Pseudodifferential Equations Over Non-Archimedean Spaces [1st ed.] 3319467379, 978-3-319-46737-5, 978-3-319-46738-2, 3319467387

Table of contents :
Front Matter....Pages i-xvi
p-Adic Analysis: Essential Ideas and Results....Pages 1-11
Parabolic-Type Equations and Markov Processes....Pages 13-41
Non-Archimedean Parabolic-Type Equations with Variable Coefficients....Pages 43-77
Parabolic-Type Equations and Markov Processes on Adeles....Pages 79-125
Fundamental Solutions for Pseudodifferential Operators, and Equations of Schrödinger Type....Pages 127-143
Pseudodifferential Equations of Klein-Gordon Type....Pages 145-165
Final Remarks and Some Open Problems....Pages 167-170
Back Matter....Pages 171-177

Citation preview

Lecture Notes in Mathematics  2174

W. A. Zúñiga-Galindo

Pseudodifferential Equations Over Non-Archimedean Spaces

Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Michel Brion, Grenoble Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg

2174

More information about this series at http://www.springer.com/series/304

W.A. ZúQniga-Galindo

Pseudodifferential Equations Over Non-Archimedean Spaces

123

W.A. ZúQniga-Galindo Department of Mathematics Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV) Mexico City, Mexico

ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-46737-5 DOI 10.1007/978-3-319-46738-2

ISSN 1617-9692 (electronic) ISBN 978-3-319-46738-2 (eBook)

Library of Congress Control Number: 2016963106 Mathematics Subject Classification (2010): 11-XX; 43-XX; 46-XX; 60-XX; 70-XX © Springer International Publishing AG 2016 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my wife Mónica, my daughter Daniela, and my son Felipe

Preface

In recent years, p-adic analysis (or more generally non-Archimedean analysis) has received a lot of attention due to its connections with mathematical physics; see, e.g., [8–14, 20, 21, 25, 26, 36, 64, 65, 67–69, 75, 76, 80, 82, 85–87, 90, 94, 106–109, 111] and references therein. All these developments have been motivated by two physical ideas. The first is the conjecture (due to I. Volovich) in particle physics that at Planck distances the space-time has a non-Archimedean structure; see, e.g., [107, 112, 113]. The second idea comes from statistical physics, in particular, in connection with models describing relaxation in glasses, macromolecules, and proteins. It has been proposed that the non-exponential nature of those relaxations is a consequence of a hierarchical structure of the state space which can in turn be put in connection with p-adic structures; see, e.g., [6, 10–12, 46, 94]. Additionally, we should mention that in the middle of the 1980s the idea of using ultrametric spaces to describe the states of complex biological systems, which naturally possess a hierarchical structure, emerged in the works of Frauenfelder, Parisi, and Stain, among others; see, e.g., [6, 36, 46]. In protein physics, it is regarded as one of the most profound ideas put forward to explain the nature of distinctive life attributes. On the other hand, stochastic processes on p-adic spaces, or more generally on ultrametric spaces, have been studied extensively in the last 30 years due to diverse physical and mathematical motivations; see, e.g., [1–4, 16, 17, 19, 28, 42, 43, 61– 63, 71, 98, 116, 118, 122] and references therein. In [10–12], Avetisov et al. introduced a new class of models for complex systems based on p-adic analysis. These models can be applied, for instance, to the study of the relaxation of certain biological complex systems. From a mathematical point of view, in these models, the time-evolution of a complex system is described by a padic master equation (a parabolic-type pseudodifferential equation) which controls the time-evolution of a transition function of a Markov process on an ultrametric space, and this stochastic process is used to describe the dynamics of the system in the space of configurational states which is approximated by an ultrametric space (Qp ). The first goal of this work is to study a very large class of heattype pseudodifferential equations over p-adic and adelic spaces, which contains as special cases many of the equations that occur in the models of Avetisov et al. It is vii

