isotropic spaces. 2.4.8 Theorem Suppose (2.4.3) applies with 1 < q < ∞ and s0 > −1 + 1/q. If E is a UMD space, then £ s0 ¤ . £ s0 ¤ . s0 s1 s1 Hq (H, E), Hq,B (H, E) θ = Hq,B (H, E), Hq,B (H, E) θ = Hq,B (H, E) . Proof (1) As explained in the notes to the preceding section, Theorem 1.1 in [MeV15] implies that (1.9.4) is true. From this and Theorem 1.6.8 we deduce that, given ` ∈ N and ` + 1/q < t < ` + 1 + 1/q, it follows that t t ˚ Hq,~ γ ` (H, E) = Hq (H, E) .

(2) Set

(2.4.19)

X t := Hqt (H, E) ,

Y t(j) := bt−j−1/q (∂H, Fj ) . q . ˚ E) are valid. Hence steps Then (2.4.19) guarantees that (2.4.12) and XCsθ = X sθ (H, (1)–(7) of the proof of Theorem 2.4.3 apply verbatim, provided we invoke Theorem VII.4.5.1 instead of the interpolation theorems for Besov spaces. ¥ 2.4.9 Remark Suppose we knew that (1.9.2) applies. Then (1.9.4) and Theorem 1.6.8 would imply that, given ` ∈ N and ν1 (` + 1/q) < t < ν1 (` + 1 + 1/q), t/ν ˚ E) . Hq,~γ ` (H, E) = Hqt/ν (H,

(2.4.20)

Then the arguments of step (2) of the preceding proof, with the obvious definitions of X t and Y t(j) , would show that the analogue of Theorem 2.4.8 holds for anisotropic Bessel potential spaces. As mentioned in the notes to Section 1, we conjecture that (2.4.20) is true if E is ν-admissible. ¥

368

2.5

VIII Traces and Boundary Operators

Notes

The first interpolation theorem for spaces with vanishing boundary conditions is due to P. Grisvard [Gri67]. He considers the isotropic q scalar case E = C and proves Theorem 2.4.8 for p = 2, s0 = 0, and s1 = r ∈ N. In [Gri69] the author extends his theorem to p 6= 2. More precisely, he proves that r r θr (Lp , Wp,B )θ,p = (Lp , Bp,B )θ,p = Bp,B .

. Note that W2r = B2r = H2r and (E0 , E1 )θ,2 = [E0 , E1 ]θ if E0 and E1 are Hilbert d

spaces with E1 ,→ E0 (e.g., [LM73, Theorem I.15.1]). An amplification of these theorems to general Besov spaces has been achieved by D. Guidetti [Gui91]. q If E = Cn , then Theorem 2.4.8, with s0 = 0 and s1 ∈ N, is due to R. Seeley [See72]. None of these authors£assumes that sθ ∈ / { mi ¤+ 1/q ; 0 ≤ i ≤ k }. If sθ is such s1 an exceptional value, then Hqs0 (H, E), Hq,B (H, E) θ is not a closed linear subspace of Hqsθ (H, E), but carries a stronger topology. Furthermore, in each of the papers mentioned, H is replaced by a smooth compact (sub-)manifold Ω (of Rd ) with boundary. However, as is well-known, the crucial step is the proof for the halfspace. From this one passes to Ω by standard localization and partition of unity procedures (see the next volume). Steps (1)–(4) of the proof of Theorem 2.4.3 follow essentially R. Seeley. As already mentioned in the notes to Section 1, the use of the J-method to derive the real interpolation results goes back to D. Guidetti. Recently, N. Lindemulder, M. Meyries, and M. Veraar [LMV18] presented a proof for the interpolation result £ s0 + ¤ . s ˚q (R , E), H ˚qs1 (R+ , E) = ˚q θ (R+ , E) H H θ if s0 > −1 + 1/p and s0 , s1 , sθ ∈ / N + 1/q, where E is a UMD space (recall Theorem 1.6.8). They also consider weighted spaces which are of no concern to us.

Appendix Vector-Valued Distributions The theory of vector-valued distributions has been almost completely developed by L. Schwartz around the middle of the last century. His results are very general since he considered locally convex target spaces. For many purposes –– in particular for use in this book –– it suffices to consider distributions with values in Banach spaces. In this case the general theory simplifies, and we develop the relevant parts in this more restricted setting in this appendix. It is subdivided into two sections. The first, rather long one, is centered around kernel theorems and their main concrete realizations, namely tensor products and convolutions. The short second section is devoted to a proof of the Riesz representation theorem in the vector-valued setting. Throughout this appendix we employ the conventions collected in ‘Notations and Conventions’ of Volume I. In particular, K stands now either for the real or for the complex number field. Corners do not appear in what follows.

© Springer Nature Switzerland AG 2019 H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4

369

370

1

Appendix

Tensor Products and Convolutions

It is crucial to possess a good theory of convolutions and ‘point-wise multiplications’. These operations are special cases of bilinear maps of vector-valued distributions. In order to handle such bilinear operations we prove in Subsection 1.5 a basic extension theorem that allows to carry over the desired results from scalar distributions to the vector-valued setting. For its proof we need a considerable amount of preparation, namely rather deep results from the theory of locally convex spaces as well as approximation and subtle continuity theorems for concrete spaces of distributions. These preparations occupy Subsections 1.1 to 1.4. Having established the basic extension theorem in Subsection 1.5, it is not too difficult to define point-wise multiplications, tensor products, and convolutions for vector-valued distributions. This is done in Subsections 1.6–1.9. Although the proofs and techniques of this appendix are rather heavy, the final results are very simple to state and very satisfactory. Indeed, given the correct interpretation, all known rules for scalar distributions carry over to the vectorvalued setting.

1.1

Locally Convex Topologies

For the reader’s convenience we collect in this subsection some more advanced topics1 from the theory of LCSs that we shall need below. Hereafter, we often use these facts without further mention. The Uniform Boundedness Principle An LCS is barreled if each absolutely convex, closed, and absorbing subset is a neighborhood of zero. Every Fr´echet space, hence every Banach space, is barreled ([Hor66, III.6]). Every reflexive LCS is barreled. In fact, an LCS is reflexive iff it is barreled and bounded subsets are relatively weakly compact. Let E and F be LCSs. A set A ⊂ L(E, F ) is equicontinuous if for each neighborhood V of zero in F there exists a neighborhood U of zero in E such that A(U ) ⊂ V for all A ∈ A. Equivalently, A is equicontinuous iff to each continuous seminorm q on F there exists a continuous seminorm p on E such that q(Ae) ≤ p(e) ,

e∈E ,

A∈A.

The set A is uniformly bounded if for each bounded subset B of E there exists a bounded set C ⊂ F such that A(B) ⊂ C for all A ∈ A. It is easily seen that each equicontinuous set is uniformly bounded. Finally, A is point-wise bounded if A(e) := { Ae ; A ∈ A } is bounded in F for each e ∈ E, that is, if A is bounded 1 Recall the ‘Notations and Conventions’ of Volume I as well as the facts collected in Subsections III.4.1 and III.4.2.

Tensor Products and Convolutions

371

in Ls (E, F ). Since L(E, F ) ,→ Ls (E, F ), every bounded subset of L(E, F ) is pointwise bounded. The uniform boundedness principle asserts that each point-wise bounded subset of L(E, F ) is equicontinuous, hence uniformly bounded, if E is barreled (e.g., [Jar81, Proposition 11.1.1] or [Sch71, Theorem III.4.2]). In particular, it follows that sup kAek < ∞ for each e ∈ E

=⇒

A∈A

sup kAk < ∞ ,

(1.1.1)

A∈A

provided E and F are Banach spaces. Suppose that E is barreled. Let (Aα ) be a net in L(E, F ) that is bounded in Ls (E, F ), and assume that Ae := lim Aα e exists in F for each e ∈ E. Then the Banach–Steinhaus theorem asserts that A ∈ L(E, F ) and that Aα → A, uniformly on compact sets (e.g., [Jar81, Theorem 11.1.3]). If E and F are Banach spaces and (Aj ) is a sequence in L(E, F ) such that (Aj e) converges in F for each e ∈ E, then it follows that kAk ≤ lim inf kAj k < ∞ j→∞

(1.1.2)

(e.g., [Yos65, Corollary II.1.2]). Hypocontinuity Let G be a third LCS and let b : E × F → G be bilinear. Clearly, b is separately continuous if b(·, f ) ∈ L(E, G) for each f ∈ F and b(e, ·) ∈ L(F, G) for each e ∈ E. Of course, every continuous bilinear map is separately continuous. A separately continuous bilinear map is continuous if E and F are Fr´echet spaces (e.g., [Rud73, Theorem 2.1.7] or [Hor66, Theorem 4.7.1]). The map b is hypocontinuous if b(·, f ) : E → G is continuous, uniformly with respect to f in bounded subsets of F , and b(e, ·) : F → G is continuous, uniformly with respect to e in bounded subsets of E. If b is hypocontinuous, then it is separately continuous and b(A, B) is bounded in G if A is bounded in E and B is bounded in F . Moreover, b is continuous on A × F and on E × B, and uniformly continuous on A × B (e.g., [Hor66, Propositions 4.7.2 and 4.7.3] or [Sch71, III.5.3]). Consequently, every hypocontinuous bilinear map is sequentially continuous. Every separately continuous bilinear map is hypocontinuous if E and F are barreled (e.g., [Hor66, Theorem 4.7.2] or [Sch71, III.5.2]). Of course, a continuous bilinear map is hypocontinuous. The following lemma will be useful for proving the hypocontinuity of certain bilinear maps. 1.1.1 Lemma Suppose that F is barreled. Then the bilinear map L(E, F ) × L(F, G) → L(E, G) , is hypocontinuous.

(S, T ) 7→ T S

372

Appendix

Proof Let q be a continuous seminorm on G and let B be a bounded subset of E. Let T ⊂ L(F, G) be bounded. It follows from the uniform boundedness principle that there exists a continuous seminorm p on F such that q(T f ) ≤ p(f ) ,

f ∈F ,

T ∈T .

Consequently, sup q(T Se) ≤ sup p(Se) , e∈B

which shows that

S ∈ L(E, F ) ,

T ∈T ,

e∈B

¡ ¢ (S 7→ T S) ∈ L L(E, F ), L(E, G) ,

uniformly with respect to T in bounded subsets of L(F, G). This proves one half of the statement. The proof for the other half is similar. ¥ Montel Spaces Recall that an LCS is a Montel space if it is barreled and bounded subsets are relatively compact. Since the unit-ball of a Banach space is relatively compact iff it is finite-dimensional, there are no infinite-dimensional Banach-Montel spaces. Every Montel space is reflexive (e.g., [Hor66, 3, §9]). The dual of a Montel space is a Montel space as well (e.g, [Hor66, Proposition 9 in Section 3, §9]). Strict Inductive Limits Let E be a vector space and let { Eα ; α ∈ A } be a family of subspaces such that: (i) Each Eα is a Fr´echet space. (ii) If Eα ⊃ Eβ , then Eα induces the original topology on Eβ . (iii) There exists a cofinal increasing sequence (En ) in { Eα ; α ∈ A }, that is, En ⊂ En+1 and to each Eα there is an En with En ⊃ Eα . S (iv) α Eα = E. Then there exists a finest locally convex Hausdorff topology τ on E such that Eα ,→ E for each α ∈ A, the (strict) inductive limit topology or LF topology induced by { Eα ; α ∈ A }. The LCS (E, τ ) is denoted by lim Eα = lim E −→ α −→

α

and said to be an LF space. Every LF space is complete and barreled, and induces on each Eα its original topology. A subset B of E is bounded iff B ⊂ Eα for some α ∈ A and B is bounded in Eα . Let F be an LCS. Then a linear map T : E → F is continuous iff T | Eα is continuous from Eα into F for each α ∈ A. This is the case iff each T | Eα is bounded. Thus an LF space E is bornological, that is, every bounded

Tensor Products and Convolutions

373

linear map from E into an LCS is continuous. Clearly, every Fr´echet space, hence every Banach space, is an LF space (for all this see [Jar81, IV], [Hor66, II.12], [Sch71, II.6–II.8]). It follows from (iii) that our inductive limits are countable. Thus, in principle, we could have restricted ourselves to the consideration of sequences (Ek ) only, instead of admitting possibly uncountable families { Eα ; α ∈ A }. However, the given formulation is well adapted to the concrete spaces we have in mind. In those cases, uncountable families of Fr´echet spaces occur naturally. Then we do not have to select a particular sequence and keep repeating that the topology is independent of that particular sequence. Smooth Functions Let X be a nonempty open subset of Rn and let E = (E, |·|) be a Banach space over K. Recall from Subsection III.4.1 that ³ ´ E(X, E) := C ∞ (X, E), { pm,K ; m ∈ N, K b X } is a Fr´echet space, where the seminorms pm,K are defined by pm,K (u) := max k∂ α uk∞,K .

(1.1.3)

|α|≤m

Test Functions Given K b X, let DK (X, E) :=

©

u ∈ E(X, E) ; supp(u) ⊂ K

ª

.

(1.1.4)

Then DK (X, E) is a closed linear subspace of E(X, E), and [ © ª DK (X, E) = u ∈ E(X, E) ; supp(u) b X . KbX

Let Xk :=

©

x ∈ X ; dist(x, X c ) > 1/k

ª

∩ kBn ,

q k∈N,

(1.1.5)

where dist(x, ∅) := ∞. Then [

Xk b Xk+1 ,

Xk = X .

(1.1.6)

k

¡ ¢ © ª Hence DXk (X, E) is a cofinal increasing sequence in DK (X, E) ; K b X . Thus D(X, E) := −−→ lim DK (X, E) , KbX

the space of E-valued test functions, is an LF space (cf. Subsection III.1.1).

374

Appendix

Rapidly Decreasing Smooth Functions Also recall from Subsection III.4.1 that the Schwartz space of smooth rapidly decreasing E-valued functions on Rn is defined by ¡ ¢ S(Rn , E) := S(Rn , E), { qk,m ; k, m ∈ N } , where qk,m (u) := sup (1 + |x|2 )k |∂ α u(x)| .

(1.1.7)

x∈Rn |α|≤m

It is a Fr´echet space. Slowly Increasing Smooth Functions Finally, we recall that OM (Rn , E) is the space of slowly increasing smooth functions on Rn . This means that u ∈ OM (Rn , E) iff u ∈ E(Rn , E) and, given α ∈ Nn , there exist mα ∈ N and cα > 0 such that |∂ α u(x)| ≤ cα (1 + |x|2 )mα ,

x ∈ Rn .

(1.1.8)

Moreover, OM (Rn , E) is given the topology induced by the family of seminorms2 u 7→ kϕ∂ α uk∞ ,

ϕ ∈ S(Rn ) ,

α ∈ Nn ,

(1.1.9)

so that it is an LCS. Suppose u ∈ S(Rn , E). Then kϕ∂ α uk∞ ≤ kϕk∞ q0,|α| (u) ,

ϕ ∈ S(Rn ) ,

α ∈ Nn .

This shows that S(Rn , E) ,→ OM (Rn , E) .

(1.1.10)

Spaces of Vector-Valued Distributions To simplify the writing we agree to put F(X, E) := F(Rn , E) if

F ∈ {S, OM } .

(1.1.11)

Then, as usual, F(X) := F(X, K) , 2 For

F ∈ {D, E, S, OM } ,

a proof of the fact that these seminorms are well-defined we refer to Proposition 1.6.1.

Tensor Products and Convolutions

375

if no confusion seems likely, and ¡ ¢ F0 (X, E) := L F(X), E

(1.1.12)

so that F0 (X) = F(X)0 for F ∈ {D, E, S, OM }. (Recall that L is always given the bounded convergence topology.) If u ∈ D0 (X, E), then, as a rule, we write u(ϕ) for the value of u at ϕ ∈ D(X). However, if u is a scalar distribution, that is, u ∈ D0 (X), then we continue to denote u(ϕ) by hu, ϕi. 1.1.2 Theorem Let F ∈ {D, E, S, OM }. Then F(X) and F0 (X) are complete Montel spaces, hence reflexive. Proof Every LF space is complete and bornological. The dual of a bornological LCS is complete (e.g., [Sch71, IV.6.1]). Hence F(X) and F0 (X) are complete for F ∈ {D, E, S}. The fact that F(X) is a Montel space for F ∈ {D, E, S} is well-known (e.g., [Hor66, Examples 3, 4, and 6 in Section 3, §9]). Thus F0 (X) is a Montel space in these cases as well. The assertion for F = OM follows from [Gro55, II.4.4]. ¥

1.2

Convolutions

In Subsection III.4.2 we have already stated the definition of the convolution of a vector-valued distribution and a scalar test function, as well as some of its basic properties. In this subsection we present proofs for some of the less obvious properties. In addition, we extend the definition to include convolutions of a vector-valued and a scalar distribution. Convolutions of Distributions and Test Functions Let E := (E, |·|) be a Banach space. Recall from Subsection III.4.2 that, given (u, ϕ) ∈ F0 (Rn , E) × F(Rn ) ,

F ∈ {D, E} ,

the convolution, u ∗ ϕ, of u and ϕ, is defined by b

u ∗ ϕ(x) := u(τx ϕ) ,

x ∈ Rn .

(1.2.1)

As already noted, the usual scalar proof (e.g., [H¨or83, Theorem 4.1.1]) carries over to the present situation to show that u ∗ ϕ ∈ E(Rn , E) and ∂ α (u ∗ ϕ) = ∂ α u ∗ ϕ = u ∗ ∂ α ϕ ,

α ∈ Nn .

(1.2.2)

376

Appendix

Moreover, supp(u ∗ ϕ) ⊂ supp(u) + supp(ϕ)

(1.2.3)

so that u ∗ ϕ ∈ D(Rn , E) if

(u, ϕ) ∈ E 0 (Rn , E) × D(Rn ) .

(1.2.4)

It is obvious that convolution is bilinear. For the reader’s convenience we prove now that it is hypocontinuous. 1.2.1 Proposition Convolution is a bilinear and hypocontinuous mapping : (i) D0 (Rn , E) × D(Rn ) → E(Rn , E); (ii) E 0 (Rn , E) × E(Rn ) → E(Rn , E); (iii) E 0 (Rn , E) × D(Rn ) → D(Rn , E). Proof To simplify the notation we put D := D(Rn ) and E := E(Rn ). (1) Let Kr := rBn for r > 0. Then u ∈ D0 (Rn , E), if and only if u belongs to L(DKr , E) for each r > 0. Let A be a bounded subset of D0 (Rn , E). Given r > 0, the uniform boundedness principle implies the existence of k ∈ N such that |u(ϕ)| ≤ cpk,Kr (ϕ) ,

ϕ ∈ D Kr ,

u∈A.

Observe that b

x ∈ Kρ , ϕ ∈ D K r

=⇒

τx ϕ ∈ DKr+ρ

(1.2.5)

pk,Kr+ρ (τx ϕ) = pk,x−Kr (τx ϕ) = pk,Kr (ϕ)

(1.2.6)

and b

b

for ρ > 0. Consequently, b

b

pKρ (u ∗ ϕ) = sup |u(τx ϕ)| ≤ cpk,Kr+ρ (τx ϕ) = cpk,Kr (ϕ)

(1.2.7)

x∈Kρ

for ϕ ∈ DKr and u ∈ A. Now we obtain from (1.2.2) that pj,Kρ (u ∗ ϕ) ≤ cpk+j,Kr (ϕ) ,

ϕ ∈ D Kr ,

u∈A,

j∈N,

ρ>0.

This shows that ¡ ¢ (ϕ 7→ u ∗ ϕ) ∈ L DKr , E(Rn , E) , u∈A, r>0. (1.2.8) ¡ ¢ Hence (ϕ 7→ u ∗ ϕ) ∈ L D, E(Rn , E) , uniformly with respect to u in bounded subsets of D0 (Rn , E).

Tensor Products and Convolutions

377

Now let B be a bounded subset of D. Then there exists r > 0 such that B is contained and bounded in DKr . It follows from (1.2.7) and (1.2.2) that pj,Kρ (u ∗ ϕ) = max sup |u ∗ ∂ α ϕ(x)| ≤ c sup |u(ψ)| , |α|≤j x∈Kρ

C :=

©

b

where

(1.2.9)

ψ∈C

τx (∂ α ϕ) ; x ∈ Kρ , |α| ≤ j, ϕ ∈ B

ª

is a bounded subset of D, thanks to (1.2.5) and (1.2.6). Consequently, ¡ ¢ (u 7→ u ∗ ϕ) ∈ L D0 (Rn , E), E(Rn , E) , uniformly with respect to ϕ in bounded subsets of D. This proves the hypocontinuity of (i). (2) Let A be a bounded subset of E 0 (Rn , E) = L(E, E). Then the uniform boundedness principle implies the existence of r > 0 and k ∈ N such that |u(ϕ)| ≤ cpk,Kr (ϕ) ,

ϕ∈E ,

u∈A.

(1.2.10)

Thus, given ρ > 0 and x ∈ Kρ , b

b

|u(τx ϕ)| ≤ cpk,Kr (τx ϕ) ≤ cpk,Kr +ρ (ϕ) ,

ϕ∈E ,

u∈A.

Hence, by (1.2.2), given ρ > 0, pj,Kρ (u ∗ ϕ) ≤ pk+j,Kr+ρ (ϕ) , j∈N, ϕ∈E , u∈A, ¡ ¢ which shows that (ϕ 7→ u ∗ ϕ) ∈ L E, E(Rn , E) , uniformly with respect to u in bounded subsets of E 0 (Rn , E). If B is a bounded subset of E, then estimate (1.2.9) is valid, ¢where C is now a ¡ bounded subset of E. Hence (u 7→ u ∗ ϕ) ∈ L E 0 (Rn , E), E(Rn , E) , uniformly with respect to ϕ in bounded subsets of E. Thus (ii) is hypocontinuous as well. (3) Suppose that A is a bounded subset of E 0 (Rn , E). Then we infer from (1.2.10) that u(ϕ) = 0 for ϕ ∈ D(Rn \Kr ) and u ∈ A. Thus supp(u) ⊂ Kr for u ∈ A. Hence we deduce from (1.2.3) and (1.2.8) that ¡ ¢ (ϕ 7→ u ∗ ϕ) ∈ L DKρ , DKr+ρ (Rn , E) , u∈A, ρ>0. Since DKr+ρ (Rn , E) ,→ D(Rn , E), we see that ¡ ¢ (ϕ 7→ u ∗ ϕ) ∈ L DKρ , D(Rn , E) ,

u∈A,

ρ>0.