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worth to mention here that the p-adic heat equation also appeared in certain works connected with the Riemann hypothesis [83]. The simplest type of master equation is the one-dimensional p-adic heat equation. This equation was introduced in the book of Vladimirov, Volovich, and Zelenov [111, Section XVI]. In [80, Chaps. 4 and 5] Kochubei presented a general theory for one-dimensional parabolic-type pseudodifferential equations with variable coefficients, whose fundamental solutions are transition density functions for Markov processes in the p-adic line; see also [97, 98, 108]. Varadarajan studied the heat equation on a division algebra over a non-Archimedean local field [108]. In [122], the author introduced p-adic analogs for the n-dimensional elliptic operators and studied the corresponding heat equations and the associated Markov processes; see also [23, 108]. In [26], building up on [25] and [80, 81], Chacón-Cortés and the author introduced a new type of nonlocal operators and a class of parabolic-type pseudodifferential equations with variable coefficients, which contains the onedimensional p-adic heat equation of [111], the equations studied by Kochubei in [80], and the equations studied by Rodríguez-Vega in [97]. The field of p-adic numbers Qp is defined as the completion of the field of rational numbers Q with respect˚ to the p-adic  norm j  jp ; see Chap.  2. The p-adic  norm satisfies jjx C yjjp  max jjxjjp ; jjyjjp , and the metric space Qnp ; jj  jjp is a complete ultrametric space. This space has a natural hierarchical structure, which is very useful in physical models that involve hierarchies. As a topological space, Qp is homeomorphic to a Cantor-like subset of the real line. The p-adic heat equation is defined as @u .x; t/ C .D˛ u/ .x; t/ D 0, x 2 Qp , t > 0 @t

(1)

where  ˛  1 .D˛ '/ .x/ D F!x jjp Fx! ' ; ˛ > 0 is the Vladimirov operator and F denotes the p-adic Fourier transform. This equation is the p-adic counterpart of the classical heat equation, which describes a particle performing a random motion (the Brownian motion); a ‘similar’ statement is valid for the p-adic heat equation. More precisely, the fundamental solution of (1) is the transition density of a bounded right-continuous Markov process without second kind discontinuities. A well-known and accepted scientific paradigm in physics of complex systems (such as glasses and proteins) asserts that the dynamics of a large class of complex systems is described as a random walk on a complex energy landscape; see, e.g., [6, 46, 47, 114], [82] and references therein. A landscape is a continuous real-valued function that represents the energy of a system. The term complex landscape means that energy function has many local minima. In the case of complex landscapes, in which there are many local minima, a “simplification method” called interbasin kinetics is applied. The idea is to study the kinetics generated by transitions between groups of states (basins). In this setting, the minimal basins correspond to local

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Fig. 1

minima of energy, and the large basins (superbasins or union of basins) have a hierarchical structure. A key idea is that a complex landscape is approximated by a disconnectivity graph (an ultrametric space) and by a function on this graph that gives the distribution function of activation energies. For the construction of the disconnectivity graph, we can imagine that the energy landscape is a water tank and that we are pumping water into it. The energy landscape of the system is flooded with water, and water forms pools around the local minima. By increasing the level of water, pools merge, until only one big pool remains. This procedure allows us to construct a directed tree of basins (pools). The activation barrier function on this tree is constructed as follows: minimal pools are assigned their depth (local minima energy), and larger pools are assigned energy levels at which these pools emerge (the activation barrier between basins). This procedure is illustrated in Fig. 1. For “real” energy landscapes and the corresponding graphs, the reader may see [114]. The next step is to construct a model of hierarchical dynamics based on the disconnectivity graph. By using the postulates of the interbasin kinetics, one gets that the transitions between basins are described by the following equations: X X @f .i; t/ D T .j; i/ f .j; t/ v.j/  T .i; j/ f .i; t/ v .j/ ; @t j j¤i

where the indices i,j number the states of the system (which correspond to local minima of energy), T .i; j/  0 is the probability per unit time of a transition from i to j, and the v.j/ > 0 are the basin volumes. Under suitable physical and mathematical hypotheses, the above master equation has the following “continuous limit”: Z @f .x; t/ Œw .x jy / f .y; t/  w .y jx / f .x; t/ dy; D @t Qp