¡ ¢ Consequently, (ϕ 7→ u ∗ ϕ) ∈ L D, D(Rn , E) , uniformly for u in bounded subsets of E 0 (Rn , E).

378

Appendix

If B is a bounded subset of D, then we deduce from (1.2.9) and (1.2.3) that ¡ ¢ (u 7→ u ∗ ϕ) ∈ L E 0 (Rn , E), D(Rn , E) , uniformly with respect to ϕ ∈ B. Hence (iii) is also hypocontinuous.

¥

Translation-Invariant Operators It is easy to verify that b

b

u(ϕ) = u ∗ ϕ(0) = u ∗ ϕ(0)

(1.2.11)

and b

b

b

(u ∗ ϕ) = u ∗ ϕ

(1.2.12)

for u ∈ F0 (Rn , E) and ϕ ∈ F(Rn ), where F ∈ {D, E}. The following theorem gives an important characterization of convolutions. ¡ ¢ 1.2.2 Theorem Suppose that F ∈ {D, E} and T ∈ L F(Rn ), C(Rn , E) and that T commutes with translations: x ∈ Rn ,

T (τx ϕ) = τx T (ϕ) ,

ϕ ∈ F(Rn ) .

Then there exists a unique u ∈ F0 (Rn , E) such that T (ϕ) = u ∗ ϕ ,

ϕ ∈ F(Rn ) .

Proof It is trivial that reflection is a toplinear automorphism of F(Rn ). Thus the continuity hypothesis implies that £ ¤ u := ϕ 7→ (T ϕ)(0) ∈ F0 (Rn , E) . b

Consequently, we infer from the commutativity hypothesis that ¡ ¢ (T ϕ)(x) = τ−x (T ϕ)(0) = T (τ−x ϕ)(0) = u (τ−x ϕ) = u(τx ϕ) = (u ∗ ϕ)(x) b

b

for x ∈ Rn ; thus T ϕ = u ∗ ϕ for ϕ ∈ F(Rn ).

¥

Convolutions of Two Distributions Now suppose that u ∈ D0 (Rn , E) and v ∈ D0 (Rn ) and u or v has compact support. Then ¡ ¢ (ϕ 7→ u ∗ ϕ) ∈ L D(Rn ), E(Rn , E)

Tensor Products and Convolutions

and

379

¡ ¢ (ϕ 7→ v ∗ ϕ) ∈ L D(Rn ), D(Rn )

if

supp(v) b Rn

by Proposition 1.2.1. The same arguments show that ¡ ¢ ¡ ¢ ϕ 7→ u ∗ (v ∗ ϕ) ∈ L D(Rn ), E(Rn , E) . Thus (III.4.2.15) and Theorem 1.2.2 guarantee the existence of a unique distribution u ∗ v ∈ D0 (Rn , E), the convolution of u and v, such that (u ∗ v) ∗ ϕ = u ∗ (v ∗ ϕ) ,

ϕ ∈ D(Rn ) .

(1.2.13)

It is clear that convolution is bilinear. The next proposition shows that it is hypocontinuous as well. 1.2.3 Proposition Convolution is a bilinear and hypocontinuous mapping : (i) E 0 (Rn , E) × D0 (Rn ) → D0 (Rn , E). (ii) D0 (Rn , E) × E 0 (Rn ) → D0 (Rn , E). (iii) E 0 (Rn , E) × E 0 (Rn ) → E 0 (Rn , E). Proof Let A and B be bounded subsets of D0 := D0 (Rn ) and D := D(Rn ), respectively. Since reflection is obviously a toplinear automorphism of D and of D0 , it follows from Proposition 1.2.1 and the boundedness properties of hypocontinuous maps that © ª C := (u ∗ ϕ) ; u ∈ A, ϕ ∈ B bb

is a bounded subset of E. Hence, given v ∈ E 0 (Rn , E) and u ∈ A, b

b

sup |v ∗ u(ϕ)| = sup |(v ∗ u) ∗ ϕ(0)| = sup |v ∗ (u ∗ ϕ)(0)| ϕ∈B ϕ∈B ¯ ¡ ¢¯ = sup ¯v (u ∗ ϕ) ¯ ≤ sup |v(ψ)| .

ϕ∈B

bb

ϕ∈B

ψ∈C

¡ ¢ This shows that (v 7→ v ∗ u) ∈ L E 0 (Rn , E), D0 (Rn , E) , uniformly with respect to u in bounded subsets of D0 . b

Let (uα ) be a net in D0 converging to zero. Then uα ∗ ϕ → 0 in E, uniformly with respect to ϕ in bounded subsets of D, thanks to Proposition 1.2.1(i). Consequently, Proposition 1.2.1(ii) guarantees that v ∗ (uα ∗ ϕ) → 0 in E(Rn , E), uniformly with respect to ϕ in bounded subsets of D and to v in bounded subsets of E 0 (Rn , E). Hence (v ∗ uα )(ϕ) = (v ∗ uα ) ∗ ϕ(0) → 0 in E, uniformly with respect to ϕ in bounded subsets ¡of D and to v ¢in bounded subsets of E 0 (Rn , E). This shows that (u 7→ v ∗ u) ∈ L D0 , D0 (Rn , E) , uniformly with respect to v in bounded subsets of E 0 (Rn , E). Hence (i) is hypocontinuous. b

b

The hypocontinuity of (ii) and (iii) follows by modifying the above arguments in an obvious way. ¥

380

Appendix

Elementary Properties of Convolutions 1.2.4 Remarks (a) Let δ ∈ E 0 (Rn ) be the Dirac distribution, that is, hδ, ϕi := ϕ(0) ,

ϕ ∈ E(Rn ) .

Then u ∗ δ = u for u ∈ D0 (Rn , E). Proof

From (1.2.1) it is obvious that δ ∗ ϕ = ϕ for ϕ ∈ E(Rn ). Hence (u ∗ δ) ∗ ϕ = u ∗ (δ ∗ ϕ) = u ∗ ϕ ,

by (1.2.13).

ϕ∈D ,

¥

(b) Suppose that u ∈ D0 (Rn , E) and v ∈ D0 (Rn ) and that u or v has compact support. Then ∂ α+β (u ∗ v) = ∂ α u ∗ ∂ β v , α, β ∈ Nn . Proof

By repeatedly applying (1.2.2) we see that ∂ α+β (u ∗ v) ∗ ϕ = (u ∗ v) ∗ ∂ α+β ϕ = u ∗ (v ∗ ∂ α+β ϕ) = u ∗ ∂ α (∂ β v ∗ ϕ) = ∂ α u ∗ (∂ β v ∗ ϕ) = (∂ α u ∗ ∂ β v) ∗ ϕ

for ϕ ∈ D(Rn ).

¥

(c) Let u ∈ D0 (Rn , E) and v ∈ D0 (Rn ) such that u or v has compact support. Then a ∈ Rn ,

τa (u ∗ v) = τa u ∗ v = u ∗ τa v , b b

and

b

(u ∗ v) = u ∗ v . Proof

This follows easily from (1.2.13) and (III.4.2.15), or (1.2.12), respectively.

(d) If { ϕε ; ε > 0 } is a mollifier, then ϕε → δ in E 0 (Rn ) as ε → 0. Consequently, ϕε ∗ u → u in D0 (Rn , E) as ε → 0 for u ∈ D0 (Rn , E). Proof

Given ψ ∈ E(Rn ), Z Z hϕε − δ, ψi = ϕε (x)ψ(x) dx − ψ(0) = Rn

¡ ¢ ϕ(y) ψ(εy) − ψ(0) dy . Rn

Hence, by the mean-value theorem, |hϕε − δ, ψi| ≤ sup |ψ(εy) − ψ(0)| ≤ ε sup |∂ψ(y)| ≤ εp1,Bn (ψ) |y|≤1

|y|≤1

for 0 < ε ≤ 1. This shows that hϕε − δ, ψi → 0 as ε → 0, uniformly with respect to ψ in bounded subsets of E(Rn ), that is, ϕε → δ in E 0 (Rn ) as ε → 0. The second part of the assertion now follows from Proposition 1.2.3 and (a). ¥

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Convolutions of Temperate Distributions Next we turn to the case of temperate (also called: tempered) distributions. For this we recall the following facts. 1.2.5 Lemma The translation group acts strongly continuously on S(Rn , E) and on S 0 (Rn , E). Proof

Theorem VII.3.3.1 in the main part of this volume.

¥

¡ ¢ It is clear that (ϕ 7→ ϕ) ∈ L S(Rn , E) . Thus, given u ∈ S 0 (Rn , E) and ϕ belonging to S(Rn ), we can again define the convolution, u ∗ ϕ, by (1.2.1), since the right side defines a continuous E-valued function of x ∈ Rn by Lemma 1.2.5. b

In order to prove continuity properties of this convolution map we need the following technical result. 1.2.6 Lemma OM (Rn , E) is a complete LCS. Proof Let (aβ ) be a Cauchy net in OM (Rn , E). From OM (Rn , E) ,→ E(Rn , E) it follows that (aβ ) is a Cauchy net in E(Rn , E). Since the latter space is complete, there exists a ∈ E(Rn , E) such that aβ → a in E(Rn , E). Given α ∈ Nn , ϕ ∈ S(Rn ), and ε > 0, there exists β0 such that kϕ∂ α aβ − ϕ∂ α aγ k∞ < ε ,

β, γ ≥ β0 .

Thus, since aγ → a in E(Rn , E), kϕ∂ α aβ − ϕ∂ α ak∞ ≤ ε ,

β ≥ β0 .

Similarly, kϕ∂ α aγ k∞ ≤ kϕ∂ α aβ0 k∞ + ε ,

γ ≥ β0 ,

implies kϕ∂ α ak∞ < ∞. Hence a ∈ OM (Rn , E) and aβ → a in OM (Rn , E).

¥

1.2.7 Proposition Convolution is a bilinear and (i) continuous map: S(Rn , E) × S(Rn ) → S(Rn , E); (ii) hypocontinuous map: S 0 (Rn , E) × S(Rn ) → OM (Rn , E). Proof (1) Given u ∈ S(Rn , E), it follows from ∂ α u ∈ BUC(Rn , E) for α ∈ Nn and from (III.4.2.10) and (III.4.2.19) that u ∗ v ∈ E(Rn , E) for v ∈ S(Rn ) ⊂ L1 . Note that 2` ³ ´ X 2` j |x|2` ≤ (|x − y| + |y|)2` = |x − y| |y|2`−j j j=0

382

Appendix

implies |x|2` |u ∗ v(x)| 2` ³ ´Z X dy 2` 2`−j ≤ |x − y|j |u(x − y)| |y| (1 + |y|2 )n |v(y)| j (1 + |y|2 )n j=0

≤ c(`)q`,0 (u)q`+n,0 (v) for u, v ∈ S(Rn ). This proves (i). (2) Given ϕ ∈ S(Rn , E) and k, m ∈ N, we find qk,m (τx ϕ) = sup (1 + |x + y|2 )k |∂ α ϕ(y)| ≤ 2k (1 + |x|2 )k qk,m (ϕ) y∈Rn |α|≤m

(1.2.14)

for x ∈ Rn , thanks to the trivial inequality 1 + |x + y|2 ≤ 2(1 + |x|2 )(1 + |y|2 ). If u ∈ S 0 (Rn , E) and ϕ ∈ D(Rn ), then u ∗ ϕ ∈ E(Rn , E) by Proposition 1.2.1. Let U be a bounded subset of S 0 (Rn , E). Then there exist k, m ∈ N with |u(ϕ)| ≤ cqk,m (ϕ) ,

u∈U ,

ϕ ∈ S(Rn ) .

Thus, for α ∈ Nn , we deduce from (1.2.14) that ¯ ¡ ¢¯ |∂ α (u ∗ ϕ)(x)| = |(u ∗ ∂ α ϕ)(x)| = ¯u τx (∂ α ϕ) ¯ ¡ ¢ ≤ cqk,m τx (∂ α ϕ) ≤ c (1 + |x|2 )k qk,m+|α| (ϕ) b

b

for ϕ ∈ D(Rn ) and u ∈ U . Consequently, given a bounded subset B of S(Rn ), sup kψ∂ α (u ∗ ϕ)k∞ ≤ ck,m sup qk,0 (ψ)qk,m+|α| (ϕ)

ψ∈B

ψ∈B

for u ∈ U and ϕ ∈ B ∩ D(Rn ). This shows that, for u ∈ S 0 (Rn , E), the linear map D(Rn ) → OM (Rn , E) ,

ϕ 7→ u ∗ ϕ ,

is continuous with respect to the topology induced on D(Rn ) by S(Rn ), uniformly with respect to u in bounded subsets of S 0 (Rn , E). Since D(Rn ) is dense in S(Rn ) and OM (Rn , E) is complete, it follows that ¡ ¢ (ϕ 7→ u ∗ ϕ) ∈ L S(Rn ), OM (Rn , E) , uniformly with respect to u in bounded subsets of S 0 (Rn , E). Now let B and C be bounded subsets of S(Rn ) and observe that, given α ∈ Nn , the image of the map b

Rn × B × C → S(Rn ) ,

(x, ψ, ϕ) 7→ ψ(x)τx (∂ α ϕ)

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383

is contained in a bounded subset D of S(Rn ), since (1.2.14) implies ¡ ¢ qk,m ψ(x)τx (∂ α ϕ) ≤ ck,m qk,m (ψ)qk,m+|α| (ϕ) , x ∈ Rn , k ∈ N . b

Thus ¯ ¡ ¢¯ sup kψ∂ α (u ∗ ϕ)k∞ = sup sup ¯u ψ(x)τx (∂ α ϕ) ¯ ≤ sup |u(χ)| , b

ψ∈B x∈Rn

ψ∈B

χ∈D

which shows that ¡ ¢ (u 7→ u ∗ ϕ) ∈ L S 0 (Rn , E), OM (Rn , E) , uniformly with respect to ϕ in bounded subsets of S(Rn ). This proves the asserted hypocontinuity. ¥

1.3

Approximations

It is the main purpose of this subsection to show that tensor products of the form D(X) ⊗ E are dense in F(X, E), where F stands for one of the letters D, E, S, OM , D0 , E 0 , and S 0 . This is a very useful approximation result that allows to reduce many results for vector-valued distributions to the corresponding scalar versions. It will be of particular importance in later subsections for extending the operations of point-wise multiplication or of convolution to the case of two vector-valued factors. Multiplications Let E and Ej , j = 0, 1, 2, be Banach spaces. If no confusion seems likely, then we denote the norms in these spaces simply by |·|. We also suppose that E1 × E2 → E0 ,

(x1 , x2 ) 7→ x1 q x2

(1.3.1)

is a multiplication. Recall from (II.1.1.4) that this means that (1.3.1) is a continuous bilinear map of norm at most 1. 1.3.1 Examples The following maps are multiplications: (a) Ordinary multiplication in a Banach algebra. (b) Multiplication with scalars: K × E → E, (α, x) 7→ αx. (c) The duality pairing E 0 × E → K, (x0 , x) 7→ hx0 , xi. (d) The evaluation map L(E1 , E0 ) × E1 → E0 , (A, x) 7→ Ax. (e) Compositions L(E1 , E2 ) × L(E0 , E1 ) → L(E0 , E2 ), (S, T ) 7→ ST . (f ) Convolution in each one of the cases (III.4.2.18)–(III.4.2.22).

384

Appendix

(g) If b ∈ L(E1 , E2 ; E0 ) and b 6= 0, then E1 × E2 → E0 ,

(x1 , x2 ) 7→

1 b(x1 , x2 ) kbk

is a multiplication. This shows that it is no restriction assuming that the norm of a multiplication is bounded by 1. It is the advantage of this normalization that we do not have to drag along norm constants. ¥ Leibniz’ Rule We also recall that, given any nonempty set S, point-wise multiplication E1S × E2S → E0S ,

(a1 , a2 ) 7→ a1 q a2

induced by (1.3.1) is defined by a1 q a2 (s) := a1 (s) q a2 (s) ,

s∈S .

1.3.2 Lemma Let p ∈ K[X1 , . . . , Xn ] be a polynomial of degree at most k in n indeterminates and let U ⊂ Rn be a nonempty open subset of Rn . Then, putting p(β) := ∂ β p ,

β ∈ Nn ,

the generalized Leibniz rule, p(∂)(a1 q a2 ) =

X 1 (∂ β a1 ) q p(β) (∂)a2 , β! β

holds for aj ∈ C k (U, Ej ), j = 1, 2. Proof

From the obvious ‘product rule’ ∂j (a1 q a2 ) = ∂j a1 q a2 + a1 q ∂j a2 ,

we deduce by induction that X p(∂)(a1 q a2 ) = ∂ β a1 q qβ (∂)a2 ,

1≤j≤n,

aj ∈ C k (U, Ej ) ,

j = 1, 2 ,

(1.3.2)

(1.3.3)

β

where qβ ∈ K[X1 , . . . , Xn ] and qβ = 0 for |β| > k. Given yj ∈ Ej and ξ, η ∈ Rn , we put a1 := ehξ,·i y1 and a2 := ehη,·i y2 . Since e−hζ,·i q(∂)ehζ,·i = q(ζ) ,

ζ ∈ Rn ,

q ∈ K[X1 , . . . , Xn ] ,

(1.3.4)

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385

it follows from (1.3.3) that p(ξ + η)y1 q y2 =

X

ξ β qβ (η)y1 q y2 .

β

Since this is true for every choice of y1 and y2 , X p(ξ + η) = ξ β qβ (η) ,

ξ, η ∈ Rn .

β

By differentiating this identity with respect to ξ and putting ξ = 0 it follows that ∂ β p(η) = β! qβ (η). This proves the assertion. ¥ Letting p(ξ) := ξ α in Lemma 1.3.2, we obtain the standard Leibniz rule: X³α ´ ∂ α (a1 q a2 ) = ∂ β a1 q ∂ α−β a2 , aj ∈ C |α| (U, Ej ) , j = 1, 2 . (1.3.5) β β≤α

Approximation by Test Functions After these preparations we can prove a useful approximation theorem for vectorvalued distributions. For this we recall convention (1.1.11) and that X is a nonempty open subset of Rn . 1.3.3 Proposition Suppose that F ∈ {D, E, S} and u ∈ F0 (X, E). Then there exists a sequence (uj ) in D(X, E) such that uj → u in F0 (X, E). Proof (1) First suppose that F ∈ {D, E}. Denote by (Xk ) a sequence S of nonempty relatively compact open subsets of Rn such that Xk ⊂ Xk+1 and Xk = X. For each k ∈ N fix χk ∈ D(X) with χk | Xk = 1. Then χk u ∈ E 0 (X, E) ⊂ E 0 (Rn , E). Denote by { ψε ; ε > 0 } a mollifier such that ψ1 is even and put q uk := (χk u) ∗ ψ1/k , k∈N. Then uk ∈ D(X, E) by Proposition 1.2.1. Given ϕ ∈ D(X), ¡ ¢ uk (ϕ) = (χk u) ∗ ψ1/k ∗ ϕ(0) = (χk u) ∗ (ψ1/k ∗ ϕ)(0) ¡ ¢ = χk u ∗ (ψ1/k ∗ ϕ) (0) = u χk (ψ1/k ∗ ϕ) , b

b

b

b

where we used (1.2.11), (1.2.12), and the evenness of ψ1/k . Since supp(ϕ) ⊂ Xj for some j ∈ N, it follows from (III.4.2.10) and (III.4.2.25) that ψ1/k ∗ ϕ → ϕ in D(X). Also χk → 1 in E(X). Consequently, χk (ψ1/k ∗ ϕ) → ϕ in D(X) if

ϕ ∈ D(X) ,

(1.3.6)

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Appendix

so that uk (ϕ) → u(ϕ) in E. This means that ¡ ¢ uk → u in Ls D(X), E .

(1.3.7)

(2) Next we choose ϕ ∈ E(X). Then the estimate |∂ α (ψε ∗ ϕ)(x) − ∂ α ϕ(x)| = |ψε ∗ ∂ α ϕ(x) − ∂ α ϕ(x)| ¯Z £ ¤ ¯¯ ¯ = ¯ ψε (y) ∂ α ϕ(x − y) − ∂ α ϕ(x) dy ¯ ¯Z £ ¤ ¯¯ ¯ = ¯ ψ(y) ∂ α ϕ(x − εy) − ∂ α ϕ(x) dy ¯

(1.3.8)

≤ sup |∂ α ϕ(x − εy) − ∂ α ϕ(x)| |y| 0 such that |∂ β u(x)| ≤ cα (1 + |x|2 )m ,

x ∈ Rn ,

β≤α.

(1.3.14)

Let (χk ) be a sequence as in (1.3.11). Then, by Leibniz’ rule, X³α´ ∂ α (χk u) − ∂ α u = ∂ α−β χk ∂ β u + (χk − 1)∂ α u . β β 0 } is an approximate identity. Thus, given ϕ ∈ D(X, E), we infer from (III.4.2.10) and (III.4.2.25) that w1/k ∗ ϕ → ϕ in E(X, E).

390

Appendix

q Let k ∈ N be fixed and put Z f (z) := w1/k (z − y)ϕ(y) dy ,

z ∈ Cn .