x

Preface

where x 2 Qp ; t  0.R The function f .x; t/ W Qp  RC ! RC is a probability density distribution, so that B f .x; t/ dx is the probability of finding the system in a domain B  Qp at the instant t. The function w .x jy / W Qp  Qp ! RC is the probability of the transition from state y to state x per unit of time. This family of parabolic-type equations contains the p-adic heat equation as a particular case. The first part of this work is dedicated to present a snapshot of the theory, still under construction, of pseudodifferential equations of parabolic type and their Markov processes on p-adic and adelic spaces. In Chap. 1, we review, without proofs, some basic aspects of p-adic analysis and p-adic manifolds that we use in this work. An interesting novelty of this work is that some of the equations studied here require using geometric methods, for instance, integration on p-adic manifolds. Chapters 2 and 3 are dedicated to the study of parabolic-type equations and their Markov processes. The results presented in these chapters continue and extend, in a considerable form, the corresponding results presented in the books [80] and [111]. Chapter 4 deals with the heat equation on the ring of adeles. We present only the essential techniques and results; many important related topics were left aside. For instance, we do not include pseudodifferential operators and wavelets on general locally compact ultrametric spaces, not necessarily having a group structure; see, e.g., [77, 82] and [66]; wavelet analysis on adeles and pseudodifferential operators; see, e.g., [74] and [73]; and p-adic Brownian motion and stochastic pseudodifferential equations; see, e.g., [21, 70, 71], among other several important topics. The second part, Chaps. 5 and 6, is dedicated to pseudodifferential equations whose symbols involve polynomials. In the 1950s Gel’fand and Shilov showed that fundamental solutions for certain types of partial differential operators with constant coefficients can be obtained by using local zeta functions [49]. The existence of fundamental solutions for general differential operators with constant coefficients was established by Atiyah [15] and Bernstein [18] using local zeta functions. A similar program can be carried out in the p-adic setting; see, e.g., [120]; see also [123]. The goal of Chap. 5 is to prove the existence of fundamental solutions for pseudodifferential operators via local zeta functions. Chapter 6 deals with a new class of non-Archimedean pseudodifferential equations of Klein-Gordon type. These equations have many similar properties to the classical Klein-Gordon equations; see, e.g., [31, 32, 103]. Finally, in Chap. 7, we present some open problems connecting non-Archimedean pseudodifferential equations with number theory, probability, and physics. These notes were intended as a one-semester doctoral course at CINVESTAV for students interested in doing research in p-adic analysis connected with physics of complex systems, probability, and number theory. There are many open problems in this area. The central goal of these notes was to prepare fastly my students to do research. I wish to thank all my coauthors (Oscar F. Casas-Sánchez, Leonardo ChacónCortes, Jeanneth Galeano-Peñaloza, John J. Rodríguez-Vega, and Sergii M. Torba) and my current doctoral students and postdocs (Victor Aguilar, Samuel Arias, Edilberto Arroyo, Maria Luisa Mendoza, and Anselmo Torresblanca) for helping

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me on this project. I also wish to thank my family (my wife Monica, my daughter Daniela, and my son Felipe) for their constant support. Finally, I wish to thank the CONACYT for supporting my research activities during the last 10 years through several grants (my latest CONACYT grant no. 250845) and through the Mexican Research Chairs Program (SNI level III since 2016). Mexico

W.A. Zúñiga-Galindo

Contents

1

p-Adic Analysis: Essential Ideas and Results . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 The Field of p-Adic Numbers .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Additive Characters .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Multiplicative Characters .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Topology of Qnp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 The Bruhat-Schwartz Space and the Fourier Transform .. . . . . . . . . . . . 1.3.1 The Fourier Transform of Test Functions .. . . . . . . . . . . . . . . . . . 1.4 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 The Fourier Transform of a Distribution .. . . . . . . . . . . . . . . . . . . 1.4.2 The Direct Product of Distributions . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.3 The Convolution of Distributions . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.4 The Multiplication of Distributions . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Essential Aspects of the p-Adic Manifolds .. . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.1 Implicit Function Theorem . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.2 p-Adic Analytic Manifolds . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.3 Integration on p-Adic Manifolds . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.4 Integration Over the Fibers . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 2 2 3 3 4 4 5 5 6 6 6 7 8 9 10

2 Parabolic-Type Equations and Markov Processes . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Operators W, Parabolic-Type Equations and Markov Processes . . . . 2.2.1 A Class of Non-local Operators . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Some Additional Results . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 p-Adic Description of Characteristic Relaxation in Complex Systems . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.4 Heat Kernels.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.5 Markov Processes Over Qnp . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.6 The Cauchy Problem.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.7 The Taibleson Operator and the p-Adic Heat Equation . . . .