Rn

It is easily verified that f is well-defined and analytic. Hence f can be represented by its Taylor series, that is easily seen to converge uniformly on compact subsets of Cn towards f . This implies that the sequence of Taylor polynomials of w1/k ∗ ϕ = f | Rn converges in E(Rn , E) towards w1/k ∗ ϕ, which gives the assertion. ¥ Separability 1.3.8 Proposition If E is separable, then D(X, E) is also separable. Proof It is an obvious consequence of Theorem 1.3.6(i) that it suffices to prove that D(X) is separable. Thus let P be the set of all polynomials in n indeterminates with rational coefficients (in K). Define (Xk ) by (1.1.5). If Xk 6= ∅, fix ϕk ∈ D(Xk+1 ) with ϕk | Xk = 1, and put ϕk := 0 if Xk = ∅. Then S := { ϕk p ; p ∈ P, k ∈ N } is a countable subset of D(X). It is an easy consequence of Lemma 1.3.7 and the properties of the LF topology of D(X) that S is dense in D(X). ¥ 1.3.9 Corollary Suppose that F ∈ { D0 , E, E 0 , S, S 0 , OM , C0 , C k , Wpk ; 1 ≤ p < ∞, k ∈ N } , and that X = Rn if F = Wpk . Then F(X, E) is separable, if E is separable. d

Proof Theorem 1.3.6 implies D(X, E) ,→ F(X, E). Hence the assertion follows from Proposition 1.3.8. ¥

1.4

Topological Tensor Products and the Kernel Theorem

In the first part of this subsection we collect some facts about tensor products of LCSs. This theory is easily accessible in standard books on linear functional analysis and topological vector spaces, in particular, in [Jar81], [Sch71], and [Tre67]. Thus we are rather brief and do not give proofs. In the second part we prove a version of the kernel theorem. This general abstract theorem will be of fundamental importance for defining bilinear operations on vector-valued distributions. Algebraic Tensor Products Let V and W be vector spaces. A tensor product of V and W is a pair (T, β) consisting of a vector space T and a bilinear map β : V × W → T such that

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391

¡ ¢ (i) T = span im(β) , (ii) β(V × W) is linearly independent in T , if V and W are linearly independent in V and W , respectively. It can be shown that there exists a tensor product of V and W and that it is unique, except for vector space isomorphisms. If (T, β) is a tensor product of V and W , then we put V ⊗ W := T ,

v ⊗ w := β(v, w) ,

(v, w) ∈ V × W ,

which is justified by ‘uniqueness’. 1.4.1 Remarks (a) The tensor product has the following important universality property: if U is a vector space and b : V × W → U is bilinear, there exists a unique linear map B : V ⊗ W → U such that the diagram V ×W

β

@ b@ @ R

- V ⊗W ¡ ¡ ¡ ª B

U

is commutative. Proof Let V and W be bases of V and W , respectively. Then β(V, W) is a basis of V ⊗ W by (i) and (ii) above. Define B on β(V, W) by B(v ⊗ w) := b(v, w) ,

(v, w) ∈ V × W ,

and extend it linearly. Then B has the desired property.

¥

(b) Let Vj and Wj be vector spaces and Aj ∈ Hom(Vj , Wj ) for j = 1, 2. Then there exists a unique A1 ⊗ A2 ∈ Hom(V1 ⊗ V2 , W1 ⊗ W2 ) , the tensor product of A1 and A2 , such that (A1 ⊗ A2 )(v1 ⊗ v2 ) = A1 v1 ⊗ A2 v2 ,

(v1 , v2 ) ∈ V1 × V2 .

Proof Since (v1 , v2 ) 7→ A1 v1 ⊗ A2 v2 is a bilinear map from V1 × V2 into W1 ⊗ W2 , the assertion follows from (a). ¥ (c) Let V1 , V2 , and V3 be vector spaces. There exists a linear isomorphism V1 ⊗ (V2 ⊗ V3 ) → (V1 ⊗ V2 ) ⊗ V3

(1.4.1)

v1 ⊗ (v2 ⊗ v3 ) 7→ (v1 ⊗ v2 ) ⊗ v3 .

(1.4.2)

such that

This means that tensor products are (canonically) associative (so that parentheses can be omitted).

392

Appendix

Proof Let Vj be a basis of Vj . Define (1.4.1) by (1.4.2) on the basis V1 ⊗ (V2 ⊗ V3 ) of V1 ⊗ (V2 ⊗ V3 ) and extend it linearly. ¥ Basic Examples 1.4.2 Examples (a) Let Km×n be the vector space of all (m × n)-matrices with entries in K and let Km × Kn → Km×n ,

(x, y) 7→ [xj yk ]1≤j≤m . 1≤k≤n

Then Km ⊗ Kn = Km×n . Note that x ⊗ y = xy > ,

x, y ∈ Km × Kn ,

if Km and Kn are identified with Km×1 and q Kn×1 , respectively, and a> ∈ Ks×r denotes the transposed of a ∈ Kr×s for r, s ∈ N. Furthermore, x ⊗ y = hy, ·ix ,

(x, y) ∈ Km × Kn ,

if Km×n and (Kn )0 are identified with L(Kn , Km ) and Kn , respectively. (b) Let S be a nonempty set, V a vector space, and Φ(S) a vector subspace of KS . Define a bilinear map Φ(S) × V → V S ,

(ϕ, v) 7→ ϕ ⊗ v

by ϕ ⊗ v(s) := ϕ(s)v. Then © ª Φ(S) ⊗ V := span ϕ ⊗ v ; ϕ ∈ Φ(S), v ∈ V is the tensor product of Φ(S) and V in V S , that is, the span lies in V S (cf. Subsections V.2.4 and 1.3). Proof Let ϕ1 , . . . , ϕn be linearly independent in Φ(S) andPv1 , . . . , vn linearly independent such that jk ξjk ϕj ⊗ vk = 0. Pin V . Suppose that P there exist ξjk ∈ K P Put ψk := j ξjk ϕj so that k ψk ⊗ vk = 0. Then ψk (s)vk = 0 for each s ∈ S. This implies ψk = 0 for k = 1, . . . , n by the linear independence of v1 , . . . , vn . Since the ϕj are linearly independent in Φ(S), we see that each ξjk is zero. ¥ (c) Suppose that F ∈ {D, E, S, OM }. Then F(X) ⊗ E, defined by (1.3.17) and (1.3.18), is the tensor product of F(X) and E in F(X, E). Similarly, if F ∈ {D, E, S} then F0 (X) ⊗ E, defined by (1.3.19) and (1.3.20), is the tensor product of F0 (X) and E in F0 (X, E). Proof

This follows from (b) by putting S := X and S := F(X), respectively.

¥

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393

(d) Let E be a Banach space, S and T nonempty sets, and Φ(S) and Ψ(T, E) vector subspaces of KS and E T , respectively. Define a bilinear map Φ(S) × Ψ(T, E) → E S×T ,

(ϕ, ψ) 7→ ϕ ⊗ ψ

by ϕ ⊗ ψ(s, t) := ϕ(s)ψ(t). Then © ª Φ(S) ⊗ Ψ(T, E) := span ϕ ⊗ ψ ; ϕ ∈ Φ(S), ψ ∈ Ψ(T, E) is the tensor product of Φ(S) and Ψ(T, E) in E S×T . Proof

Identify E S×T and (E T )S by means of £ ¤ £ ¤ (s, t) 7→ u(s, t) ←→ s 7→ u(s, ·) .

Then the assertion follows from (b).

¥

Projective Tensor Products Let now F := (F, P) and G := (G, Q) be LCSs. For (p, q) ∈ P × Q we put ©P ª P p ⊗π q(z) := inf p(fj )q(gj ) ; z = fj ⊗ gj , z ∈F ⊗G . This defines a seminorm on F ⊗ G, the tensor product of the seminorms p and q. In fact, it is the Minkowski functional of the absolutely convex hull of Bp ⊗ Bq where Bp := [p < 1]. It has the property p ⊗π q(f ⊗ g) = p(f )q(g) ,

(f, g) ∈ F × G .

(1.4.3)

Thus the family of the tensor product seminorms is separating. Consequently, it defines a locally convex Hausdorff topology on F ⊗ G, the projective topology, and ¡ ¢ F ⊗π G := F ⊗ G, { p ⊗π q ; p ∈ P, q ∈ Q } ∼

is the projective tensor product of F and G. We denote by F ⊗ G the completion of the LCS F ⊗π G. Nuclear Maps and Spaces A linear map N from the LCS F into a Banach space E is nuclear if there exist an equicontinuous sequence (fj0 ) in F 0 , a bounded sequence (bj ) in E, and a summable sequence (λj ) in K such that X Nf = λj hfj0 , f ibj , f ∈F . (1.4.4) j

The LCS F is nuclear if each continuous linear map from F into a Banach space is nuclear. It is conuclear if its dual is nuclear.

394

Appendix

For the following theorem we recall convention (1.1.11). 1.4.3 Theorem Let X be a nonempty open subset of Rn and suppose that F belongs to {D, E, S, OM }. Then F(X) is nuclear and conuclear. Proof If F ∈ {D, E, S}, this can be found in [Tre67] (also cf. [Jar81] and [Sch71]). As for F = OM , we refer to [Gro55, II.4.4]. ¥ Projective Tensor Products and Maps of Finite Rank Lastly, we define an injective linear map by τ : F ⊗ G → L(F 0 , G) ,

f ⊗ g 7→ h·, f ig ,

(1.4.5)

the canonical injection. Then we prepare for the proof of the abstract kernel theorem by deriving a series of lemmas. 1.4.4 Lemma Let F be reflexive and nuclear. Then τ is a toplinear isomorphism from F ⊗π G onto the subspace of L(F 0 , G) of maps of finite rank. Proof

(1) Given a continuous seminorm q on G and a bounded subset B 0 of F 0 ,

¡ ¢ X sup q τ z(f 0 ) ≤ sup |hf 0 , fj i| q(gj ) ,

f 0 ∈B 0

j

z=

X

f 0 ∈B 0

fj ⊗ gj ∈ F ⊗ G . (1.4.6)

Since f 7→ p(f ) := sup |hf 0 , f i| f 0 ∈B 0

defines a continuous seminorm on F 00 = F , it follows from (1.4.6) that ¡ ¢ sup q τ z(f 0 ) ≤ p ⊗π q(z) ,

z ∈F ⊗G .

f 0 ∈B 0

Hence

¡ ¢ τ ∈ L F ⊗π G, L(F 0 , G) .

It is obvious that τ z has finite rank for z ∈ F ⊗ G. Conversely, suppose that T ∈ L(F 0 , G) has finite rank, and let g1 , . . . , gm be a basis of im(T ). Then Tf0 =

m X

ξj (f 0 )gj ,

f0 ∈ F0 ,

j=1

where ξj (f 0 ) = hgj0 , T f 0 iG = hT 0 gj0 , f 0 iF 0 ,

f0 ∈ F0 ,

Tensor Products and Convolutions

395

the gj0 ∈ G0 satisfying hgj0 , gk i = δjk . Since T 0 ∈ L(G0 , F ) by the reflexivity of F , it Pm follows that fj := T 0 gj0 ∈ F and T = j=1 h·, fj iF gj . Hence T = τz

with

z :=

m X

fj ⊗ gj ∈ F ⊗ G .

j=1

This shows that τ is a bijection from F ⊗ G onto the subspace of L(F 0 , G) of finite rank operators. (2) Now let p and q be continuous seminorms onPF and G, respectively. Given T ∈ L(F 0 , G) of finite rank, let z := τ −1 T , and let fj ⊗ gj be a representation of z in F ⊗ G. Denote by © ª B◦p := f 0 ∈ F 0 ; |hf 0 , f i| ≤ 1 for f ∈ Bp the polar of Bp . Then p ⊗ε q(z) :=

sup ◦ (f 0 ,g 0 )∈B◦ p ×Bq

¯X ¯ ¯ ¯ ¯ hf 0 , fj ihg 0 , gj i¯ j

¯­ ®¯ ¡ ¢ = sup sup ¯ g 0 , τ z(f 0 ) ¯ = sup q τ z(f 0 ) 0 ◦ f 0 ∈B◦ p g ∈Bq

(1.4.7)

f 0 ∈B◦ p

by the bipolar theorem. Since F is reflexive, B◦p is weakly compact by the AlaogluBourbaki theorem. Hence it is weakly bounded and thus, thanks to Mackey’s theorem, bounded in F 0 (cf. [Hor66], [Jar81], [Tre67], or [Sch71] for these standard theorems from the theory of LCSs). Observe that p ⊗ε q, as defined in (1.4.7), is a seminorm on F ⊗ G. In the theory of LCSs it is shown that the family { p ⊗ε q ; p ∈ P, q ∈ Q } is a separating family of seminorms on F ⊗ G defining the ‘injective tensor product topology’ on F ⊗ G. Since F is nuclear, this injective tensor product topology coincides with the projective tensor product topology of F ⊗ G (e.g., [Tre67, Theorem 50.1(f)]). Thus we infer from (1.4.7) that τ −1 ∈ L(R, F ⊗π G), where R is the linear subspace of L(F 0 , G) of finite rank operators. ¥ Approximation by Maps of Finite Rank Let p be a continuous seminorm on F . Then ker(p) is a closed linear subspace of F . Hence Fp := F/ ker(p) is a normed vector space with respect to the quotient norm © ª x b := x bp := x + ker(p) 7→ pb(b x) := inf p(y) ; y ∈ x b . (1.4.8) Observe that pb(b x) = p(y) for y ∈ x b. Let q be a second continuous seminorm on F such that q ≥ p. Then ker(q) ⊂ ker(p) implies that x bq 7→ x bp is a well-defined continuous linear map from Fq into Fp of norm at most one. Let Fep be the completion of Fp . Then we consider x bq 7→ x bp to be a linear map from Fq into Fep , the canonical e map Fq → Fp .

396

Appendix

1.4.5 Lemma Suppose that F is nuclear and p is a continuous seminorm on F . Then there exists a continuous seminorm q ≥ p on F such that the canonical map Fq → Fep is nuclear. Proof Since F is nuclear, the quotient map πp : F → Fep is nuclear. Hence πp has a representation of the form (1.4.4) where (bj ) is a bounded sequence in Fep . Consequently, p(f ) = pb(πp f ) ≤ c sup |hfj0 , f i| =: q(f ) ,

f ∈F ,

(1.4.9)

j

P with c := j |λj | supj kbj k. The equicontinuity of the sequence (fj0 ) implies that q is a continuous seminorm on F . By (1.4.8) and (1.4.9), each fj0 defines naturally a continuous linear form h0j on Fq of norm at most 1/c. Thus the canonical map X ­ ® Fq → Fep , fbq 7→ λj h0j , fbq Fq bj j

is nuclear.

¥

By means of the preceding lemma we can give sufficient conditions for linear operators to be approximable by operators of finite rank. 1.4.6 Lemma If F is nuclear, then the maps of finite rank are dense in L(F, G). Proof Let p be a continuous seminorm on F . Lemma 1.4.5 guarantees the existence of a continuous seminorm q ≥ p on F such that the canonical map S : Fq → Fep is nuclear. Hence there exist a summable sequence (λj ) in K and bounded sequences ¡ 0¢ ¡ ¢ fbj and fbj in (Fq )0 and Fep , respectively, such that X ­ ® S fb = λj fbj0 , fb fbj , fb ∈ Fq . j

Thus, letting Sn :=

n X

­ ® λj fbj0 , · fbj ,

n∈N,

j=0

it is obvious that Sn → S

¡ ¢ in L Fq , Fep .

(1.4.10)

Observe that the diagram F

id

-

F

πq

πp

? Fq

S

-

? Fp

(1.4.11)

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397

is commutative, where πq and πp are the quotient maps. Choose fj ∈ fbj and put Tn f :=

n X

­ ® λj fbj0 , πq f fj ,

f ∈F .

j=0

Then, thanks to (1.4.10) and (1.4.11), p(Tn f − f ) → 0 ,

n→∞,

(1.4.12)

uniformly with respect to f in bounded subsets of F . Now let R ∈ L(F, G) and a continuous seminorm r on G be given. Then p := r ◦ R is a¡ continuous¢ seminorm on F . Put Rn := R ◦ Tn . It follows from (1.4.12) that r (Rn − R)f → 0 as n → ∞, uniformly with respect to f in bounded subsets of F . Since Rn ∈ L(F, G) has finite rank, the assertion follows. ¥ Completeness of Spaces of Linear Operators In our last preparatory lemma we give conditions for L(F 0 , G) to be complete. 1.4.7 Lemma Let F and G be complete and let F be reflexive. Then L(F 0 , G) is complete. Proof Let (Tα ) be a Cauchy net in L(F 0 , G). Since G is complete and (Tα f 0 ) is a Cauchy net in G, there exists T ∈ Hom(F 0 , G) such that Tα f 0 → T f 0 in G for each f 0 ∈ F 0 . Given g 0 ∈ G0 , hf 0 , Tα0 g 0 iF = hg 0 , Tα f 0 iG → hg 0 , T f 0 iG = g 0 ◦ T (f 0 )

(1.4.13)

for each f 0 ∈ F 0 , thanks to F 00 = F . Observe that (Tα0 g 0 ) is a Cauchy net in F . Hence we deduce from (1.4.13) and the completeness of F that g 0 ◦ T ∈ F for each g 0 ∈ G0 . w∗

Suppose that gα0 −→ g 0 . Then gα0 ◦ T (f 0 ) = hgα0 , T f 0 iG → hg 0 , T f 0 iG ,

f0 ∈ F0 .

This shows that t T ∈ L(G0w∗ , Fw ), where t T is the algebraic dual of T . Let q be a continuous seminorm on G. Then B◦q is w∗ -compact by the Alaoglu-Bourbaki theorem. Hence K := t T (B◦q ) is weakly compact as well, thus weakly bounded. Consequently, K is bounded in F . Hence V 0 := K ◦ is a neighborhood of zero in F 0 . Observe that f 0 ∈ V 0 iff |hf 0 , t T g 0 iF | = |hg 0 , T f 0 iG | ≤ 1 for g 0 ∈ B◦q , that is, iff T f 0 ∈ ◦ (B◦q ), where, given C 0 ⊂ G0 , © ª ◦ 0 C := g ∈ G ; |hg 0 , gi| ≤ 1 for g 0 ∈ C 0 is the polar of C 0 in G. By the bipolar theorem ◦ (B◦q ) = Bq . Hence © ª sup q(T f 0 ) ; f 0 ∈ V 0 ≤ 1 .

(1.4.14)

398

Appendix

Since V 0 is the polar of K, it is absolutely convex and w∗ -closed. Thus it is weakly closed by the reflexivity of F . Hence it is closed. Denoting by p the Minkowski functional of V 0 , it follows that f 0 ∈ V 0 iff p(f 0 ) ≤ 1. From this and (1.4.14) we infer that q(T f 0 ) ≤ p(f 0 ) , f0 ∈ F0 , which shows that T ∈ L(F 0 , G), thanks to the fact that p is a continuous seminorm on F 0 . ¥ The Abstract Kernel Theorem After these preparations we can prove the abstract kernel theorem in a form that is most suitable for our purposes. Recall that τ is defined in (1.4.5). 1.4.8 Theorem Let F and G be complete LCSs such that F is reflexive, nuclear, and conuclear. Then τ is (that is, extends to) a toplinear isomorphism ∼

τ : F ⊗ G → L(F 0 , G) ,

(1.4.15)

the canonical isomorphism. Proof It follows from Lemma 1.4.7 that L(F 0 , G) is complete. Since F is conuclear, we infer from Lemma 1.4.6 that the linear subspace R of maps of finite rank is dense in L(F 0 , G). Since F is also nuclear, Lemma 1.4.4 guarantees that τ is a toplinear isomorphism from F ⊗π G onto R. Now the assertion is an easy ∼ consequence of the density of F ⊗π G in F ⊗ G and a well-known result about continuous extensions of continuous linear maps (e.g., [Jar81, Theorem 3.4.2]). ¥ Tensor Product Characterizations of Some Distribution Spaces As a first application of this general theorem we can prove the following characterizations for spaces of vector-valued distributions. They are the basis for defining bilinear maps of vector-valued distributions in the next subsection. 1.4.9 Theorem Let E be a Banach space and F ∈ {D, E, S, OM }. Then ¡ ¢ ∼ F(X) ⊗ E = F(X, E) ∼ = L F0 (X), E and

¡ ¢ ∼ F0 (X) ⊗ E ∼ = F0 (X, E) = L F(X), E ,

where ∼ = denotes the canonical toplinear isomorphism. Proof From Theorem 1.1.2 we know that F(X) and F0 (X) are complete and reflexive. Furthermore, F(X) and F0 (X) are both nuclear and conuclear, thanks to

Tensor Products and Convolutions

399

Theorem 1.4.3. Hence it follows from Theorem 1.4.8 that, given F ∈ {D, E, S, OM }, ¡ ¢ ∼ τ : F(X) ⊗ E → L F0 (X), E and

¡ ¢ ∼ τ : F0 (X) ⊗ E → L F(X), E = F0 (X, E)

are toplinear isomorphisms. This proves the second assertion and part of the ∼ first one. It remains to show that F(X) ⊗ E = F(X, E). Since F(X) ⊗ E is dense in F(X, E) by Theorem 1.3.6, we have to verify that F(X, E) induces on F(X) ⊗ E the projective topology. Suppose that F ∈ {E, S, OM } and let p be one of the seminorms (1.1.3), (1.1.7), or (1.1.9), respectively. Then p(ϕ ⊗ e) = p(ϕ) |e| ,

ϕ ∈ F(X) ,

e∈E ,

where on the left side p is the seminorm on F(X, E), and on the right side p denotes the corresponding seminorm on F(X). From this we easily deduce that F(X, E) induces on F(X) ⊗ E a topology that is weaker than the projective tensor product topology. P Conversely, let ϕj ⊗ ej be a representation of z ∈ F(X) ⊗ E. Then, given ϕ0 ∈ F0 (X) and e0 ∈ E 0 , D X E D D X EE X hϕ0 , ϕj ihe0 , ej i = e0 , hϕ0 , ϕj iej = e0 , ϕ0 , ϕj ⊗ e j . Consequently (cf. (1.4.7)), p ⊗ε |·|E (z) = sup |hϕ0 , zi|E ≤ p(z) ϕ0 ∈B0p

by the bipolar theorem. This shows that the topology induced by F(X, E) on F(X) ⊗ E is stronger than the injective tensor product topology. Since F(X) is nuclear, the latter coincides with the projective topology. Finally, given K b X, the last string of arguments shows that DK (X, E) induces on DK (X) ⊗ E the projective topology. Since D(X, E) induces on DK (X, E) its original topology, it follows that D(X, E) induces on DK (X) ⊗ E the projective topology. From D(X) ⊗ E = −−→ lim DK (X) ⊗ E KbX

(e.g., [Jar81, Corollary 4 to Theorem 15.5.3]) it follows that D(X, E) induces on D(X) ⊗ E the projective tensor product topology. Thus, since D(X) ⊗ E is dense ∼ in D(X, E) by Theorem 1.3.6, it follows that D(X) ⊗ E = D(X, E). ¥

400

Appendix

Suppose that F and G are Fr´echet spaces such that F is nuclear. Then ∼

∼

(F ⊗ G)0 = F 0 ⊗ G0

(1.4.16)

by means of the duality pairing induced by hf 0 ⊗ g 0 , f ⊗ gi := hf 0 , f ihg 0 , gi

(1.4.17)

for (f, g) ∈ F × G and (f 0 , g 0 ) ∈ F 0 × G0 (e.g., [Sch71, Theorem IV.9.9]). By means of this fact we can identify the dual of F(X, E) for F ∈ {E, S}. 1.4.10 Corollary Let E be a Banach space and suppose that F ∈ {E, E 0 , S, S 0 }. Then ¡ ¢0 ∼ ∼ F(X, E)0 = F(X) ⊗ E = F0 (X) ⊗ E 0 = F0 (X, E 0 ) by means of the duality pairing induced by (1.4.17). If E is reflexive, then F(X, E) is reflexive as well. In Theorem 1.7.5 below it is shown that this duality coincides with the one introduced in Subsection VI.1.3. There are various versions of the abstract kernel theorem in the textbook literature (e.g., [Jar81, Theorem 21.5.9], [Tre67, Section 50]). However, it is always assumed that F and G are Fr´echet spaces. Since we need to apply Theorem 1.4.8 in the special case that F = OM (Rn ) and since OM (Rn ) is not metrizable, this assumption does not fit our purposes. For this reason we have included a complete proof of Theorem 1.4.8.

1.5

Extending Bilinear Maps

In this subsection we prove a general extension theorem for bilinear maps on tensor products. For this we need a technical lemma guaranteeing a suitable kind of uniformity for equicontinuous sets of nuclear maps. 1.5.1 Lemma Let F be a nuclear LCS and E a Banach space. Suppose that T is an equicontinuous subset of L(F, E). Then there exist a summable sequence (λj ) in K, an equicontinuous sequence (fj0 ) in F 0 , and a bounded map T → B(N, E) , such that Tf =

X j

¡ ¢ T 7→ ej (T ) j∈N

λj hfj0 , f iej (T ) ,

f ∈F ,

T ∈T .

Tensor Products and Convolutions

401

T© −1 Proof By the equicontinuity of T , the set T (BE ) ; T ∈ T } is a closed absolutely convex neighborhood of zero. Let p be its Minkowski functional. Then p is a continuous seminorm on F and |T f |E ≤ p(f ) ,

f ∈F ,

T ∈T .

(1.5.1)

The nuclearity of F implies, thanks to Lemma 1.4.5, the existence of a continuous seminorm q ≥ p on F such that the canonical map S : Fq → Fep is nuclear. Hence ¡ ¢ there exist a summable sequence (λj ) in K and bounded sequences fbj0 and (b gj ) 0 e in (Fq ) and Fp , respectively, such that X ­ ® S fb = λj fbj0 , fb gbj , fb ∈ Fq . (1.5.2) j

¡ ¢ It is easy to verify that for each T ∈ T there exists Te ∈ L Fep , E such that the diagram πq - Fq F T

? E

S

¾

?

T

Fp

is commutative, where πq is the quotient map. Hence, letting fj0 := (πq )0 fbj0 and ° ° ej (T ) := Tegbj , the assertion follows since (1.5.1) and (1.4.8) imply °Te° ≤ 1. ¥ Now we are in a position to prove the following general extension theorem for bilinear maps. Throughout the remainder of this subsection we assume that Ej , j = 0, 1, 2, are Banach spaces and E1 × E2 → E0 ,

(e1 , e2 ) 7→ e1 q e2

(1.5.3)

is a multiplication. Furthermore, given an LCS F , we put F (E) := L(F 0 , E) and we remind the reader of the definition in Subsection II.1.1 of point-wise multiplication induced by (1.5.3). 1.5.2 Proposition Let Fj , j = 0, 1, 2, be LCSs such that (i) F1 is reflexive, complete, nuclear, and conuclear. (ii) F0 (E0 ) is complete. Suppose that there is given a hypocontinuous bilinear map F1 × F2 (E2 ) → F0 (E2 ) ,

(f1 , v) 7→ f1 ¡ v .

Define a bilinear map ¡ q : (F1 ⊗ E1 ) × F2 (E2 ) → F0 (E0 )

(1.5.4)

402

Appendix

by (f1 ⊗ e1 , v) 7→ e1 q (f1 ¡ v) ,

(e1 , f1 ) ∈ E1 × F1 ,

(1.5.5)

and by bilinear extension. Then ¡ q possesses a unique hypocontinuous bilinear extension ∼

¡ q : (F1 ⊗ E1 ) × F2 (E2 ) → F0 (E0 ) ,

(u, v) 7→ u ¡ q v . ∼

Proof Uniqueness follows immediately from the density of F1 ⊗ E1 in F1 ⊗ E1 . P Given u := fj ⊗ ej ∈ F1 ⊗ E1 and v ∈ F2 (E2 ), X u ¡q v = ej q (fj ¡ v) ∈ F0 (E0 ) . (1.5.6) Let B00 be a bounded subset of F00 . Then p(w) := sup |wf 0 |E0 ,

w ∈ F0 (E0 ) ,

p2 (v) := sup |vf 0 |E2 ,

v ∈ F0 (E2 ) ,

f 0 ∈B00

and

f 0 ∈B00

define continuous seminorms on F0 (E0 ) and F0 (E2 ), respectively. Hence X p(u ¡ q v) ≤ p2 (fj ¡ v) |ej |E1 .

(1.5.7)

j

Let B2 be a bounded subset of F2 (E2 ). Then (1.5.7) and the hypocontinuity of (1.5.4) imply the existence of a continuous seminorm p1 on F1 such that X p(u ¡ q v) ≤ p1 (fj ) |ej |E1 , v ∈ B2 . j

Since p1 is independent of the particular representation of u, it follows that p(u ¡ q v) ≤ (p1 ⊗π |·|)(u) ,

u ∈ F1 ⊗ E1 ,

v ∈ B2 .

This shows that ¡ ¢ (u 7→ u ¡ q v) ∈ L F1 ⊗π E1 , F0 (E0 ) ,

(1.5.8)

uniformly with respect to v in bounded subsets of F2 (E2 ). Given v ∈ F2 (E2 ), there exists a unique continuous extension ¡ ∼ ¢ Uv ∈ L F1 ⊗ E1 , F0 (E0 )

(1.5.9)

Tensor Products and Convolutions

403

of (1.5.8), thanks to assumption (ii). Then £ ¤ ∼ (u, v) 7→ u ¡ q v := Uv (u) : (F1 ⊗ E1 ) × F2 (E2 ) → F0 (E0 )

(1.5.10) ∼

is an extension of (1.5.5) that is trivially linear in the first variable. For u ∈ F1 ⊗ E1 ∼ choose a net (uα ) in F1 ⊗ E1 such that uα → u in F1 ⊗ E1 . Then, given λj ∈ K and vj ∈ F2 (E2 ) for j = 1, 2, £ ¤ u ¡ q (λ1 v1 + λ2 v2 ) = lim uα ¡ q (λ1 v1 + λ2 v2 ) α

= λ1 lim(uα ¡ q v1 ) + λ2 lim(uα ¡ q v2 ) α

α

= λ1 (u ¡ q v1 ) + λ2 (u ¡ q v2 ) . Thus (1.5.10) is bilinear and it remains to show that it is hypocontinuous. The uniformity assertion contained in (1.5.8) implies that the family of linear operators (1.5.8) is equicontinuous if v stays in bounded subsets of F2 (E2 ). Thus, given a closed neighborhood V0 of zero in F0 (E0 ) and a bounded subset B2 of F2 (E2 ), there exists a neighborhood V1 of zero in F1 ⊗π E1 such that u ¡ q v ∈ V0 for (u, v) ∈ V1 × B2 . Consequently, Uv (V1 ) ⊂ V0 ,

v ∈ B2 ,

∼

where V1 is the closure of V1 in F1 ⊗ E1 . This shows that { Uv ; v ∈ B2 } is equicon¡ ∼ ¢ ∼ tinuous in L F1 ⊗ E1 , F0 (E0 ) . Hence u ¡ q v → 0 in F0 (E0 ) as u → 0 in F1 ⊗ E1 , uniformly with respect to v in bounded subsets of F2 (E2 ). ∼

Now let B be a bounded subset of F1 ⊗ E1 . Thanks to assumption (i) and Theorem 1.4.8, ∼ F1 ⊗ E 1 ∼ = L(F10 , E1 ) = F1 (E1 ) .

(1.5.11)

Put B1 := τ (B), where τ is the isomorphism of (1.5.11). Then B1 is bounded in F1 (E1 ). Since F10 is barreled, being the dual of a reflexive space, hence reflexive, the uniform boundedness principle shows that B1 is equicontinuous. Thus Lemma 1.5.1 guarantees the existence of a summable sequence (λj ) in K, a bounded sequence (fj ) in F1 , and a bounded set C in E1 such that each u ∈ B1 has a repP ∼ resentation u = j λj fj ⊗ ej , where ej ∈ C and the series converges in F1 ⊗ E1 . Hence, given v ∈ F2 (E2 ), it follows from (1.5.9), (1.5.10), and (1.5.6) that X u ¡q v = λj ej q (fj ¡ v) . j

Consequently, p(u ¡ q v) ≤

X j

|λj | p2 (fj ¡ v) |ej |E1 ,

u ∈ B1 ,

v ∈ F2 (E2 ) .

(1.5.12)

404

Appendix

From the hypocontinuity of (1.5.4) and the boundedness of the sequence (fj ) in F1 we deduce the existence of a continuous seminorm q on F2 (E2 ) such that p2 (fj ¡ v) ≤ q(v) ,

v ∈ F2 (E2 ) ,

j∈N.

(1.5.13)

Since ej ∈ C and C is bounded in E1 , we infer from (1.5.12) and (1.5.13) that p(u ¡ q v) ≤ cq(v) , This proves that

v ∈ F2 (E2 ) ,

u ∈ B1 .

¡ ¢ (v 7→ u ¡ q v) ∈ L F2 (E2 ), F0 (E0 ) , ∼

uniformly with respect to u in bounded subsets of F1 ⊗ E1 . Thus the map (1.5.10) is hypocontinuous. ¥ By specializing we deduce from the preceding proposition the following fundamental extension result: 1.5.3 Theorem Let Fj , j = 0, 1, 2, be reflexive, complete, nuclear, and conuclear LCSs. Suppose that there are a bilinear map F1 × F2 → F0 ,

(f1 , f2 ) 7→ f1 ¯ f2

(1.5.14)

and a hypocontinuous bilinear map ∼

∼

F1 × (F2 ⊗ E2 ) → F0 ⊗ E2 ,

(f1 , v) 7→ f1 } v

(1.5.15)

such that f1 } (f2 ⊗ e2 ) = (f1 ¯ f2 ) ⊗ e2 ,

fj ∈ Fj ,

j = 1, 2 ,

e2 ∈ E2 . (1.5.16)

Then there exists a unique hypocontinuous bilinear map ∼

∼

∼

¯ q : (F1 ⊗ E1 ) × (F2 ⊗ E2 ) → F0 ⊗ E0 ,

(u, v) 7→ u ¯ q v

(1.5.17)

satisfying (f1 ⊗ e1 ) ¯ q (f2 ⊗ e2 ) = (f1 ¯ f2 ) ⊗ (e1 q e2 )

(1.5.18)

for fj ∈ Fj and ej ∈ Ej , j = 1, 2. ∼

Proof Since Ej ⊗ Fj is dense in Ej ⊗ Fj for j = 1, 2, there exists at most one ∼ ∼ ∼ separately continuous bilinear map ¯ q from (F1 ⊗ E1 ) × (F2 ⊗ E2 ) to F0 ⊗ E0 satisfying (1.5.18).

Tensor Products and Convolutions

405 ∼

Thanks to Theorem 1.4.8 we can identify Fj ⊗ Ej with Fj (Ej ) by means of the respective canonical isomorphisms. Then it follows from (1.5.16) that, given f00 ∈ F00 and v := f2 ⊗ e2 ∈ F2 ⊗ E2 , £

¤ £ ¤ e1 q (f1 } v) (f00 ) = e1 q (f1 ¯ f2 ) ⊗ e2 (f00 ) = e1 q hf00 , f1 ¯ f2 ie2 £ ¤ = hf00 , f1 ¯ f2 i(e1 q e2 ) = (f1 ¯ f2 ) ⊗ (e1 q e2 ) (f00 )

for f1 ∈ F1 and e1 ∈ E1 . Hence e1 q (f1 } v) = (f1 ¯ f2 ) ⊗ (e1 q e2 )

(1.5.19)

for f1 ⊗ e1 ∈ F1 ⊗ E1 and v := f2 ⊗ e2 ∈ F2 ⊗ E2 . By Proposition 1.5.2 there exists a unique hypocontinuous bilinear extension (1.5.17) of (1.5.15) satisfying (f1 ⊗ e1 ) ¯ q v = e1 q (f1 } v) , Thus (1.5.19) implies (1.5.18).

f1 ⊗ e1 ∈ F1 ⊗ E1 ,

∼

v ∈ F2 ⊗ E2 .

¥

The results of this subsection are due to L. Schwartz. In fact, they are special cases of much more general theorems given in [Schw57b, chap. II] (also see [Schw57a]). General Hypothesis Throughout the remainder of this section we suppose that E, F , and Ej , j = 0, 1, 2, are Banach spaces and E1 × E2 → E0 , (e1 , e2 ) 7→ e1 q e2 is a multiplication. Moreover, X is a nonempty open subset of Rn and convention (1.1.11) is effective.

1.6

(1.5.20)

Point-Wise Multiplication

As a first application of the general extension result of Theorem 1.5.3 we define point-wise multiplication of a vector-valued smooth function with a vector-valued distribution and study some of its properties. It is an immediate consequence of Leibniz’ rule that for each m ∈ N point-wise multiplication induced by (1.5.20), C m (X, E1 ) × C m (X, E2 ) → C m (X, E0 ) , is well-defined, bilinear and continuous.

(a1 , a2 ) 7→ a1 q a2 ,

(1.6.1)

406

Appendix

Given a ∈ E(X) and v ∈ D0 (X, E), we recall that av ∈ D0 (X, E) is defined by av(ϕ) := v(aϕ) ,

ϕ ∈ D(X) .

(1.6.2)

It is easily verified that

¡ ¢ (ϕ 7→ aϕ) ∈ L D(X) ,

a ∈ E(X) .

A Characterization of OM First we prove a characterization of OM (Rn , E) that extends (III.4.1.9). In addition, it explains the name ‘space of multipliers’ for OM (Rn , E). 1.6.1 Proposition Suppose that a ∈ E(Rn , E). Then ¡ ¢ a ∈ OM (Rn , E) iff (ϕ 7→ ϕa) ∈ L S(Rn ), S(Rn , E) . Given a ∈ OM (Rn , E) and k, m ∈ N, Leibniz’ rule and (1.1.8) imply X qk,m (ϕa) ≤ c max sup (1 + |x|2 )k |∂ β ϕ(x)| |∂ α−β a(x)| ≤ cq`,m (ϕ) (1.6.3)

Proof

|α|≤m

β≤α

x∈Rn

for ϕ ∈ S(Rn ), where ` := k + max{ mα−β ; β ≤ α, |α| ≤ m } and where mα−β ∈ N are such that there exists cα−β satisfying |∂ α−β a(x)| ≤ cα−β (1 + |x|2 )mα−β for x ∈ Rn . This shows that ¡ ¢ (ϕ 7→ ϕa) ∈ L S(Rn ), S(Rn , E) . (1.6.4) Conversely, let (1.6.4) be true. Then, given α ∈ N, there exist k, m ∈ N and a positive constant c such that k∂ α (ϕa)k∞ ≤ q0,|α| (ϕa) ≤ cqk,m (ϕ) ,

ϕ ∈ S(Rn ) .

Fix ψ ∈ D(Rn ) such that ψ equals 1 near zero. Since τx ψ ∈ D(Rn ) ⊂ S(Rn ) for x ∈ Rn , it follows that ¯ ¡ ¢ ¯ |∂ α a(x)| = ¯∂ α (τx ψ)a (x)¯ ≤ cqk,m (τx ψ) ≤ c sup (1 + |x − y|2 )k |∂ α ψ(y)| ≤ c (1 + |x|2 )k

(1.6.5)

|α|≤m y∈Rn

for x ∈ Rn , thanks to the trivial inequality 1 + |x − y|2 ≤ 2(1 + |x|2 )(1 + |y|2 ). Hence a ∈ OM (Rn , E). ¥ 1.6.2 Corollary Suppose that a ∈ E(Rn ). Then a ∈ OM (Rn )

iff

¡ ¢ (u 7→ au) ∈ L S 0 (Rn ) .

Proof For simplicity, we put OM = OM (Rn ) etc. If a ∈ OM , it follows from Proposition 1.6.1 that (ϕ 7→ ϕa) ∈ L(S). Hence the dual of this linear map, which is

Tensor Products and Convolutions

407

given by u 7→ au for u ∈ S 0 , satisfies (u 7→ au) ∈ L(S 0 ). Conversely, (u 7→ au) ∈ L(S 0 ) implies (u 7→ au)0 ∈ L(S) by the reflexivity of S. Since the latter dual is the map ϕ 7→ ϕa for ϕ ∈ S, the assertion follows. ¥ The General Theorem Now it is easy to prove the following technical lemma that will be used in the proof of the next theorem, one of the main results of this subsection. 1.6.3 Lemma Suppose that (F1 , F2 ; F0 ) is one of the triplets (E, D0 ; D0 ), (E, E 0 ; E 0 ), (S, OM ; S), (OM , OM ; OM ), (OM , S 0 ; S 0 ), (E, E; E), or (E, D; D). Then ‘point-wise multiplication’ F1 (X) × F2 (X, E) → F0 (X, E) ,

(a, u) 7→ au

(1.6.6)

is a well-defined hypocontinuous bilinear map. Proof (1) We see from (1.1.8) and (1.1.9) that B is bounded in OM (Rn , E) iff, given α ∈ Nn , there exist mα ∈ N and cα > 0 such that |∂ α u(x)| ≤ cα (1 + |x|2 )mα ,

x ∈ Rn ,

u∈B .

(1.6.7)

Hence (1.6.3) shows that ¡ ¢ (a 7→ au) ∈ L S(Rn ), S(Rn , E) , uniformly with respect to u in bounded subsets of OM (Rn , E). Since the function defined by x 7→ (1 + |x|2 )k ∂ β ϕ(x) belongs to S(Rn ) for ϕ ∈ S(Rn ), k ∈ N, and β ∈ Nn , it follows from (1.6.3) that, given k, m ∈ N and a ∈ S(Rn ), there exists ak,α ∈ S(Rn ) such that qk,m (au) ≤ c max kak,α ∂ α uk∞ . |α|≤m

It follows ¡ ¢ (u 7→ au) ∈ L OM (Rn , E), S(Rn , E) , a ∈ S(Rn ) . (1.6.8) ¡ ¢ Thus, letting Mu := (a 7→ au) ∈ L S(Rn ), S(Rn , E) , we see from (1.6.8) that ³ ¡ ¢´ (u 7→ Mu ) ∈ L OM (Rn , E), Ls S(Rn ), S(Rn , E) . Consequently, S(Rn ) being a Montel space, the Banach–Steinhaus theorem guarantees that ³ ¡ ¢´ (u 7→ Mu ) ∈ L OM (Rn , E), L S(Rn ), S(Rn , E) , that is, the map u 7→ au is continuous from OM (Rn , E) into S(Rn , E), uniformly with respect to a in bounded subsets of S(Rn ). This proves the assertion for the case (F1 , F2 ; F0 ) = (S, OM ; S).

408

Appendix

(2) Given a ∈ OM (Rn ) and u ∈ OM (Rn , E), it follows from Leibniz’ rule that ° X ° ° ° kϕ∂ α (au)k∞ ≤ c °ϕ ∂ β a∂ α−β u° , ϕ ∈ S(Rn ) . β≤α

∞

Thus (1.6.7) implies the existence of m ∈ N such that X kϕ∂ α (au)k∞ ≤ c k(1 + |x|2 )m ϕ∂ β ak∞ ,

u∈B .

β≤α

This shows that a 7→ au maps OM (Rn ) continuously into OM (Rn , E), uniformly with respect to u in bounded subsets of OM (Rn , E). Similarly, we see that u 7→ au maps OM (Rn , E) continuously into itself, uniformly with respect to a in bounded subsets of OM (Rn ). This proves the assertion if (F1 , F2 ; F0 ) = (OM , OM ; OM ). (3) Since supp(au) ⊂ supp(a) ∩ supp(u) for a ∈ E(X) and u ∈ D(X, E), Leibniz’ rule and the properties of LF spaces easily imply that point-wise multiplication is separately continuous from E(X) × D(X, E) into D(X, E). Now hypocontinuity in the case (E, D; D) follows from the fact that E(X) and D(X, E) are barreled. (4) If (F1 , F2 ; F0 ) = (E, E; E), then the assertion follows from (1.6.1). (5) Suppose © ª (F1 , F2 ; F0 ) ∈ (E, D0 ; D0 ), (E, E 0 ; E 0 ), (Om , S 0 ; S 0 ) and set F := F2 = F0 . By Theorem 1.4.9, ¡ ¢ F(X, E) = L F0 (X), E , © ª and reflexivity guarantees F0 (X) ∈ D(X), E(X), S(X) . Hence it follows from step (3), resp. step (4), that ¡ ¢ Ma := (ϕ 7→ aϕ) ∈ L F0 (X) , a ∈ F1 (X) , (1.6.9) and ¡ ¡ ¢ (a 7→ Ma ) ∈ L F1 (X), L F0 (X)

(1.6.10)

if (F1 , F) = (E, D0 ), resp. (F1 , F) = (E, E 0 ). If (F1 , F) = (Om , S 0 ), then (1.6.9) and (1.6.10) are implied by an obvious modification of step (1). Consider the bilinear map ¡ ¢ ¡ ¢ ¡ ¢ L F0 (X), F0 (X) × L F0 (X), E → L F0 (X), E , (M, u) → 7 uM . Lemma 1.1.1 guarantees that it is hypocontinuous. From this and since, by (1.6.10), the map a 7→ Ma map is bounded on bounded sets, we deduce that ¡ ¢ ¡ ¢ F1 (X) × L F0 (X), E → L F0 (E), E , a 7→ uMa = Ma u is hypocontinuous. This proves the assertion if (1.6.9) applies.