13 13 14 14 17 18 20 22 24 26

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2.3

Elliptic Pseudodifferential Operators, Parabolic-Type Equations and Markov Processes . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Decaying of the Fundamental Solution at Infinity.. . . . . . . . . 2.3.3 Positivity of the Fundamental Solution .. . . . . . . . . . . . . . . . . . . . 2.3.4 Some Additional Results . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.5 The Cauchy Problem.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.6 Markov Processes Over Qnp . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

28 28 31 35 37 39 40

3 Non-Archimedean Parabolic-Type Equations with Variable Coefficients.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 A Class of Non-local Operators .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Parabolic-Type Equations with Constant Coefficients . . . . . . . . . . . . . . . 3.3.1 Claim u.x; t/ 2 M . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 Claim u.x; t/ Satisfies the Initial Condition . . . . . . . . . . . . . . . . 3.3.3 Claim u.x; t/ Is a Solution of Cauchy Problem (3.4) .. . . . . . 3.4 Parabolic-Type Equations with Variable Coefficients.. . . . . . . . . . . . . . . 3.4.1 Parametrized Cauchy Problem . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.2 Heat Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.3 Construction of a Solution .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Uniqueness of the Solution .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Markov Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 The Cauchy Problem Is Well-Posed . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

43 43 43 45 46 51 51 59 60 61 64 70 73 75

4 Parabolic-Type Equations and Markov Processes on Adeles . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 Adeles on Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Fourier Transform on Adeles . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Metric Structures, Distributions and Pseudodifferential Operators on Af . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 A Structure of Complete Metric Space for the Finite Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 The Fourier Transform of Radial Functions . . . . . . . . . . . . . . . . 4.3.3 Distributions on Af . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.4 Pseudodifferential Operators and the Lizorkin Space on Af . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Metric Structures, Distributions and Pseudodifferential Operators on A .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 A Structure of Complete Metric Space for the Adeles . . . . . 4.4.2 Distributions on A . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.3 Pseudodifferential Operators and the Lizorkin Space on A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 The Adelic Heat Kernel on Af . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Markov Processes on Af . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

79 79 80 80 82 84 84 91 98 101 104 104 104 105 108 111

Contents

Cauchy Problem for Parabolic Type Equations on Af . . . . . . . . . . . . . . . 4.7.1 Homogeneous Equations with Initial Values in D.Af / . . . . 4.7.2 Homogeneous Equations with Initial Values in L2 .Af / . . . . 4.7.3 Homogeneous Equations with Initial Values in L0 .Af / .. . . 4.7.4 Non Homogeneous Equations .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8 The Adelic Heat Kernel on A . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.9 Markov Processes on A . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.10 Cauchy Problem for Parabolic Type Equations on A . . . . . . . . . . . . . . . . 4.10.1 Homogeneous Equations . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.10.2 Non Homogeneous Equations .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of Schrödinger Type . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 The Method of Analytic Continuation of Gel’fand-Shilov . . . . . . . . . . 5.3 Igusa’s Local Zeta Functions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Fundamental Solutions for Pseudodifferential Operators .. . . . . . . . . . . 5.5 Fundamental Solutions for Quasielliptic Pseudodifferential Operators .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Schrödinger-Type Pseudodifferential Equations .. . . . . . . . . . . . . . . . . . . . 5.6.1 The Homogeneous Equation . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6.2 The Inhomogeneous Equation .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6 Pseudodifferential Equations of Klein-Gordon Type . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 Fourier Transform on Finite Dimensional Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 The p-Adic Minkowski Space . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.3 Invariant Measures Under the Orthogonal Group O.Q/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.4 Some Remarks About p-Adic Analytic Manifolds .. . . . . . . . 6.2.5 Some Additional Results on ı .Q .k/  t/. . . . . . . . . . . . . . . . . . . 6.2.6 The p-Adic Restricted Lorentz Group . .. . . . . . . . . . . . . . . . . . . . 6.3 A p-Adic Analog of the Klein-Gordon Equation.. . . . . . . . . . . . . . . . . . . . 6.4 The Cauchy Problem for the Non-Archimedean Klein-Gordon Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.1 Twisted Vladimirov Pseudodifferential Operators . . . . . . . . . 6.4.2 The Cauchy Problem for the p-Adic Klein-Gordon Equation .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Further Results on the p-Adic Klein-Gordon Equation .. . . . . . . . . . . . .