¥

Tensor Products and Convolutions

409

Now we can extend point-wise multiplication to the case that both factors are vector-valued. 1.6.4 Theorem There exists a unique hypocontinuous bilinear map E(X, E1 ) × D0 (X, E2 ) → D0 (X, E0 ) ,

(a, u) 7→ a q u ,

(1.6.11)

called point-wise multiplication induced by (1.5.20), such that (ϕ ⊗ e1 ) q (ψ ⊗ e2 ) = ϕψ ⊗ (e1 q e2 )

(1.6.12)

for a := ϕ ⊗ e1 ∈ D(X) ⊗ E1 and u := ψ ⊗ e2 ∈ D(X) ⊗ E2 . It restricts to a hypocontinuous bilinear map F1 (X, E1 ) × F2 (X, E2 ) → F0 (X, E0 ) , where (F1 , F2 ; F0 ) is any one of the triplets (E, E 0 ; E 0 ), (S, OM ; S), (OM , OM ; OM ), (OM , S 0 ; S 0 ), (E, E; E), or (E, D; D). Proof Let (F1 , F2 ; F0 ) be as stated or (F1 , F2 ; F0 ) = (E, D0 ; D0 ), and write Fj for Fj (X), j = 0, 1, 2. Then the spaces Fj are reflexive, complete, nuclear, and conuclear by Theorems 1.1.2 and 1.4.3. Moreover3 , ∼

Fj ⊗ Ej t = Fj (X, Ej ) ,

j = 0, 1, 2 ,

thanks to Theorem 1.4.9. Define the bilinear maps (1.5.14) and (1.5.15) by pointwise multiplication. It is easily verified that (1.5.16) is true and it follows from Lemma 1.6.3 that the map (1.5.15) is hypocontinuous. Hence Theorem 1.5.3 guarantees the existence of a unique hypocontinuous bilinear map F1 (X, E1 ) × F2 (X, E2 ) → F0 (X, E0 ) ,

(a, u) 7→ a q u

(1.6.13)

satisfying (1.6.12) for a := ϕ ⊗ e1 ∈ F1 (X) ⊗ E1 and u := v ⊗ e2 ∈ F2 (X) ⊗ E2 . The assertion is now an obvious consequence of the almost trivial fact that the separately continuous is uniquely determined by its restriction ¡ bilinear map ¢ (1.6.13) ¡ ¢ to the subspace D(X) ⊗ E1 × D(X) ⊗ E2 which, by Theorem 1.3.6, is dense in F1 (X, E1 ) × F2 (X, E2 ). ¥ Basic Properties of Multiplications 1.6.5 Remarks (a) Since E2 × E1 → E0 , (e2 , e1 ) 7→ e1 q e2 is a multiplication as well, the assertions of Theorem 1.6.4 are ‘symmetric’ with respect to E1 and E2 , that is, the roles of E1 and E2 can be interchanged. This fact will often be employed in the following, usually without further mention. 3 Here

∼

and below we identify F(X) ⊗ E with F(X, E) by means of the canonical isomorphism of Theorem 1.4.9.

410

Appendix

(b) Point-wise multiplication induced by (1.5.20), as defined in Theorem 1.6.4, coincides on regular distributions with point-wise multiplication in the usual sense. In other words, if a ∈ E(X, E1 ) and u ∈ L1,loc (X, E2 ) then a q u ∈ L1,loc (X, E0 ) and a q u(·) = a(·) q u(·) .

(1.6.14)

Observe that this justifies the use of the name ‘point-wise multiplication’ for the bilinear map of Theorem 1.6.4. Proof Since L1,loc (X, E2 ) ,→ D0 (X, E2 ), it follows from Theorem 1.6.4 and an obvious estimate that both sides of (1.6.14) are separately continuous bilinear maps E(X, E1 ) × L1,loc (X, E2 ) → D0 (X, E0 ) . Theorem 1.3.6(viii) immediately implies d

D(X) ⊗ E2 ,→ L1,loc (X, E2 ) .

(1.6.15)

Thus we ¡deduce from¢ (1.6.12) that both sides of (1.6.14) coincide on the dense ¡ ¢ subspace D(X) ⊗ E1 × D(X) ⊗ E2 of E(X, E1 ) × L1,loc (X, E2 ). Hence the assertion follows. ¥ (c) Leibniz’ rule is valid in the general case as well. More precisely, if p is a polynomial in n indeterminates then X 1 p(∂)(a q u) = (∂ β a) q p(β) (∂)u (1.6.16) β! β

for a ∈ E(X, E1 ) and u ∈ D0 (X, E2 ), where p(β) := ∂ β p. In particular: X³α ´ ∂ α (a q u) = (∂ β a) q ∂ α−β u β

(1.6.17)

β≤α

for a ∈ E(X, E1 ) and u ∈ D0 (X, E2 ). ¡ ¢ ¡ ¢ Proof It follows from ∂ α ∈ L E(X, E1 ) and p(β) (∂) ∈ L D0 (X, E2 ) that both sides of (1.6.16) define separately continuous bilinear maps: E(X, E1 ) × D0 (X, E2 ) → D0 (X, E0 ) . Thanks (1.6.12) they coincide on the linear sub¡ to Lemma¢1.3.2 ¡ and to property ¢ space D(X) ⊗ E1 × D(X) ⊗ E2 . Since, by Theorem 1.3.6(ii) and (v), this subspace is dense in E(X, E1 ) × D0 (X, E2 ), they coincide everywhere. ¥ (d) If (a, u) ∈ E(X, E1 ) × D0 (X, E2 ), then supp(a q u) ⊂ supp(a) ∩ supp(u) . © ª Proof Let K := supp(a) and denote by Kε := x ∈ X ; dist(x, K) < ε the open ε-neighborhood of K in X for ε > 0. Choose ψε ∈ E(X) with supp(ψε ) ⊂ Kε and

Tensor Products and Convolutions

411

ψ | K = 1. Since E(X) ⊗ E1 is dense in E(X, E1 ), there exists a sequence (aj ) in E(X) ⊗ E1 converging in E(X, E1 ) towards a. Then bj := ψε aj ∈ E(X) ⊗ E1 and supp(bj ) ⊂ Kε . It is easily verified that supp(bj q u) ⊂ K ε ∩ supp(u) =: Mε and that © ª 0 DM (X, E0 ) := v ∈ D0 (X, E0 ) ; supp(v) ⊂ Mε ε is a closed linear subspace of D0 (X, E0 ). Now we deduce from Theorem 1.6.4 that 0 bj q u → a q u in D0 (X, E0 ), hence in DM (X, E0 ). Consequently, supp(a q u) ⊂ Mε ε for each ε > 0, which implies the assertion. ¥ (e) Let E3 , E4 , and E5 be further Banach spaces and suppose that there are multiplications E1 × E2

E2 × E3

?

?

E0 × E3

E1 × E4

H HH j

E5

© © © ¼

all denoted by q , that are associative, that is, (e1 q e2 ) q e3 = e1 q (e2 q e3 ) ,

ej ∈ Ej ,

j = 1, 2, 3 .

(1.6.18)

Then point-wise multiplication is associative as well, when defined, that is, (u1 q u2 ) q u3 = u1 q (u2 q u3 ) , uj ∈ Fj (X, Ej ) , © ª where (F1 , F2 , F3 ) = (E, E, D0 ), (OM , OM , S 0 ) .

j = 1, 2, 3 ,

(1.6.19)

Proof It follows from Theorem 1.6.4 that both sides of (1.6.19) define separately continuous trilinear maps 3 Y

Fj (X, Ej ) → F3 (X, E5 )

j=1

that coincide on the dense linear subspace 0

0

Q3 j=1

D(X) ⊗ Ej , hence everywhere.

¥

0

(f ) Suppose that F ∈ {D, E, S, OM , D , E , S }. Then the map E1 × F(X, E2 ) → F(X, E0 ) ,

(e1 , u) 7→ e1 q u := (1 ⊗ e1 ) q u

(1.6.20)

is bilinear and continuous, and ∂ α (e1 q u) = e1 q ∂ α u ,

α ∈ Nn ,

e1 ∈ E1 ,

u ∈ F(X, E2 ) .

(1.6.21)

Proof Since 1 ⊗ e1 ∈ OM (Rn , E1 ) ,→ E(Rn , E1 ), it follows from Theorem 1.6.4 and Remark (a) that (1.6.20) is a well-defined separately continuous bilinear map.

412

Appendix

Moreover, given u = v ⊗ e2 ∈ F(X) ⊗ E2 and ϕ ∈ F0 (X), £ ¤ ¡ ¢ (e1 q u)(ϕ) = (1 ⊗ e1 ) q (v ⊗ e2 ) (ϕ) = hv, ϕiF0 e1 q e2 = e1 q hv, ϕiF0 e2 = e1 q u(ϕ) . Thus the density of F(X) ⊗ E2 in F(X, E2 ) implies (e1 q u)(ϕ) = e1 q u(ϕ) ,

e1 ∈ E1 ,

u ∈ F(X, E2 ) ,

ϕ ∈ F0 (X) .

(1.6.22)

Consequently, given a bounded subset B of F0 (X), sup |(e1 q u)(ϕ)|E0 ≤ |e1 |E1 sup |u(ϕ)|E2 ,

ϕ∈B

ϕ∈B

e 1 ∈ E1 ,

u ∈ F(X, E2 ) .

This proves the continuity of (1.6.20), thanks to Theorem 1.4.9. Lastly, (1.6.21) is a consequence of (c). ¥ (g) Let F ∈ {D, E, S, OM , D0 , E 0 , S 0 } and consider the multiplication L(E, F ) × E → F ,

(T, e) 7→ T q e := T e .

(1.6.23)

Then it follows from (f) that (1.6.23) induces a continuous bilinear map L(E, F ) × F(X, E) → F(X, F ) ,

(T, u) 7→ T q u := T u .

In particular, each T ∈ L(E, F ) induces a continuous linear map via (1.6.23), ¡ ¢ (u 7→ T q u) ∈ L F(X, E), F(X, F ) . (1.6.24) We denote it by T as well and call it the linear map induced by T ∈ L(E, F ) via pointwise multiplication. Moreover, (1.6.21) implies ∂ α ◦ T = T ◦ ∂ α for α ∈ Nn , that is, the diagram T - F(X, F ) F(X, E) ∂α

?

F(X, E)

∂α

T

-

?

F(X, F )

is commutative for α ∈ Nn . If T ∈ L(E, F ) is injective, then (1.6.24) is injective as well. Consequently, i : E ,→ F

implies

i : F(X, E) ,→ F(X, F ) .

Proof We have to prove only the injectivity assertion. Thus suppose that T u = 0 for some u ∈ F(X, E) ¡ ¢ and that T ∈ L(E, F ) is injective. Then we obtain from (1.6.22) that T u(ϕ) = 0 for each ϕ ∈ F0 (X). Hence u(ϕ) = 0 for each ϕ ∈ F0 (X), that is, u = 0. ¥

Tensor Products and Convolutions

413

(h) Suppose that E1 = E2 =: E and multiplication (1.5.20) is symmetric, that is, e1 q e2 = e2 q e1 for ej ∈ E. Then a q u = u q a for a ∈ E(X, E) and u ∈ D0 (X, E). Proof This is an immediate consequence of (b), the density of D(X, E) × D(X, E) in E(X, E) × D0 (X, E), and the separate continuity of point-wise multiplication. ¥ 1.6.6 Corollary Suppose that (E, q) is a [commutative] Banach algebra and that F ∈ {D, E, S, OM }. Then F(X, E) is a locally convex [commutative] algebra with respect to point-wise multiplication, a multiplication algebra. Multiplication is continuous if F ∈ {D, E, S}, and hypocontinuous if F = OM . If E possesses a unit, e0 , then E(X, E) and OM (Rn , E) possess a unit as well, namely 1 ⊗ e0 . Proof This follows from Theorem 1.6.4, Remarks 1.6.5(e) and (h), appropriate continuous injections given in Theorem 1.3.6, and the fact that hypocontinuous maps are continuous on barreled spaces. ¥

1.7

Scalar Products and Duality Pairings

In this subsection we show, in particular, that, given F ∈ {D, E, S}, there exists a unique hypocontinuous bilinear map F0 (X, E 0 ) × F(X, E) → K

(1.7.1)

that extends in a natural way the duality pairing F0 (X) × F(X) → K. In fact, we consider more general situations that will be needed in the remaining subsections. 1.7.1 Lemma Let F ∈ {D, E, S, D0 , E 0 , S 0 }. Then the bilinear map F(X, E) × F0 (X) → E ,

(u, ϕ) 7→ u(ϕ)

(1.7.2)

is hypocontinuous. ¡ ¢ Proof Since F00 = F, it follows from Theorem 1.4.9 that F(X, E) = L F0 (X), E . Hence the map (1.7.2) is well-defined and continuous in u, uniformly on bounded 0 subsets Thanks to the fact that F0 (X) is barreled, bounded subsets ¡ 0 of F (X). ¢ of L F (X), E are equicontinuous by the uniform boundedness principle. Thus (1.7.2) is continuous in ϕ, uniformly with respect to u varying in bounded subsets of F(X, E). ¥ It is now easy to prove the following general existence theorem that will imply, in particular, the desired extension (1.7.1). 1.7.2 Theorem Suppose that F ∈ {D, E, S, D0 , E 0 , S 0 }. Then there exists a unique hypocontinuous bilinear map F0 (X, E1 ) × F(X, E2 ) → E0 ,

(u0 , u) 7→ hu0 q uiF ,

(1.7.3)

414

Appendix

the scalar product induced by the multiplication (1.5.20), such that ­ ® (ϕ ⊗ e1 ) q (ψ ⊗ e2 ) F = hϕ, ψiD (e1 q e2 )

(1.7.4)

for ϕ, ψ ∈ D(X) and ej ∈ Ej , j = 1, 2. Proof Put F0 := K, F1 := F0 (X), and F2 := F(X). Then the Fj are reflexive, complete, nuclear, and conuclear LCSs. Consequently, Theorem 1.4.9 guarantees that ∼ ∼ F2 ⊗ E2 = F(X, E2 ), and it is trivially true that F0 ⊗ E0 = F0 ⊗ E0 = E0 . Define the bilinear maps (1.5.14) and (1.5.15) by f1 ¯ f2 := hf1 , f2 iF2 and f1 } v := v(f1 ), respectively. Then Lemma 1.7.1 shows that (1.5.15) is hypocontinuous. Since it is obvious that condition (1.5.16) is satisfied, Theorem 1.5.3 gives the assertion. ¥ 1.7.3 Remark Suppose that u ∈ L1,loc (X, E1 ) and v ∈ D(X, E2 ). Then Z hu q viD = u(x) q v(x) dx .

(1.7.5)

X

Proof

d

Since L1,loc (X, E1 ) ,→ D0 (X, E1 ), Theorem 1.7.2 implies that the map L1,loc (X, E1 ) × D(X, E2 ) → E0 ,

(u, v) 7→ hu q viD

(1.7.6)

is bilinear and separately continuous. From (1.6.15) and Theorem 1.3.6 we infer that ¡ ¢ ¡ ¢ d D(X) ⊗ E1 × D(X) ⊗ E2 ⊂ L1,loc (X, E1 ) × D(X, E2 ) . (1.7.7) It is obvious that (1.7.5) is true for u ∈ D(X) ⊗ E1 and v ∈ D(X) ⊗ E2 . Hence the assertion follows from (1.7.7) and the separate continuity of the map (1.7.6). ¥ Parseval’s Formula As a first application of Theorem 1.7.2 we obtain a natural extension of Parseval’s formula for the Fourier transform. 1.7.4 Proposition Parseval’s formula is valid for vector-valued distributions, that is, ­ ® hu q ϕiS = (2π)−n u b qϕ b S , u ∈ S 0 (Rn , E1 ) , ϕ ∈ S(Rn , E2 ) . (1.7.8) b

Proof Thanks to Theorem 1.7.2 and the fact that the Fourier transform and reflection are toplinear automorphisms, it follows that both sides of (1.7.8) are separately continuous bilinear maps from S 0 (Rn , E1 ) × S(Rn , E2 ) into E0 . By (1.7.4) and the well-known Parseval formula for scalar distributions the two expressions on either side of (1.7.8) coincide on the dense linear subspace ¡ 0 n ¢ ¡ ¢ S (R ) ⊗ E1 × S(Rn ) ⊗ E2 . Hence they are equal.

¥

Tensor Products and Convolutions

415

Duality Pairings By specializing the above results to the case where (1.5.20) is a duality pairing we obtain the desired map (1.7.1). 1.7.5 Theorem Let E be a Banach space and let F ∈ {D, E, S}. Then there exists a unique hypocontinuous bilinear map F0 (X, E 0 ) × F(X, E) → K ,

(u0 , u) 7→ hu0 , uiF(X,E) ,

the duality pairing between F0 (X, E 0 ) and F(X, E), such that Z 0

hu , uiF(X,E) =

X

­ 0 ® u (x), u(x) E dx

(1.7.9)

for u0 ∈ D(X, E 0 ) and u ∈ D(X, E). Proof Let E1 := E 0 , E2 := E, and E0 := K, and put e1 q e2 := he1 , e2 iE . Then the assertion follows from Theorem 1.7.2 and Remark 1.7.3. ¥ 1.7.6 Remarks (a) It is also true that Z hu0 , uiD(X,E) =

X

­ 0 ® u (x), u(x) E dx ,

(u0 , u) ∈ L1,loc (X, E 0 ) × D(X, E) ,

and Z

­

0

hu , uiS(Rn ,E) =

u0 (x), u(x)

Rn

® E

dx ,

(u0 , u) ∈ Lp (Rn , E 0 ) × S(Rn , E) ,

where 1 ≤ p < ∞. Proof The first part of the assertion is a consequence of Remark 1.7.3. The second one follows from d

d

D(Rn , E 0 ) ,→ Lp (Rn , E 0 ) ,→ S 0 (Rn , E 0 ) ,

1≤p 0 } be a mollifier on Rn2 . Then, putting ` := n2 and ψε := ϕε/4 ∗ χ(ε/2)B` ,

ε>0,

it is easily verified that ψε ∈ D(Rn2 ) satisfies ψε (x) = 1 for x ∈ (ε/4)B` , has its support in εB` , and k∂ α ψε k∞ ≤ c(α, n2 )ε−|α| ,

α ∈ Nn2 ,

ε>0.

(1.8.19)

Thus, given ϕj ∈ E(Rnj ), j = 1, 2, we see that ϕ1 ⊗ (1 − ψε )ϕ2 ∈ E(Rn1 × Rn2 ) q has its support in Rn1 × (Rn2 ) . Consequently, ¡ ¢ u(ϕ1 ⊗ ϕ2 ) = u(ϕ1 ⊗ ψε ϕ2 ) + u ϕ1 ⊗ (1 − ψε )ϕ2 = u(ϕ1 ⊗ ψε ϕ2 ) , ε>0, thanks to (1.8.17). Hence we infer from (1.8.18) |u(ϕ1 ⊗ ϕ2 )| ≤ cpm,K (ϕ1 ⊗ ψε ϕ2 ) ≤c

max

|α1 |+|α2 |≤m

¡ ¢ pK1 (∂ α1 ϕ1 )pεB` ∂ α2 (ψε ϕ2 ) ,

(1.8.20)

where K1 is the canonical projection of K into Rn1 . Now suppose that ∂ α ϕ2 (0) = 0 for |α| ≤ m. Using (1.8.19) and Leibniz’ rule, we deduce from (1.8.16) that X X ¯ ¯ ¯ |β|−|γ| ¯ max ¯∂ α2 (ψε ϕ2 )(x2 )¯ ≤ c ε|β|−|α2 | εm−|γ| max ¯∂F ψ(x2 )¯ |x2 |≤ε

β≤α2

≤c

X X

|x2 |≤ε

γ≤β

¯ |β|−|γ| ¯ ε|β−γ|+|α2 | max ¯∂F ψ(x2 )¯

β≤α2 γ≤β

|x2 |≤ε

for |α2 | ≤ m − |α1 |. Now we infer from (1.8.15) that ¡ ¢ lim pεB` ∂ α2 (ψε ϕ2 ) = 0 , |α2 | ≤ m − |α1 | . ε→0

Consequently, by (1.8.20), u(ϕ1 ⊗ ϕ2 ) = 0

for

ϕj ∈ E(Rnj ) with

∂ α ϕ2 (0) = 0 ,

|α| ≤ m . (1.8.21)

Given any ϕ2 ∈ E(Rn2 ), we obtain from (1.8.14) and (1.8.15) that ϕ2 (x2 ) =

X 1 ∂ α ϕ2 (0)xα 2 + χ(x2 ) , α!

|α|≤m

x2 ∈ Rn2 ,

Tensor Products and Convolutions

425

where χ ∈ E(Rn2 ) satisfies ∂ α χ(0) = 0 for |α| ≤ m. Thus (1.8.21) implies ³ ´ X 1 X u(ϕ1 ⊗ ϕ2 ) = u ϕ1 ⊗ ∂ α ϕ2 (0)xα (−1)|α| uα (ϕ1 )∂ α ϕ2 (0) 2 = α! |α|≤m

|α|≤m

for ϕj ∈ E(Rnj ), where uα (ϕ1 ) :=

(−1)|α| u(ϕ1 ⊗ xα 2) , α!

|α| ≤ m .

Now Theorem 1.8.4 gives u=

X

uα ⊗ ∂ α δ(x2 )

|α|≤m

with uα ∈ E 0 (Rn1 , E). (3) Finally, suppose that there are m e ∈ N and u eα ∈ E 0 (Rn1 , E) such that X u= u eα ⊗ ∂ α δ(x2 ) . |α|≤m

By replacing m and m e by max{m, m}, e we can assume that m = m. e Then X 0= (uα − u eα ) ⊗ ∂ α δ(x2 )(ϕ1 ⊗ xβ2 ) = (−1)|β| β! (uβ − u eβ )(ϕ1 ) |α|≤m

for |β| ≤ m and ϕ1 ∈ E(Rn1 ). This proves the asserted uniqueness.

¥

It should be noted that the proofs of the preceding examples are adaptions of corresponding results for scalar distributions (e.g., [H¨or83]). Topological Tensor Products of Distributions Now we consider the special case of tensor products of a scalar and a vector-valued distribution. 1.8.7 Theorem Suppose that F ∈ {E, S}. Then ∼

F(X1 ) ⊗ F(X2 , E) = F(X1 × X2 , E) . Proof Thanks to Corollary 1.8.2 it suffices to show that F(X1 × X2 , E) induces on F(X1 ) ⊗ F(X2 , E) the projective topology. Given a continuous seminorm p on F(X1 × X2 , E), belonging to the families (1.1.3) and (1.1.7) respectively, it is easily verified that there exist continuous

426

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seminorms r on F(X1 ) and s on F(X2 , E), respectively, such that p(ϕ ⊗ ψ) ≤ r(ϕ)s(ψ) , ϕ ∈ F(X1 ) , ψ ∈ F(X2 , E) . P For a representation ϕj ⊗ ψj of z ∈ F(X1 ) ⊗ F(X2 , E) it follows that X X p(z) ≤ p(ϕj ⊗ ψj ) ≤ r(ϕj )s(ψj ) . This implies that p(z) ≤ r ⊗π s(z) for z ∈ F(X1 ) ⊗ F(X2 , E). Thus F(X1 × X2 , E) induces on F(X1 ) ⊗ F(X2 , E) a topology weaker than the projective topology. Conversely, let r and s be continuous seminorms on F(X1 ) and F(XP 2 , E), respectively, belonging to the respective families (1.1.3) and (1.1.7), and let ϕj ⊗ ψj be a representation of z ∈ F(X1 ) ⊗ F(X2 , E). Observe that D D X EE X hϕ0 , ϕj ihψ 0 , ψj i = ϕ0 , ψ 0 , ϕj ⊗ ψj , ϕ0 ∈ F0 (X1 ) , ψ 0 ∈ F0 (X2 , E) . Hence, cf. (1.4.7),

¯X ¯ ³ ³X ´´ ¯ ¯ r ⊗ε s(z) = sup ¯ hϕ0 , ϕj ihψ 0 , ψj i¯ = r s ϕj ⊗ ψj ϕ0 ∈B0r ψ 0 ∈B0s

by the bipolar theorem, where Br is the open r-unit-ball in F(X1 ) and Bs is the open s-unit-ball in F(X2 , E). From this we infer the existence of a continuous seminorm p on F(X1 × X2 , E) such that r ⊗ε s(z) ≤ p(z) for z ∈ F(X1 ) ⊗ F(X2 , E). Thus F(X1 × X2 , E) induces on F(X1 ) ⊗ F(X2 , E) a topology that is stronger than the injective tensor product topology. Since F(X1 ) is nuclear, the latter coincides with the projective topology (e.g. [Tre67, Theorem 50.1(f)]). ¥ 1.8.8 Corollary If F ∈ {E, S}, then ∼

F0 (X1 ) ⊗ F0 (X2 , E 0 ) = F(X1 × X2 , E)0 . Proof

From Corollary 1.4.10 and Theorem 1.8.7 we know that ¡ ¢0 ∼ F0 (X1 × X2 , E 0 ) = F(X1 × X2 , E)0 = F1 (X1 ) ⊗ F(X2 , E) .

Hence the assertion follows from (1.4.16) and by applying Corollary 1.4.10 once more. ¥ It should be remarked that ∼

D(X1 ) ⊗ D(X2 ) 6= D(X1 × X2 ) as topological vector spaces, although these spaces are equal as vector spaces. Moreover, ¡ ¢0 ∼ ∼ D0 (X1 ) ⊗ D0 (X2 ) 6= D(X1 ) ⊗ D(X2 ) (cf. [Schw57b, chap. I, p. 95]).

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Kernel Theorems After these preparations we can prove some ‘vector-valued kernel theorems’: 1.8.9 Theorem Let E be a Banach space and let Xj be open in Rnj , j = 1, 2. Then ¡ ¢ E(X1 × X2 , E) ∼ = L E 0 (X1 ), E(X2 , E) , ¡ ¢ ∼ L E(X1 ), E 0 (X2 , E 0 ) , E 0 (X1 × X2 , E 0 ) = ¡ ¢ S(Rn1 × Rn2 , E) ∼ = L S 0 (Rn1 ), S(Rn2 , E) , ¡ ¢ S 0 (Rn1 × Rn2 , E 0 ) ∼ = L S(Rn1 ), S 0 (Rn2 , E 0 ) , where ∼ = denotes the canonical toplinear isomorphism. Proof These assertions are immediate consequences of Theorems 1.4.8 and 1.8.7, and of Corollary 1.8.8. ¥ For completeness we recall the ‘classical’ Schwartz kernel theorem for scalar distributions. ¡ ¢ 1.8.10 Theorem D0 (X1 × X2 ) ∼ = L D(X1 ), D0 (X2 ) by means of the canonical toplinear isomorphism. Proof

For example, [H¨or83] or [Tre67].

¥

1.8.11 Remark It may be useful to reinterprete the Kernel Theorems 1.8.9 and 1.8.10. Suppose, for instance, that k ∈ E 0 (X1 × X2 , E) and define K by the relation (Kϕ1 )(ϕ2 ) = k(ϕ1 ⊗ ϕ2 ) ,

ϕj ∈ E(Xj ) ,

Then Theorem 1.8.9 guarantees that ¡ ¢ K ∈ L E(X1 ), E 0 (X2 , E) .

j = 1, 2 .

(1.8.22)

(1.8.23)

Conversely, for any K satisfying (1.8.23) there exists a unique k ∈ E 0 (X1 × X2 , E) such that (1.8.22) is true. Moreover, the map ¡ ¢ E 0 (X1 × X2 , E) → L E(X1 ), E 0 (X2 , E) , k 7→ K (1.8.24) is a toplinear isomorphism. The distribution k is said to be the kernel of the map K, and K is associated to the kernel k. In symbolic notation the isomorphism (1.8.24) is written as Z Kϕ1 = k(x1 , ·)ϕ1 (x1 ) dx1 , ϕ1 ∈ E(X1 ) . X1

In this case (1.8.22) takes the suggestive form Z Z (Kϕ1 )(x2 )ϕ2 (x2 ) dx2 = k(x1 , x2 )ϕ1 (x1 )ϕ2 (x2 ) d(x1 , x2 ) . X2

X1 ×X2

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The kernel k ∈ E 0 (X1 × X2 , E) is said to be regularizing if its associated map K can be extended from D(X1 ) to a continuous linear map of E 0 (X1 ) into E(X2 , E). Thanks to Theorem 1.8.9 this is the case iff k ∈ E(X1 × X2 , E). Similar definitions and conventions apply to tempered distributions and to the classical case of the Schwartz kernel theorem 1.8.10. ¥

1.9

Convolutions of Vector-Valued Distributions

In Subsection 1.2 we have already defined convolutions of a vector-valued and a scalar distribution. In this subsection we extend these definitions to the case of two vector-valued distributions. In addition, we weaken the support restrictions. The Basic Theorem 1.9.1 Theorem Suppose that either

or

u1 ∈ D0 (Rn , E1 )

and

u2 ∈ E 0 (Rn , E2 ) ,

u1 ∈ S(Rn , E1 )

and

u2 ∈ S 0 (Rn , E2 ) .

Then there exists a unique distribution in D0 (Rn , E0 ) or OM (Rn , E0 ), respectively, the convolution of u1 and u2 with respect to the multiplication (1.5.20), written u1 ∗q u2 , such that (v1 ⊗ e1 ) ∗ q (v2 ⊗ e2 ) = (v1 ∗ v2 ) ⊗ (e1 q e2 )

(1.9.1)

for v1 , v2 ∈ D(Rn ), ej ∈ Ej , and j = 1, 2, and such that the ‘convolution maps’ (u1 , u2 ) 7→ u1 ∗ q u2 are bilinear and hypocontinuous: D0 (Rn , E1 ) × E 0 (Rn , E2 ) → D0 (Rn , E0 )

(1.9.2)

S(Rn , E1 ) × S 0 (Rn , E2 ) → OM (Rn , E0 ) ,

(1.9.3)

and

respectively. In addition, the convolution maps restrict to hypocontinuous bilinear maps: E 0 (Rn , E1 ) × E 0 (Rn , E2 ) → E 0 (Rn , E0 ) , E(Rn , E1 ) × E 0 (Rn , E2 ) → E(Rn , E0 ) , D(Rn , E1 ) × D0 (Rn , E2 ) → E(Rn , E0 ) , D(Rn , E1 ) × E 0 (Rn , E2 ) → D(Rn , E0 ) , S(Rn , E1 ) × S(Rn , E2 ) → S(Rn , E0 ) . Proof Let (F1 , F2 ; F0 ) be either of the triplets (D0 , E 0 ; D0 ), (S, S 0 ; OM ), (E 0 , E 0 ; E 0 ), (E, E 0 ; E), (D, D0 ; E), (D, E 0 ; D), or (S, S; S), and put Fj := Fj (Rn ), j = 0, 1, 2.

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Then the Fj are reflexive, complete, nuclear, and conuclear by Theorems 1.1.2 ∼ and 1.4.3. Moreover, Fj ⊗ Ej = Fj (Rn , Ej ) by Theorem 1.4.9. Define the maps (1.5.14) and (1.5.15) by f1 ¯ f2 := f1 ∗ f2 and f1 } v := f1 ∗ v, respectively. From Propositions 1.2.1, 1.2.3, and 1.2.7 we know that they are well-defined and that (1.5.15) is hypocontinuous. Definition (1.2.1) of convolution immediately implies that condition (1.5.16) is satisfied. Hence Theorem 1.5.3 guarantees the existence of a unique hypocontinuous bilinear map F1 (X, E1 ) × F2 (X, E2 ) → F0 (X, E0 ) ,

(u1 , u2 ) 7→ u1 ∗q u2

(1.9.4)

satisfying (1.9.1)¡ for uj ∈ Fj¢(X)¡ and ej ∈ E¢j , j = 1, 2. By Theorem 1.3.6 the linear subspace D(X) ⊗ E1 × D(X) ⊗ E2 is dense in F1 (X, E1 ) × F2 (X, E2 ). Thus the map (1.9.4) is determined by its restriction to this subspace. This proves the theorem. ¥ 1.9.2 Remarks (a) It should be observed that Theorem 1.9.1 is ‘symmetric in E1 and E2 ’, that is, the roles of E1 and E2 can be interchanged. This fact will be used throughout, usually without further mention. (b) Suppose that either uj ∈ D0 (Rn , Ej ), j = 1, 2, and that u1 or u2 has compact support, or u1 ∈ S 0 (Rn , E1 ) and u2 ∈ S(Rn , E2 ). Then ­ ® u1 ∗ q u2 (ϕ) = u1 ∗ q (u2 ∗ ϕ)(0) = u1 q (u2 ∗ ϕ) D (1.9.5) b

b

for ϕ ∈ D(Rn ). Proof We can assume that u2 ∈ E 0 (Rn , E2 ) in the first case. Then by Theorems 1.7.2 and 1.9.1 each one of the three expressions in (1.9.5) defines a separately continuous trilinear map D0 (Rn , E1 ) × E 0 (Rn , E2 ) × D(Rn ) → E0 in the first case, and S 0 (Rn , E1 ) × S(Rn , E2 ) × D(Rn ) → E0 in the second one. Suppose that uj = vj ⊗ ej ∈ D(Rn ) × Ej , j = 1, 2. Then we deduce from (1.9.1), (1.2.11), and (1.2.13) that u1 ∗ q u2 (ϕ) = (v1 ⊗ e1 ) ∗ q (v2 ⊗ e2 )(ϕ) = v1 ∗ v2 (ϕ)(e1 q e2 ) £ ¤ = v1 ∗ (v2 ∗ ϕ)(0)(e1 q e2 ) = v1 ∗ (v2 ∗ ϕ) ⊗ (e1 q e2 )(0) b

b b

= u1 ∗ q (u2 ∗ ϕ)(0) . Moreover, by these calculations, (1.2.11), Remark 1.2.4(c), and (1.7.4), ­ ® u1 ∗ q u2 (ϕ) = v1 ∗ (v2 ∗ ϕ)(0)(e1 q e2 ) = v1 , (v2 ∗ ϕ) D (e1 q e2 ) ­ £ ¤® = hv1 , v 2 ∗ ϕiD (e1 q e2 ) = (v1 ⊗ e1 ) q (v 2 ∗ ϕ) ⊗ e2 D ­ ® = u1 q (u2 ∗ ϕ) D . bb

b

b

b

b

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Thus all three terms in (1.9.5) agree on the dense linear subspace ¡ ¢ ¡ ¢ D(Rn ) ⊗ E1 × D(Rn ) ⊗ E2 × D(Rn ) , and so everywhere.

¥

Lp Functions with Compact Supports Given K b X, we put Lp,K (X, E) :=

©

u ∈ Lp (X, E) ; supp(u) ⊂ K

ª

,

1≤p≤∞.

Then Lp,K (X, E) is a closed linear subspace of the Banach space Lp (X, E), hence a Banach space as well. Moreover, [ © ª Lp,K (X, E) = u ∈ Lp (X, E) ; supp(u) b X . KbX

Thus, recalling (1.1.5) and (1.1.6), Lp,c (X, E) := −−→ lim Lp,K (X, E) ,

1≤p≤∞,

(1.9.6)

KbX

is a well-defined LF space, the space of E-valued Lp functions with compact support. It is obvious that Lp,c (X, E) ,→ Lp (X, E) ,→ L1,loc (X, E) ,→ D0 (X, E)

(1.9.7)

for 1 ≤ p ≤ ∞, and it is not difficult to see that d

D(X, E) ,→ Lp,c (X, E) ,

1≤p 0 such that ¡ ¢ ¡ ¢ xj ∈ supp(uj ) , x1 + x2 ∈ X =⇒ xj ∈ ρBn , j = 1, 2 . (1.9.11) Thus, given ψj ∈ D(Rn ) with

ψj | ρBn = 1 ,

j = 1, 2 ,

(1.9.12)

the convolution (ψ1 u1 ) ∗ q (ψ2 u2 ) is well-defined, since ψj uj ∈ E 0 (Rn , Ej ). Moreover, for χj ∈ D(Rn ) with χj | ρBn = 1 it follows from Remarks 1.8.5(a) and (e),

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433

the fact that ψj − χj vanish on ρBn , and from (1.9.11) that, given ϕ ∈ D, £

¤ (ψ1 u1 ) ∗ q (ψ2 u2 ) − (χ1 u1 ) ∗ q (χ2 u2 ) (ϕ) £ ¤¡ ¢ = (ψ1 u1 ) ⊗ q (ψ2 u2 ) − (χ1 u1 ) ⊗ q (χ2 u2 ) ϕ(x1 + x2 ) £ ¤¡ ¢ = (ψ1 − χ1 )u1 ⊗ q (ψ2 u2 ) + (χ1 u1 ) ⊗ q (ψ2 − χ2 )u2 ϕ(x1 + x2 ) n£ o¡ ¤ £ ¤ ¢ = (ψ1 − χ1 ) ⊗ ψ2 (u1 ⊗ q u2 ) + χ1 ⊗ (ψ2 − χ2 ) (u1 ⊗ q u2 ) ϕ(x1 + x2 ) n£ o ¤ = (u1 ⊗ q u2 ) (ψ1 − χ1 )(x1 )ψ2 (x2 ) + χ1 (x1 )(ψ2 − χ2 )(x2 ) ϕ(x1 + x2 ) = 0 .

This shows that (ψ1 u1 ) ∗ q (ψ2 u2 ) defines a distribution in D0 (X, E0 ), independently of the choice of the ψj satisfying (1.9.12). From this and the fact that a distribution is determined by its values on the open subsets of Rn (e.g., [H¨or83, Theorem 2.2.4] and note that the proof carries over to the E-valued case) we see that  given uj ∈ D0 (Rn , Ej ), j = 1, 2, having convolutive supports,    there exists a unique distribution u1 ∗q u2 ∈ D0 (Rn , E0 ), the    convolution of u1 and u2 with respect to multiplication (1.5.20),  (1.9.13) satisfying   n  ˚ u1 ∗q u2 | X = (ψ1 u1 ) ∗q (ψ2 u2 ) | X , X=XbR ,     where ψj satisfies (1.9.12) for j = 1, 2.

Basic Properties In the following remarks we collect some of the most important properties of convolutions. 1.9.6 Remarks Unless explicit restrictions are given, we suppose that either uj ∈ D0 (Rn , Ej ) for j = 1, 2, with convolutive supports, or that u1 ∈ S 0 (Rn , E1 ) and u2 ∈ S(Rn , E2 ). (a) It is an obvious consequence of (1.9.13) and Remark 1.9.2(b) that ­ ® u1 ∗q u2 (ϕ) = u1 ∗ (u2 ∗ ϕ)(0) = u1 q (u2 ∗ ϕ) D b

b

for ϕ ∈ D(Rn ). (b) If uj ∈ L1,loc (Rn , Ej ) satisfy condition (Σ), then u1 ∗q u2 ∈ L1,loc (Rn , E0 ) and Z Z q u1 ∗ q u2 (x) = u1 (x − y) u2 (y) dy = u1 (y) q u2 (x − y) dy , a.a. x ∈ Rn . Rn

Rn

Proof Thanks to (1.9.13) we can assume that uj has compact support for j = 1, 2. Then the assertion follows from Proposition 1.9.3. ¥

434

Appendix

(c)(Associativity) Let the associativity hypotheses of Remark 1.6.5(e) be satisfied and suppose that either uj ∈ D0 (Rn , Ej ), j = 1, 2, 3, have convolutive supports, or u1 ∈ S 0 (Rn , E1 ) and uk ∈ S(Rn , Ek ), k = 2, 3. Then u1 ∗ q (u2 ∗q u3 ) = (u1 ∗ q u2 ) ∗ q u3 .

(1.9.14)

Proof If condition (Σ) is satisfied,4 then it suffices, by (1.9.13), to prove the assertion for uj ∈ E 0 (X, Ej ), j = 1, 2, 3. Then Theorem 1.9.1 implies that both sides of (1.9.14) define separately continuous trilinear maps E 0 (Rn , E1 ) × E 0 (Rn , E2 ) × E 0 (Rn , E3 ) → E 0 (Rn , E0 ) if condition (Σ) is satisfied, and S 0 (Rn , E1 ) × S(Rn , E2 ) × S(Rn , E3 ) → OM (Rn , E0 ) otherwise. Thanks to (1.9.1) and (1.2.13) they coincide on the dense linear subspace Q3 j=1 D(X) ⊗ Ej . Thus they are equal. ¥ (d) (Commutativity) Suppose that E1 = E2 =: E and multiplication (1.5.20) is symmetric. Then u1 ∗q u2 = u2 ∗ q u1 . Proof By (1.9.13) we can assume that uj ∈ E 0 (Rn , E) if condition (Σ) is satisfied. Thus, by Theorem 1.9.1 and the density of D(Rn , E) in E 0 (Rn , E) and in S 0 (Rn , E) and S(Rn , E), respectively, we can assume that uj ∈ D(Rn , E). Now the assertion follows from (b). ¥ (e)(Distributivity) Let the distributivity hypotheses of Remark 1.8.5(e) be satisfied and suppose that uj ∈ D0 (Rn , Ej ) as well as vj ∈ D0 (Rn , Ej+2 ), j = 1, 2, have convolutive supports. Then (u1 ∗q u2 ) ⊗ q (v1 ∗ q v2 ) = (u1 ⊗ q v1 ) ∗ q (u2 ⊗ q v2 ) .

(1.9.15)

Proof First we show that uj ⊗ q vj ∈ D0 (Rn × Rn , Ej+5 ) satisfy condition (Σ). By Remark 1.8.5(a) supp(uj ⊗ q vj ) ⊂ supp(uj ) × supp(vj ) ,

j = 1, 2 .

(1.9.16)

Let X ⊂ Rn be bounded and open. Then there exists ρ > 0 such that xj ∈ supp(uj ) and yj ∈ supp(vj ) with x1 + x2 and y1 + y2 belonging to X imply xj , yj ∈ ρBn . Thus, if (x1 , y1 ) + (x2 , y2 ) ∈ X × X, it follows from (1.9.16) that (xj , yj ) ∈ ρBn × ρBn ⊂ ρ1 B2n ,

j = 1, 2 ,

for some ρ1 > 0. Since every bounded open subset of R2n is contained in a product set X × X, we see that uj ⊗ vj satisfy condition (Σ). Hence both sides of (1.9.15) are well-defined. 4 We leave it to the reader to verify that the convolutions on either side of (1.9.14) are welldefined if (Σ) is satisfied (also cf. (f)).

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Given ψj , ψj+2 ∈ D(Rn ), it follows from Remark 1.8.5(e) that (ψj uj ) ⊗ q (ψj+2 vj ) = (ψj ⊗ ψj+2 )(uj ⊗ q vj ) ,

j = 1, 2 .

Using this and (1.9.13) it is easily seen that it suffices to prove the assertion for uj ∈ E 0 (Rn , Ej ) and vj ∈ E 0 (Rn , Ej+2 ), j = 1, 2. Now Theorems 1.8.4 and 1.9.1 together with a density argument show that we can assume that uj ∈ D(Rn , Ej ) and vj ∈ D(Rn , Ej+2 ). Thus, by (b) and Remark 1.8.5(d), the left side of (1.9.15) equals Z Z u1 (x1 − y1 ) q u2 (y1 ) dy1 q v1 (x2 − y2 ) q v2 (y2 ) dy2 , (x1 , x2 ) ∈ Rn × Rn . Rn

Rn

By Fubini’s theorem and (1.8.11) this product takes the form Z ¡ ¢ ¡ ¢ u1 (x1 − y1 ) q v1 (x2 − y2 ) q u2 (y1 ) q v2 (y2 ) d(y1 , y2 ) , R2n

which is the right side of (1.9.15).