xv

113 114 116 117 118 118 119 122 122 125 127 127 128 129 130 133 137 140 143 145 145 146 146 147 148 149 151 154 155 158 159 161 165

7 Final Remarks and Some Open Problems . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167 7.1 Zeta Functions, Adelic Riesz Kernels and Heat Equations . . . . . . . . . . 167 7.2 Fundamental Solutions and Operators of Bernstein-Sato Type . . . . . 169

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Contents

7.3 7.4

Parabolic-Type Equations and Markov Processes. . . . . . . . . . . . . . . . . . . . 170 Non-Archimedean Quantum Fields . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 170

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171

Chapter 1

p-Adic Analysis: Essential Ideas and Results

In this chapter, we present, without proofs, the essential aspects and basic results on p-adic functional analysis needed in the book. For a detailed exposition on p-adic analysis the reader may consult [5, 105, 111].

1.1 The Field of p-Adic Numbers Along this book p will denote a prime number. The field of p-adic numbers Qp is defined as the completion of the field of rational numbers Q with respect to the p-adic norm j  jp , which is defined as jxjp D

8

=

  p pj y   0 d n y : > ;

By using the formula 8 n ˆ ˆ 1  p if j  0 ˆ ˆ ˆ Z <  j  n p p y   0 d y D pn if j D 1 ˆ ˆ ˆ ˆ Sn0 ˆ : 0 if j < 1;

(2.8)

see e.g. [105, Lemma 4.1], we get Aw ./ D .1  pn / D .1  pn /

1 X pn Cn pn.Cj/ C w. p Cj / w. p C1 / jD2 1 X

pn Cn pnj C : w. pj / w. p C1 / jD C2

(2.9)

From (2.9) follows that   Aw ./ is radial, positive, continuous outside of the origin, and that Aw pord./ is a decreasing function of ord./. To show that Aw . p /  . C1/   Aw . p / D is a decreasing function of

 , we note that, by (2.9), Aw p pnCn

1 w. pC2 /



1 w. pC1 /

< 0. The continuity at the origin follows from

Aw .0/ WD lim .1  pn /  !1

1 X

pnj pn Cn C lim D 0; w. pj /  !1 w. p C1 / jDC2

2.2 Operators W, Parabolic-Type Equations and Markov Processes

P since 1 jDM cf. (2.1).

pnj w. pj /

< 1, cf. (2.2 ), and

1 C0



p˛ 1 . C1/ w. pC1 /



pnCn w. pC1 /

17

for  big enough, 

Remark 6 We denote by C.U; C/, respectively by C.U; R/, the vector space of C-valued, respectively of R-valued, continuous functions defined on an open subset U of Qnp . In some cases we use the notation C.U/, or just C, if there is no danger of confusion.     1 Proposition 7 (i) .W'/ .x/ D F!x Aw .kkp /Fx! ' for ' 2 D Qnp , and     W' 2 C Qnp \ L Qnp , for 1   1. The Operator W extends to an   unbounded and densely defined operator in L2 Qnp with domain  ˚ Dom.W/ D ' 2 L2 I Aw .kkp /F ' 2 L2 :

(2.10)

(ii) .W; Dom.W// is self-adjoint and positive operator. (iii) W is the infinitesimal generator of a contraction C0 semigroup .T .t//t0 . Moreover, the semigroup .T .t//t0 is bounded holomorphic with angle =2. Proof (i) It follows from Lemma 4 and the fact that Aw .kkp / is continuous, cf. Lemma 5. (ii) follows from the fact that W is a pseudodifferential operator and that the Fourier transform preserves the inner product of L2 . (iii) It follows of well-known results, see e.g. [41, Chap. 2, Sect. 3] or [24]. For the property of the semigroup of being holomorphic, see e.g. [41, Chap. 2, Sect. 4.7]. 

2.2.2 Some Additional Results Lemma 8 Assume that there exist positive constants ˛ 1 , ˛ 2 , C0 , C1 , with ˛ 1 > n, ˛ 2 > n, and ˛ 3  0, such that    ˛  ˛ ˛  0 C0  0 p 1  w. 0 p /  C1  0 p 2 e 3 k kp , for any  0 2 Qnp :

(2.11)

Then there exist positive constants C2 , C3 , such that 1

C2 kkp˛2 n e˛ 3 pkkp  Aw .kkp /  C3 kkp˛ 1 n 1

1

for any  2 Qnp , with the convention that e˛ 3 pk0kp WD limkkp !0 e˛ 3 pkkp D 0. Furthermore, if ˛ 3 > 0, then ˛ 1  ˛ 2 , and if ˛ 3 D 0, then ˛ 1 D ˛ 2 . Proof By using the lower bound for w given in (2.11), and kkp D p , Aw .kkp / 

1 .1  pn / X pnj pnCn C ˛ . C1/  C3 kkp˛1 n : j˛ C0 p 1 p 1 jD C2

18

2 Parabolic-Type Equations and Markov Processes

  On the other hand, Aw kkp  in (2.11),   Aw kkp 

pnCn w. pC1 /

, and by using the upper bound for w given

1 pnCn pnCn  C2 kk˛p 2 n e˛ 3 pkkp :   C1 C1 ˛ . C1/ ˛ p w. p / C1 p 2 e 3

 Definition 9 We say that W (or Aw ) is of exponential type if inequality (2.11) is only possible for ˛ 3 > 0 with ˛ 1 , ˛ 2 , C0 , C1 positive constants and ˛ 1 > n, ˛ 2 > n. If (2.11) holds for ˛ 3 D 0 with ˛ 1 , ˛ 2 , C0 , C1 positive constants and ˛ 1 > n, ˛ 2 > n, we say that W (or Aw ) is of polynomial type. We note that if W is of polynomial type then ˛ 1 D ˛ 2 > n and C0 , C1 are positive constants with C1  C0 . Lemma 10 With the hypotheses of Lemma 8, etAw .kkp / 2 L .Qnp / for 1  < 1 and t > 0: Proof Since etAw .kkp / is a continuous function, it is sufficient to show that there exists M 2 N such that Z IM .t/ WD e tAw .kkp / dn  < 1; for t > 0. kkp >pM

Take M 2 N, by Lemma 8, we have 1

MC1

C2 kkp˛2 n e˛ 3 pkkp > C2 kkp˛2 n e˛ 3 p MC1

and (with B D C2 e˛ 3 p Z IM .t/ 

for kkp > pM ;

), ˛ 2 n

etBkkp

n

dn   C.M; ; /t ˛2 n ; for t > 0.

kkp >pM



2.2.3 p-Adic Description of Characteristic Relaxation in Complex Systems In [11] Avetisov et al. developed a new approach to the description of relaxation processes in complex systems (such as glasses, macromolecules and proteins) on the basis of p-adic analysis. The dynamics of a complex system is described by a

2.2 Operators W, Parabolic-Type Equations and Markov Processes

19

random walk in the space of configurational states, which is approximated by an ultrametric space (Qp ). Mathematically speaking, the time-evolution of the system is controlled by a master equation of the form @f .x; t/ D @t

Z fv .x j y/ f .y; t/  v .y j x/ f .x; t/g dy, x 2 Qp , t 2 RC ;

(2.12)