¥

(f )(Support Theorem) Suppose that uj ∈ D0 (Rn , Ej ), j = 1, 2, have convolutive supports. Then supp(u1 ∗q u2 ) ⊂ supp(u1 ) + supp(u2 ). Proof From Remark 1.9.5(a) we know that C := supp(u1 ) + supp(u2 ) is closed in Rn . Let ϕ ∈ D(C c ). Then the support of the function (x1 , x2 ) 7→ ϕ(x1 + x2 ) does not meet supp(u1 ) × supp(u2 ), hence not¡ supp(u1 ⊗ ¢q u2 ) by Remark 1.8.5(a). Thus (ψ1 u1 ) ∗q (ψ2 u2 )(ϕ) = (ψ1 u1 ) ⊗ q (ψ2 u2 ) ϕ(x1 + x2 ) = 0 for all ψj ∈ D(X), j = 1, 2. Consequently, (1.9.13) implies u1 ∗q u2 (ϕ) = 0, that is, the assertion. ¥ (g) ∂ α+β (u1 ∗ q u2 ) = ∂ α u1 ∗q ∂ β u2 , α, β ∈ Nn . Proof

First suppose that u1 and u2 satisfy condition (Σ). Since supp(∂ γ u) ⊂ supp(u) ,

u ∈ D0 (Rn , E) ,

γ ∈ Nn ,

it follows that ∂ α u1 and ∂ β u2 satisfy condition (Σ). Hence ∂ α u1 ∗ q ∂ β u2 is welldefined. Given ϕ ∈ D(Rn ), ∂ α+β (u1 ∗q u2 )(ϕ) = (−1)|α|+|β| (u1 ∗ q u2 )(∂ α+β ϕ) . Let X be a bounded open subset of Rn containing supp(ϕ) and choose ψj ∈ D(Rn ) with ψj | ρX Bn = 1. Then (−1)|α|+|β| (u1 ∗ q u2 )(∂ α+β ϕ) = (−1)|α|+|β| (ψ1 u1 ) ∗q (ψ2 u2 )(∂ α+β ϕ) ¡ ¢ = (−1)|α|+|β| (ψ1 u1 ) ⊗ q (ψ2 u2 ) ∂ α+β ϕ(x1 + x2 ) ¡ ¢ = ∂ α (ψ1 u1 ) ⊗ q ∂ β (ψ2 u2 ) ϕ(x1 + x2 ) ¡ ¢ = (ψ1 ∂ α u1 ) ⊗ q (ψ2 ∂ β u2 ) ϕ(x1 + x2 ) = (ψ1 ∂ α u1 ) ∗q (ψ2 ∂ β u2 )(ϕ) = ∂ α u1 ∗ q ∂ β u2 (ϕ) , thanks to Remark 1.8.5(b), Leibniz’ rule, and the properties of ψj .

436

Appendix

If u1 ∈ S 0 (Rn , E1 ) and u2 ∈ S(Rn , E2 ), then we can restrict ourselves as usual to the case uj ∈ D(Rn ) ⊗ Ej . Then the assertion follows from Remark 1.2.4(b). ¥ (h) τa (u1 ∗ q u2 ) = (τa u1 ) ∗q u2 = u1 ∗ q τa u2 , a ∈ Rn , and b

b

b

(u1 ∗q u2 ) = u1 ∗ q u2 . Proof It is easily verified that these convolutions are all well-defined. If u1 and u2 satisfy condition (Σ), we deduce from (1.9.13) that we can assume that u1 and u2 have compact supports. Thus in either case it suffices by Theorem 1.9.1 to consider the case where uj ∈ D(Rn ) ⊗ Ej . Now the assertion follows from Remark 1.2.4(c). ¥ (i) Suppose that a ∈ E1 and u ∈ D0 (Rn , E2 ). Then £ ¤ ∂ α (δ ⊗ a) ∗ q u = (∂ α δ ⊗ a) ∗ q u = a q ∂ α u = ∂ α (a q u) for α ∈ Nn . Proof Since ∂ β δ ⊗ a ∈ E 0 (Rn , E1 ), we see that the above convolutions are welldefined. Thanks to (g) and Remark 1.6.5(f) we can assume that α = 0. By Theorems 1.6.4 and 1.9.1 it suffices to consider u ∈ D(Rn ) × E2 . Then the assertion follows from (1.9.1) and Remark 1.2.4(a). ¥ Convolution Algebras Given a nonempty subset K of Rn , we put © ª 0 DK (Rn , E) := u ∈ D0 (Rn , E) ; supp(u) ⊂ K

(1.9.17)

0 0 and, as usual, DK (Rn ) := DK (Rn , K), if no confusion seems likely. We also set 0 D+ (E) := DR0 + (R, E) .

(1.9.18)

0 Note that DK (Rn , E) is a closed linear subspace of D0 (Rn , E).

In the following theorem we collect some of the properties of convolutions in a particularly important setting. 1.9.7 Theorem Let Γ be a proper closed convex cone in Rn and let F ∈ {D, S, E 0 , DΓ0 } . Then convolution is a well-defined hypocontinuous bilinear map F(Rn , E1 ) × F(Rn , E2 ) → F(Rn , E0 ) ,

(u1 , u2 ) 7→ u1 ∗q u2

which possesses the associativity and commutativity properties of Remarks 1.9.6(c) and (d), respectively. In particular, if (E, q) is a [commutative] Banach algebra,

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437

¡ ¢ then F(Rn , E), ∗ q is a [commutative] algebra,¢ a convolution algebra. If (E, q) has ¡ 0 0 n a unit e0 and F ∈ {E , DΓ }, then F(R , E), ∗ q has a unit as well, namely δ ⊗ e0 . Proof Everything, except the hypocontinuity of the convolution map if F = DΓ0 , follows easily from Theorem 1.9.1, the injection D(Rn , E2 ) ,→ E 0 (Rn , E2 ), and Remarks 1.9.5(c), 1.9.6(f), and 1.2.4(a). Recall that DΓ0 (Rn , E) is a closed linear subspace of D0 (Rn , E). Thus, if (uα ) is a net in DΓ0 (Rn , E1 ) converging to zero, it converges to zero in D0 (Rn , E1 ). Let V be a bounded subset of DΓ0 (Rn , E2 ), hence of D(Rn , E2 ). Then, given any bounded subset B of D(Rn ), there exists K := KB b Rn such that supp(ϕ) ⊂ K for ϕ ∈ B. Hence condition (Σ) guarantees the existence of ρ > 0 such that x, y ∈ Γ with x + y ⊂ K implies x, y ∈ ρBn . Thus, letting ψj ∈ D(Rn ) satisfy ψ | ρBn = 1, it follows from (1.9.13) that uα ∗ q v(ϕ) = (ψ1 uα ) ∗q (ψ2 v)(ϕ) ,

ϕ∈B ,

v∈V .

(1.9.19)

Note that ψ2 v(χ) = v(ψ2 χ) and ψ2 χ ∈ Dsupp(ψ2 ) (Rn ) for χ ∈ E(Rn ) imply that { ψ2 v ; v ∈ V } is bounded in E 0 (Rn , E2 ). Now the hypocontinuity of the map (1.9.2) and (1.9.19) entail uα ∗q v(ϕ) → 0 in E0 , uniformly with respect to v ∈ V and ϕ ∈ B. Consequently, uα ∗q v → 0 in D0 (Rn , E0 ), hence in DΓ0 (Rn , E0 ), uniformly with respect to v in bounded subsets of DΓ0 (Rn , E0 ). This proves, by symmetry, the asserted hypocontinuity. ¥ 1.9.8 Remarks (a) Let Γ be a proper, closed, and convex cone in Rn . Then D(Rn ), S(Rn ), E 0 (Rn ), and DΓ0 (Rn ) are commutative convolution algebras. Moreover, E 0 (Rn ) and DΓ0 (Rn ) possess the unit δ. (b) Since D(Rn , E) and S(Rn , E) are barreled, the convolution product is, in fact, continuous on these spaces. 0 0 (c) Suppose that uj ∈ D+ (Ej ) are regular distributions. Then u1 ∗q u2 ∈ D+ (E0 ) is a regular distribution as well, and Z t u1 ∗ q u2 (t) = u1 (t − s) q u2 (s) ds , a.a. t ∈ R+ . 0

Proof

This is an easy consequence of Remarks 1.9.6(b) and (f).

¥

Convolutions of Bounded and Integrable Functions For the reader’s convenience we now collect the properties of convolution on certain Banach spaces of vector-valued convolutions, thus extending, in particular, assertions (III.4.2.19)–(III.4.2.22) (note that (III.4.2.18) is a special case of (III.4.2.21)). 1.9.9 Theorem Suppose that (F1 , F2 ; F0 ) is any one of the triplets (BUC, L1 ; BUC), (C0 , L1 ; C0 ), (Lp , L1 ; Lp ), (L∞ , L1 ; BUC), (Lq , Lq0 ; C0 ) ,

438

Appendix

where 1 ≤ p < ∞ and 1 < q < ∞. Then the convolution with respect to the multiplication (1.5.20) extends from F1 (Rn , E1 ) × D(Rn , E2 ) to a multiplication F1 (Rn , E1 ) × F2 (Rn , E2 ) → F0 (Rn , E0 ) . It is given by

Z

u1 ∗q u2 (x) =

Z Rn

u1 (x − y) q u2 (y) dy =

Rn

u1 (y) q u2 (x − y) dy

(1.9.20)

for a.a. x ∈ Rn . Proof Since F1 (Rn , E1 ) ,→ L1,loc (Rn , E1 ) ∩ S 0 (Rn , E1 ) (see (VII.1.2.1)), we infer from (1.9.3) that u1 ∗ q u2 ∈ OM (Rn , E0 ) ,

u1 ∈ F1 (Rn , E1 ) ,

u2 ∈ D(Rn , E2 ) ,

and Proposition 1.9.3 guarantees that u1 ∗ q u2 is given by the above integral in this case. Since D(Rn , E2 ) is dense in F2 (Rn , E2 ), it suffices to show that u1 ∗q u1 ∈ F0 (Rn , E0 ) and that the estimate ku1 ∗q u2 kF0 (Rn ,E0 ) ≤ ku1 kF1 (Rn ,E1 ) ku2 kF2 (Rn ,E2 )

(1.9.21)

is valid for u1 ∈ F1 (Rn , E1 ) and u2 ∈ D(Rn , E2 ). Then the assertion follows by continuous extension. If u1 ∈ Lr (Rn , E1 ), 1 ≤ r ≤ ∞, and u2 ∈ D(Rn , E2 ), then Z |u1 ∗ u2 (x)| ≤ |u1 (x − y)| |u2 (y)| dy , x ∈ Rn , Rn

and the classical scalar Young inequality imply ku1 ∗q u2 kLr (Rn ,E0 ) ≤ ku1 kLr (Rn ,E1 ) ku2 kL1 (Rn ,E2 )

(1.9.22)

ku1 ∗ q u2 kL∞ (Rn ,E0 ) ≤ ku1 kLr (Rn ,E1 ) ku2 kLr0 (Rn ,E2 )

(1.9.23)

and

(cf. (III.4.2.20)–(III.4.2.22)). If uj ∈ D(Rn , Ej ) for j = 1, 2, we know from the next to last line of Theorem 1.9.1 that u1 ∗ q u2 ∈ D(Rn , E0 ). Since D(Rn , E1 ) is dense in F1 (Rn , E1 ) if F1 belongs to {C0 , Lp , Lq } and since C0 (Rn , E0 ) is a closed linear subspace of L∞ (Rn , E0 ) containing D(Rn , E0 ), we infer from (1.9.22) and (1.9.23) that the assertion is true for F1 ∈ {C0 , Lp , Lq }. If u1 ∈ BUC(Rn , E1 ) and u2 ∈ D(Rn , E2 ), then, given ε > 0, there exists δ > 0 such that |u1 (x) − u1 (y)| ≤ ε for x, y ∈ Rn with |x − y| ≤ δ. Hence Z |u1 ∗ q u2 (x) − u1 ∗ q u2 (y)| ≤ |u1 (x − z) − u1 (y − z)| |u2 (z)| dz ≤ ε ku2 k1 Rn

n

for x, y ∈ R with |x − y| ≤ δ. This shows that u1 ∗q u2 ∈ BUC(Rn , E0 ) in this case. Hence the assertion follows for F1 = BUC from (1.9.22) and the fact that BUC(Rn , E0 ) is a closed linear subspace of L∞ (Rn , E0 ).

Tensor Products and Convolutions

439

Finally, suppose that u1 ∈ L∞ (Rn , E1 ) and u2 ∈ D(Rn , E2 ). Then we infer from Remark 1.9.6(h) and (1.9.22) that kτa (u1 ∗q u2 ) − u1 ∗ q u2 k∞ = ku1 ∗ q (τa u2 − u2 )k∞ ≤ ku1 k∞ kτa u2 − u2 k1 for a ∈ Rn . Since the translation group is strongly continuous on L1 (Rn , E2 ), it follows that u1 ∗ q u2 ∈ BUC(Rn , E0 ) in this case. Thus the assertion holds for F1 = L∞ as well. ¥ The Convolution Theorem Lastly, we extend the convolution theorem to vector-valued distributions. 1.9.10 Theorem (u1 ∗q u2 )b = u b1 q u b2 for u1 ∈ S 0 (Rn , E1 ) and u2 ∈ S(Rn , E2 ). Proof Since F is a toplinear automorphism of S(Rn , E) and of S 0 (Rn , E), it follows from Theorems 1.6.4 and 1.9.1, together with S(Rn , E) ,→ OM (Rn , E) ,→ S 0 (Rn , E) , that the bilinear maps (u1 , u2 ) 7→ (u1 ∗ q u2 )b and (u1 , u2 ) 7→ u b1 q u b2 are well-defined 0 n n 0 and hypocontinuous from S (R , E1 ) × S(R , E2 ) into S (Rn , E0 ). Thanks to u ⊗ e ∈ S 0 (Rn ) ⊗ E ,

(u ⊗ e)b = u b⊗e ,

we see from (1.6.12), (1.9.1), and theorem that they coincide ¡ the scalar ¢convolution ¡ ¢ on the dense linear subspace S 0 (Rn ) ⊗ E1 × S(Rn ) ⊗ E2 , hence everywhere. ¥ 1.9.11 Remarks (a) Let 0 OC (Rn , E) := F −1 OM (Rn , E) :=

©

u ∈ S 0 (Rn , E) ; Fu ∈ OM (Rn , E)

ª

,

endowed with the unique locally convex topology such that ¡ 0 ¢ F ∈ Lis OC (Rn , E), OM (Rn , E) . Then it follows from Theorem 1.9.10, the continuity properties of F , and Theorem 1.6.4 that we can define a hypocontinuous bilinear map 0 S 0 (Rn , E1 ) × OC (Rn , E2 ) → S 0 (Rn , E0 ) ,

(u1 , u2 ) 7→ u1 ∗q u2 ,

(1.9.24)

the convolution with respect to multiplication (1.5.20), by putting u1 ∗ q u2 := F −1 (b u1 q u b2 ) ,

u1 ∈ S 0 (Rn , E1 ) ,

0 u2 ∈ OC (Rn , E2 ) .

(1.9.25)

We leave it to the interested reader to carry over the properties of Remarks 1.9.6 to this more general case.

440

Appendix

(b) Given a ∈ OM (Rn , E1 ), we put a(D)u := F −1 a qF u := F −1 (a q u b) ,

u ∈ S 0 (Rn , E2 ) .

Then the isomorphism properties of the Fourier transform and Theorem 1.6.4 imply that ¡ ¢ a(D) ∈ L S 0 (Rn , E2 ), S 0 (Rn , E0 ) . Moreover,

a(D)u = F −1 (a) ∗ q u ,

u ∈ S(Rn , E2 ) ,

thanks to the Convolution Theorem 1.9.10. Motivated by these facts we put a(D)u := F −1 (a qF )u := F −1 (a) ∗q u

(1.9.26)

whenever a ∈ S 0 (Rn , E1 ) and u ∈ S 0 (Rn , E2 ) are such that the convolution product on the right side is well-defined. In these general situations a(D) is said to be a translation-invariant (pseudodifferential) operator with symbol a (related to the multiplication (1.5.20)). Of course, this is motivated by Theorem 1.2.2. It should be observed that this definition is consistent with the definition of Fourier multipliers in Subsection VI.3.4 of the main text (see, in particular, Theorem VI.3.4.1. ¥

Vector Measures and the Riesz Representation Theorem

2

441

Vector Measures and the Riesz Representation Theorem

In connection with duality theory and Besov spaces we need some explicit information on C0 (Rn , E)0 . For this reason we present in this section some elements of the theory of vector measures and prove a generalization of the well-known Riesz representation theorem. Measures of Bounded Variation Throughout this section E := (E, |·|) is a Banach space, X is a σ-compact metrizable space, and BX denotes the Borel σ-algebra of X. By an E-valued vector measure µ on X we mean a σ-additive map µ : BX → E satisfying µ(∅) = 0. For such a vector measure µ we define the variation |µ| : BX → R+ ∪ {∞} by X |µ| (B) := sup |µ(A)| , B ∈ BX , π(B)

A∈π(B)

where the supremum is taken over all partitions π(B) of B into a finite number of pair-wise disjoint Borel subsets. Then µ is said to be of bounded variation if kµkBV := |µ| (X) < ∞ . The set of all E-valued vector measures on X of bounded variation is denoted by ¡ ¢ MBV (X, E) := MBV (X, E), k·kBV . It is clear that MBV (X, E) is a normed vector space with the obvious linear structure (as a vector subspace of E BX ). Given µ ∈ MBV (X, E), it follows from |µ(B)| ≤ |µ| (B) for B ∈ BX that |µ| is a positive Borel measure on X. We denote its completion, defined on the σ-algebra A|µ| , consisting of all A ⊂ X of the form A = B ∪ N where B ∈ BX and N is a subset of a |µ|-null set, again by |µ|. Then |µ| is a positive finite Radon measure on X. 2.0.1 Remark Denote by dx a positive Radon measure on X. Then, given any function u ∈ L1 (X, dx, E), Z (u dx)(B) := u dx , B ∈ BX , B

defines an E-valued measure of bounded variation and ku dxkBV = kuk1 (cf. Theorem 4(iv) on p. 46 in [DU77] or [Lan69, Theorem XI.9]). Hence (u 7→ u dx) : L1 (X, dx, E) → MBV (X, E) is a linear isometry. Occasionally, it will be convenient to identify u with u dx, that is, to consider L1 (X, dx, E) as a closed linear subspace of MBV (X, E). ¥

442

Appendix

Integrals with Respect to Vector Measures Let E0 , E1 , and E2 be Banach spaces and suppose that E1 × E2 → E0 ,

(e1 , e2 ) 7→ e1 q e2

(2.0.1)

is a multiplication. Let B(X, E1 ) be the closure in B(X, E1 ) of the linear subspace S(X, E1 ) of all simple functions X u= χB ⊗ e B , e B ∈ E1 , (2.0.2) B∈π

where π := π(X) runs through all partitions of X. If u ∈ S(X, E1 ) is given by (2.0.2) and µ ∈ MBV (X, E2 ), then we put Z Z X u dµ := u dµ := eB q µ(B) ∈ E0 . X

B∈π

It follows that

Z S(X, E1 ) × MBV (X, E2 ) → E0 ,

(u, µ) 7→

u dµ

is a well-defined bilinear map satisfying ¯Z ¯ X X ¯ ¯ |eB | |µ(B)| ≤ |eB | |µ| (B) ≤ kuk∞ kµkBV . ¯ u dµ¯ ≤ E0

B∈π

(2.0.3)

B∈π

Hence it possesses a unique continuous bilinear extension of norm at most one over B(X, E1 ) × MBV (X, E2 ). Note that C0 (X, E1 ) is a closed linear subspace of B(X, E1 ). Thus, by restriction, we obtain a well-defined multiplication Z Z C0 (X, E1 ) × MBV (X, E2 ) → E0 , (u, µ) 7→ u dµ := u dµ (2.0.4) X

R

and X u dµ is said to be the integral of u over X with respect to µ (and the multiplication (2.0.1)). In fact, from (2.0.3) we deduce that Z ¯Z ¯ ¯ ¯ (2.0.5) ¯ u dµ¯ ≤ |u| d |µ| ≤ kuk∞ kµkBV E0

for u ∈ C0 (X, E1 ) and µ ∈ MBV (X, E2 ). Now suppose that there are further Banach spaces E3 , E4 , and E5 , and multiplications satisfying the associativity assumption of Remark 1.6.5(e). Then R it follows from the definition of u dµ that, for each u of the form (2.0.2), Z X ¡ ¢ u dµ q e = eB q µ(B) q e ∈ E5 , e ∈ E3 . (2.0.6) B∈π

Vector Measures and the Riesz Representation Theorem

443

Note that, given e ∈ E3 , µ q e := µ(·) q e : BX → E4 ,

B 7→ µ(B) q e

is σ-additive. Moreover, |µ q e(B)| ≤ |µ(B)| |e| ≤ |µ| (B) |e| ,

B ∈ BX ,

implies that µ q e is an E4 -valued vector measure on X of bounded variation, and that (µ, e) 7→ µ q e

MBV (X, E2 ) × E3 → MBV (X, E4 ) ,

is a multiplication. From this and (2.0.6) we infer that Z ³Z ´ u dµ q e = u d(µ q e)

(2.0.7)

(2.0.8)

and that Z C0 (X, E1 ) × MBV (X, E2 ) × E3 → E5 ,

(u, µ, e) 7→

u d(µ q e)

(2.0.9)

is a trilinear multiplication, that is, a continuous trilinear map of norm at most 1. Similarly as above, we see that (2.0.1) induces a multiplication E1 × MBV (X, E2 ) → MBV (X, E0 ) , Thus, given ϕ=

X

(e, µ) 7→ e q µ .

(2.0.10)

αB χB ∈ S(X, K) ,

B∈π

it follows from ϕ⊗e=

X

αB χB ⊗ e =

B∈π

X

χB ⊗ (αB e) ∈ S(X, E1 )

B∈π

for e ∈ E1 that Z ϕ ⊗ e dµ =

X

Z αB e q µ(B) =

ϕ d(e q µ) .

B∈π

From this we deduce that Z

Z ϕ ⊗ e dµ =

ϕ d(e q µ)

for ϕ ⊗ e ∈ C0 (X) ⊗ E1 and µ ∈ MBV (X, E2 ).

(2.0.11)

444

Appendix

In the following, we leave it to the reader to identify the choices for the spaces E0 , . . . , E5 and the multiplications that we are using in concrete situations. Vector Measures as Distributions Now suppose that X is open in Rn . Since D(X) is dense in C0 (X), it follows that the multiplication Z C0 (X) × MBV (X, E) → E , (ϕ, µ) 7→ ϕ dµ is completely determined by its restriction to D(X) × MBV (X, E). Put Z Tµ ϕ := ϕ dµ , (ϕ, µ) ∈ D(X) × MBV (X, E) . Then D(X) ,→ C0 (X) implies

¡ ¢ (µ 7→ Tµ ) ∈ L MBV (X, E), D0 (X, E) .

(2.0.12)

Suppose that Tµ = 0 in D0 (X, E) for some µ ∈ MBV (X, E). Then we infer from (2.0.8) that Z 0 he , Tµ ϕiE = ϕ dhe0 , µiE = 0 , ϕ ∈ D(X) , e0 ∈ E 0 . 0 0 Since he0 , µiE ∈ MBV­(X) := M ® BV (X, K),0 this 0implies he , µiE = 0 for each e be0 0 longing to E . Thus e , µ(B) E = 0 for e ∈ E and B ∈ BX , which gives µ = 0. This shows that (2.0.12) is an injection. Thus the following convention is justified. Let X be a nonempty open subset of Rn . Then we identify µ ∈ MBV (X, E) with Tµ ∈ D0 (X, E) so that

and

MBV (X, E) ,→ D0 (X, E)

(2.0.13)

µ ∈ MBV (X, E) ,

(2.0.14)

Z µ(ϕ) =

ϕ dµ ,

ϕ ∈ D(X) .

X

Convolutions Involving Vector Measures d

Observe that S(Rn ) ,→ C0 (Rn ) implies MBV (Rn , E) ,→ S 0 (Rn , E) .

(2.0.15)

Hence we know from Proposition 1.2.7 that the convolution product ϕ ∗ µ is a welldefined element of OM (Rn , E) for ϕ ∈ S(Rn ) and µ ∈ MBV (Rn , E). Moreover, Z (ϕ ∗ µ)(x) = µ(τx ϕ) = ϕ(x − y) dµ(y) , x ∈ Rn , (2.0.16) b

Rn

Vector Measures and the Riesz Representation Theorem

445

for ϕ ∈ S(Rn ) and µ ∈ MBV (Rn , E). Hence, by (2.0.5), Z |ϕ ∗ µ(x)|E ≤ |ϕ(x − y)| d |µ| (y) = |ϕ| ∗ |µ| (x) ,

x ∈ Rn ,

Rn

so that, by Fubini’s theorem, kϕ ∗ µk1 ≤ kϕk1 kµkBV ,

ϕ ∈ S(Rn ) ,

µ ∈ MBV (Rn , E) .

(2.0.17)

2.0.2 Proposition Convolution is a well-defined multiplication L1 (Rn ) × MBV (Rn , E) → L1 (Rn , E) . Proof

This follows from (2.0.17) by continuous extension.

¥

If E = K then, thanks to Remark 2.0.1, Proposition 2.0.2 reduces to the wellknown fact that L1 (Rn ) is an ideal in the convolution algebra of all K-valued Borel measures (e.g., [Pet83, Theorem I.4.5]). 2.0.3 Lemma Suppose ϕ ∈ L1 (Rn ), u ∈ C0 (Rn , E), and µ ∈ MBV (Rn , E 0 ). Then 5 Z Z Z ϕ ∗ u dµ = u d(ϕ ∗ µ) = hϕ ∗ µ, uiE dx = hϕ ∗ µ, ui . (2.0.18) b

b

Rn

b

Rn

Rn

Proof From Theorem 1.9.9, Remark 2.0.1, Proposition 2.0.2, and (2.0.4) we infer that each one of the maps that send (ϕ, u, µ) ∈ L1 (Rn ) × C0 (Rn , E) × MBV (Rn , E 0 ) into one of the integrals in (2.0.18) is a trilinear K-valued multiplication. Thus, since C0 (Rn ) ⊗ E is dense in C0 (Rn , E) by Theorem 1.3.6(x), it suffices to consider the case where u = ψ ⊗ e ∈ C0 (Rn ) ⊗ E. Since ϕ ∗ (ψ ⊗ e) = (ϕ ∗ ψ) ⊗ e, we deduce from (2.0.11), (2.0.16), and Fubini’s theorem that Z Z ZZ ϕ ∗ u dµ = ϕ ∗ ψ dhµ, eiE = ϕ(x − y) dhµ, eiE (x)ψ(y) dy Z (2.0.19) ¡ ¢ = ϕ ∗ hµ, eiE ψ dx . b

Moreover, thanks to (2.0.8), Z DZ E ϕ ∗ hµ, eiE (x) = ϕ(x − y) dhµ, eiE (y) = ϕ(x − y) dµ(y), e E ­ ® = ϕ ∗ µ(x), e E , b

b

b

the definition of h·, ·i in Subsection VII.1.2.

b

5 Recall

446

Appendix

so that the last integral in (2.0.19) equals Z Z ­ ® ϕ ∗ µ(x), ψ(x) ⊗ e E dx = hϕ ∗ µ, uiE dx = hϕ ∗ µ, ui . b

b

b

This proves the lemma.

¥

The Riesz Representation Theorem Finally, we prove a generalization of the classical (scalar) Riesz representation theorem. 2.0.4 Theorem Let X be a σ-compact metrizable space and let E be a Banach space. Then C0 (X, E)0 = MBV (X, E 0 ) with respect to the duality pairing Z hµ, uiC0 := u dµ , µ ∈ MBV (X, E 0 ) ,

u ∈ C0 (X, E) .

X

Proof

Suppose that µ ∈ MBV (X, E 0 ). Then it follows from (2.0.4) that Z ¡ ¢ u 7→ u dµ ∈ C0 (X, E)0

and ¯Z ¯ ¯ ¯ ¯ u dµ¯ ≤ kµkBV kuk∞ ,

u ∈ C0 (X, E) .

(2.0.20)

Conversely, let w ∈ C0 (X, E)0 be fixed. Then, given any e ∈ E, the scalar Riesz representation theorem (e.g., [Rud70, Theorems 2.14, 6.2, and 6.19]) implies the existence of a unique regular K-valued Radon measure µe on X satisfying Z w(ϕ ⊗ e) = ϕ dµe , ϕ ∈ C0 (X) , (2.0.21) and kµe kBV =

¯Z ¯ ¯ ¯ sup ¯ ϕ dµe ¯ = kϕk∞ ≤1

sup |w(ϕ ⊗ e)| ≤ kwkC00 |e|E .

kϕk∞ ≤1

Since, by uniqueness, the map E → (BX → K), e 7→ µe is linear, it follows that ­ ® µ(B), e := µe (B) , B ∈ BX , (2.0.22) defines a map µ : BX → E 0 that is easily seen to be finitely additive.

Vector Measures and the Riesz Representation Theorem

447

In order to show that µ has bounded variation we first consider a family {O1 , . . . , Om } of pair-wise disjoint open subsets of X. For each j ∈ {1, . . . , m} we choose ϕj ∈ C0 (X) with supp(ϕj ) ⊂ Oj and kϕj k∞ ≤ 1, as well as ej ∈ BE . Then, putting εj := sign w(ϕj ⊗ ej ) ∈ K and using (2.0.21), m ¯Z X ¯ ¯ j=1

Oj

m m ¯ X X ¯ ϕj dµej ¯ = |w(ϕj ⊗ ej )| = εj w(ϕj ⊗ ej ) j=1

=w

m ³X

j=1

´ ϕj ⊗ εj ej ≤ kwkC00 .

j=1

Consequently, m X ¯ ¯ ¯µe ¯ (Oj ) ≤ kwkC 0 , j 0

e j ∈ BE ,

1≤j≤m.

(2.0.23)

j=1

Now let π := {B1 , . . . , Bm } be an arbitrary partition of X into Borel sets. Then, given ε > 0, we find for each j ∈ {1, . . . , m} an element ej ∈ BE such that ¯­ ® ¯ |µ(Bj )|E 0 ≤ ¯ µ(Bj ), ej E ¯ + ε/m . (2.0.24) The regularity of µej implies the existence of a compact subset Kj of Bj with ¯ ¯ ¯ ¯ ¯µej ¯ (Bj ) ≤ ¯µej ¯ (Kj ) + ε/m . (2.0.25) Since Kj ∩ Kk = ∅ for j 6= k we find pair-wise disjoint open sets O1 , . . . , Om in X such that Kj ⊂ Oj for 1 ≤ j ≤ m. Now we infer from (2.0.23)–(2.0.25) that m X

|µ(Bj )|E 0 ≤

j=1

≤

m m X X ¯­ ¯ ¯ ® ¯ ¯ µ(Bj ), ej ¯ + ε ≤ ¯µe ¯ (Bj ) + ε j E j=1 m X

¯ ¯ ¯µe ¯ (Kj ) + 2ε ≤ j

j=1

j=1 m X

¯ ¯ ¯µe ¯ (Oj ) + 2ε ≤ kwkC 0 + 2ε . j 0

j=1

Hence µ is of bounded variation. B :=

Let S now (Bj ) be a sequence in BX such that Bj ∩ Bk = ∅ for j 6= k, and put Bj . Then it is clear that X j

|µ(Bj )| ≤ kµkBV < ∞ .

448

Appendix

Thus

­P

DX j

j

µ(Bj ), e

® E

E µ(Bj ), e

E

=

is well-defined for e ∈ E and

X­ X ® ­ ® µ(Bj ), e E = µe (Bj ) = µe (B) = µ(B), e E (2.0.26) j

j

by the σ-additivity of µe . Since (2.0.26)Pis true for every e ∈ E, and E separates the points of E 0 , it follows that µ(B) = j µ(Bj ). Now (2.0.11), (2.0.21), (2.0.22), and the density of C0 (X) ⊗ E in C0 (X, E) imply Z hw, uiC0 = u dµ , u ∈ C0 (X, E) . From this and (2.0.20) the assertion follows.

¥

The above proof of the generalized Riesz representation theorem is due to W. Arendt (personal communication).

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M.S. Agranovich, M.I. Vishik. Elliptic problems with a parameter and parabolic problems of general type. Russ. Math. Surveys, 19 (1964), 53– 157.

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List of Symbols Special Symbols448

Maps

448 ∗ q . . . . . . . . . . . . . . . . . . . . . . . . . . 428, sθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/rθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [`, d, ν] . . . . . . . . . . . . . . . . . . . . . . . . . . . [η] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m, ν]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . x i ; xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . hxi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = ............................... η

t q ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 rQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 λy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 MQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 h·, ·i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 h ·, ·ii . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 21 hu0 , uiS . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 hu0 q uiF . . . . . . . . . . . . . . . . . . . . . . . . . 413 ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . 388, 391 ⊗ q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 ⊗π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 ⊗ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 ∼ ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

439 28 28 32 43 33 7 5 273

[[·]]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 q ωi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 σt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34, 37 σtν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 f ∼ g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 k(ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 t(j, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Sets J∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 JK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Sϑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Hd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 K∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Kdk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Sd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ∂j K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 r ∂(j) K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 dK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 dI K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Topological Concepts BQ (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . SQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n ............................... nS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188 44 441 299 353

Operator Families Hzk (Z, E) . . . . . . . . . . . . . . . . . . . . . . . . . Mη (Rd , E) . . . . . . . . . . . . . . . . . . . . . . . P L(E1 , E0 ); κ, ϑ . . . . . . . . . . . . . . . . Ps Z, L(E); κ . . . . . . . . . . . . . . . . . . . . Ps Z, L(E1 , E0 ); κ, ϑ . . . . . . . . . . . . .

45 57 55 66 58

Linear Operators 1X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 s Tq,[η] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 (RK , EK ) . . . . . . . . . . . . . . . . . . . . . . . 8, 14 γjk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 γ∂H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 j γ∂H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

© Springer Nature Switzerland AG 2019 H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4

457

458 γdK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂nj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂nk j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~γ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~I ~γIm .............................

List of Symbols 334 354 300 333 301 334

224

Point-Wise Defined Functions B ................................ BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BC ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/ν BCη . . . . . . . . . . . . . . . . . . . . . . . . . . s C .............................. m/ν C0 ............................ s/ν C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . BUC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BUC m/ν . . . . . . . . . . . . . . . . . . . . . . . . . BUC s/ν . . . . . . . . . . . . . . . . . . . . . . . . . m/ν BUCη ........................ D ............................... E................................ OM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S ............................... `r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . `sr . . p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ρ/ν b∞,surj . . . . . . . . . . . . . . . . . . . . . . . . . . bucs/ν . . . . . . . . . . . . . . . . . . . . . . . . . . . c ................................. cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cs0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78 79 80 272 257 80 157 79 80 157 273 373 373 374 374 20 26 345 157 19 258 20 26 19

Integrable Functions t/ν

Br . . . . . . . . . . . . . . . . . . . . . . . . . . . . s/ν Bq,r;η . . . . . . . . . . . . . . . . . . . . . . . . . . . s/ν Hq;η . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ Lr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lp,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lp,c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lq,ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/ν W∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . m/ν Wq;η . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 157 210 100 100

Distributions

Spaces of Linear Operators L(E0 , E1 ; E2 ) . . . . . . . . . . . . . . . . . . .

m/ν

Wq ............................ s/ν Wq ........................... s/ν Wp . . . . . . . . . . . . . . . . . . . . . . . . . . . s/ν `r X . . . . . . . . . . . . . . . . . . . . . . . . . . . s/ν c0 X . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 274 274 141 430 430 310 226 272

s Bq,r ............................. s/ν Bq,r . . . . . . . . . . . . . . . . . . . . . . . 104, s/ν bq,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −m/ν C0 ........................... s/ν Fq,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . s/ν Hq . . . . . . . . . . . . . . . . . . . . . . . . . . . . s(~ k,p)/ν

p

Wp ...................... D0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 D+ .............................. 0 DK ............................. MBV . . . . . . . . . . . . . . . . . . . . . . . . 82, 0 OC .............................. 0 S ................................ s/ν ˚q,r B ............................ s/ν ˚ Fq . . . . . . . . . . . . . . . . . . . . . . . . . . . . t/ν Fq,B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s/ν

258 129 107 83 196 168 326 256 436 436 441 439 11 107 304 360

Fq,~γ k . . . . . . . . . . . . . . . . . . . . . . . . . . . .

304

s/ν Fq,r (H+ , H− ; E) . . . . . . . . . . . . . . . . s/ν ˚ Fq (H+ , H− ; E) . . . . . . . . . . . . . . . . s(~ k,q)/ν ∂Fq ...................... t/ν ∂Fq . . . . . . . . . . . . . . . . . . . . . . . . . . .

354 357

p

p

301 300

Differential Operators ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ∂ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ∂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ∂a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B(C m ). . . . . . . . . . . . . . . . . . . . . . . . . . . 355 a(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 Fourier Transforms, Multipliers, and Convolutions J z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Jηz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Hz (Z, E) . . . . . . . . . . . . . . . . . . . . . . . . . 56 M(Rd , E) . . . . . . . . . . . . . . . . . . . . . . . . 54 Ma(Rd , E) . . . . . . . . . . . . . . . . . . . . . . 171

List of Symbols

459

Mi(Rd , E) . . . . . . . . . . . . . . . . . . . . . . 171 m(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 u ∗ ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Interpolation Theory J(θ, E0 , E1 ) . . . . . . . . . . . . . . . . . . . . .

212

Norms and Seminorms Λ ................................ k·ks,q,J . . . . . . . . . . . . . . . . . . . . . . . . . . k·ks,q,r,J . . . . . . . . . . . . . . . . . . . . . . . . k·k∗s/ν ,p,r . . . . . . . . . . . . . . . . . . . . . . . . |||·|||m/ν ,q;η . . . . . . . . . . . . . . . . . . . . . . |||·|||m/ν ,r . . . . . . . . . . . . . . . . . . . . . . . . . |||·|||∗s/ν ,p,r . . . . . . . . . . . . . . . . . . . . . . . |||·|||s/ν,p,r . . . . . . . . . . . . . . . . . . . . . . . [[·]]θ,p,r;i . . . . . . . . . . . . . . . . . . . . . . . . . [·]θ,q,J . . . . . . . . . . . . . . . . . . . . . . . . . . . [·]θ,q,r,J . . . . . . . . . . . . . . . . . . . . . . . . . [·]δϑ,∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . [·]s,p,r . . . . . . . . . . . . . . . . . . . . . . 149, N ................................ |µ| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pm,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . qk,m . . . . . . . . . . . . . . . . . . . . . . . . . . 5,

43 257 258 149 272 79 154 150 154 257 257 159 213 39 441 373 374

Index adapted, Ω-, 67 adjoint corner, 4 admissible ν-, 67 ν– Banach space, 173 annihilator, 94 augmentation, parametric, 43 Banach ν-admissible – space, 173 – space of bounded functions, 78 barreled, 370 Besov space, 104 –H¨ older space, 155 –Slobodeckii space, 155 Fourier multiplier theorem for –s, 113 little, 107 very little, 107 Bessel ν-anisotropic – kernel, 53 – potential space, 168 bornological, 372 clustering, d-, 33 cofinal, 372 condition (Σ), 432 distributions satisfying –, 432 cone, 42 dual, 432 proper, 432 conuclear, 393 convolution, 428, 439 – of distributions, 379 core, 137 corner adjoint, 4 face of a, 5 general, 6

standard, 4 type of a, 5 covering, ν-dyadic, 67 derivative directional, 146 distributional, 11 jump of a normal, 357 dilation associated with a weight system, 33 Dirac distribution, 380 directional derivative, 146 distribution – satisfying condition (Σ), 432 Dirac, 380 tempered, 10 dual exponent, 21 duality – pairing, 11, 20, 21, 415 distributional – pairing, 85 embedding Rellich–Kondrachov - theorem, 255 equivalent η-uniformly, 58, 273 uniformly, 273 Euclidean ν-quasinorm, 40 extension point-wise, 224 trivial, 8 face of a corner, 5 Fatou property, 200 Fourier multiplier, 52 – space, 54 – theorem for Besov spaces, 113 Fubini’s rule, 417 function Banach space of bounded –s, 78

© Springer Nature Switzerland AG 2019 H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics 106, https://doi.org/10.1007/978-3-030-11763-4

461

462

Index Heaviside, 423 space of bounded and uniformly continuously differentiable –s, 80

Hardy inequalities, 312 Heaviside function, 423 H¨ older, generalized – inequality, 217 H¨ older space, 157 – scale, 157 Besov–, 155 little, 157 homogeneous ν– of degree z, 39 positively z–, 39 inductive limit topology, 372 inequality generalized H¨ older, 217 Hardy –ies, 312 Sobolev, 217 integral with respect to vector measures, 442 interface, 353 jump, 355, 357 – of a normal derivative, 357 kernel ν-anisotropic Bessel, 53 – theorem, 398 regularizing, 428 Kondrachov, Rellich– embedding theorem, 255 Kronecker symbol, 19 Leibniz rule, 385, 410 linear representation, 34 locally convex direct sum, 19 map – associated to a kernel, 427 canonical, 395 proper, 432 matrix, transposed, 392 measure vector, 441 weighted counting, 27 metric, ν-parabolic, 42

multiplication, 47, 383 – algebra, 413 point-wise, 384, 409 multiplier Fourier, 52 Fourier – space, 54 Fourier – theorem for Besov spaces, 113 universal point-wise – space, 226 norm quotient, 26 uniformly equivalent, 273 normal – boundary operator, 344 – inner (unit), 299 ν-admissible, 67 – Banach space, 173 ν-dyadic – covering, 67 – partition of unity, 68 ν-starshaped, 67 ν-homogeneous weight system, 33 ν-parabolic – metric, 42 – quasinorm, 42 nuclear, 393 operator normal boundary, 344 normal transmission, 356 trace, 283 translation-invariant, 440 transmission, 355 order sequence, 348 pair r-c, 4 r-e, 9, 14 restriction-extension, 9, 14 retraction-coretraction, 4 universal retraction-coretraction, 104 pairing distributional duality, 85 duality, 11, 20, 21, 415 parametric augmentation, 43 partition of unity, ν-dyadic, 68

Index

point-wise – extension, 224 – multiplication, 384, 409 – multiplier space, 226 – restriction, 7 polar set, 94 product injective tensor, 395 projective tensor, 393 scalar – induced by multiplication, 414 tensor, 388, 390, 417 proper, 432 property – (α), 171 Fatou, 200 pull-back, 11 quasinorm ν-, 39 Euclidean ν-, 40 natural ν-, 39, 43 ν-parabolic, 42 quasisphere, Q-, 44 quotient norm, 26 Rellich–Kondrachov embedding theorem, 255 restriction-extension pair, 9, 14 retraction, Q-, 44 retraction-coretraction pair, 4 universal – pair, 104 rule Fubini’s, 417 Leibniz, 385, 410 scale H¨ older space, 157 little H¨ older space, 157 Sobolev–Slobodeckii space, 157 very little H¨ older space, 157 Schwartz kernel theorem, 427 Slobodeckii Besov – space, 155 Sobolev– space scale, 157 space, 157

463 Sobolev – space of negative order, 86 –Slobodeckii space scale, 157 anisotropic Lq – space, 84 inequality, 217 space – of Lp functions with compact support, 430 – of bounded and uniformly continuously differentiable functions, 80 – of tempered distributions, 10 – of test functions, 373 anisotropic Lq Sobolev, 84 Banach – of bounded functions, 78 Besov, 104 Besov – on corners, 129 Besov–H¨ older, 155 Besov–Slobodeckii, 155 Bessel potential, 168 Fourier multiplier, 54 H¨ older, 157 H¨ older – scale, 157 LF –, 372 little Besov, 107 little H¨ older, 157 Montel, 372 ν-admissible Banach, 173 Schwartz, 374 Slobodeckii, 157 Sobolev – of negative order, 86 Sobolev–Slobodeckii – scale, 157 universal point-wise multiplier, 226 very little Besov, 107 very little H¨ older, 157 weighted sequence, 26 standard corner, 4 starshaped, ν-, 67 support, convolutive, 432 tempered distribution, 10 tensor product, 388, 390, 417 theorem Banach–Steinhaus, 371 convolution, 439 Fourier multiplier, 113 kernel, 398

464 Rellich–Kondrachov embedding, 255 Schwartz kernel, 427 topology inductive limit, 372 injective tensor product, 395 LF –, 372 projective tensor product, 393 trace operator, 283 translation semigroup, 146 translation-invariant operator, 440 transmission operator, 355 normal, 356 trivial – extension, 8 – weight system, 42 type – of a boundary operator, 343 – of a corner, 5 uniformly – equivalent, 273 – surjective, 345 η-, 58, 273 universal – point-wise multiplier space, 226 – r-e pair, 15 – retraction-coretraction pair, 104 -ity property, 391 variation, 441 bounded, 441 vector measure, 441 weight system, 32 dilation associated with a, 33 non-reduced, 33 ν-homogeneous, 33 parabolic, 33 reduced, 33 trivial, 42 weighted – counting measure, 27 – sequence space, 26

Index