Qp

where the function f .x; t/ W Qp RC ! RC is a probability density distribution, and the function v .x j y/ W Qp  Qp ! RC is the probability of transition from state y to the state x per unit time. The transition from a state y to a state x can be perceived as overcoming the energy barrier separating these states. In [11] an Arrhenius type relation was used:   U .x j y/ ; v .x j y/ A.T/ exp  kT where U .x j y/ is the height of the activation barrier for the transition from the state y to state x, k is the Boltzmann constant and T is the temperature. This formula establishes a relation between the structure of the energy landscape U .x j y/ and the transition function v .x j y/. The case v .x j y/ D v .y j x/ corresponds to a degenerate energy landscape. In this case the master equation (2.12) takes the form @f .x; t/ D @t   where v jx  yjp D

Z

  v jx  yjp f f .y; t/  f .x; t/g dy,

Qp

 A.T/ jxyjp

exp

U .jxyj /  kT p

 . By choosing U conveniently, several

energy landscapes can be obtained. Following [11], there are three basiclandscapes:   (i) (logarithmic) v jx  yjp D jxyj ln˛ 11Cjxyj , ˛ > 1; (ii) (linear) v jx  yjp D . / p p   1 e˛jxyjp D , ˛ > 0; (iii) (exponential) v  yj jx ˛C1 p jxyj , ˛ > 0. jxyjp

p

Thus, it is natural to study the following Cauchy problem: 8 @u.x;t/ R D  u.xy;t/u.x;t/ dn y, x 2 Qnp ; t 2 RC ; ˆ ˆ @t w.y/ < n Q p

ˆ ˆ :

  u .x; 0/ D ' 2 D Qnp ;

where w .y/ is a radial function belonging to a class of functions that contains functions like: , here pn ./ is the n-dimensional p-adic Gamma (i) w.kykp / D pn .˛/ kyk˛Cn p function, and ˛ > 0; (ii) w.kykp / D kykˇp e˛kykp , ˛ > 0.

20

2 Parabolic-Type Equations and Markov Processes

By imposing condition (2.11) to w, we include the linear and exponential energy landscapes in our study. On the other hand, take w.kykp / satisfying (2.11) and take   f kykp a continuous and increasing function such that     0 < sup f kykp < 1 and 0 < infn f kykp < 1: y2Qp

y2Qnp

  Then f kykp w.kykp / satisfies (2.11). This fact shows that the class of operators W is very large. Finally we note that kykˇp ln˛ .1Ckykp /, ˇ > n, ˛ 2 N, does not satisfies kyk˛p 1  kykˇp ln˛ .1 C kykp / for any y 2 Qnp , and hence our results do not include the case of logarithmic landscapes.

2.2.4 Heat Kernels In this section we assume that function w satisfies conditions (2.11 ). We define Z Z.x; tI w; / WD Z.x; t/ D etAw .kkp / p .x  /dn  for t > 0 and x 2 Qnp . Qnp 1 ŒetAw .kkp /  2 C \ L2 for t > 0. We call a Note that by Lemma 10, Z.x; t/ D F!x such function a heat kernel. When considering Z.x; t/ as a function of x for t fixed we will write Zt .x/.

Lemma 11 (i) There exists a positive constant C, such that 1 , for x 2 Qnp X f0g and t > 0. Z.x; t/ < Ct kxk˛ p

  (ii) Zt .x/ 2 L1 Qnp for every t > 0. Proof (i) Let kxkp D pˇ . Since Z.x; t/ 2 L1 .Qnp / for t > 0, by using Qnp X f0g D F j n j2Z p S0 and formula (2.8), we get 2 4.1  pn / Z.x; t/ D kxkn p

1 X

3 eAw

. pˇj /t

pnj  eAw

jD0 ˇj /t

By using that eAw . p

 1 for j 2 N , we have

h i Aw . pˇC1 /t 1  e : Z.x; t/  kxkn p

. pˇC1 /t

5:

2.2 Operators W, Parabolic-Type Equations and Markov Processes

21

We now apply the mean value theorem to the real function f .u/ D ˇC1 /u eAw . p on Œ0; t with t > 0, and Lemma 8, ˇC1 1 /  Ct kxk˛ : Z.x; t/  C0 kxkn p tAw . p p

(ii) Notice that Z

Z Zt .x/dn x D

Qnp

Z Zt .x/dn x C

Bn0

Zt .x/dn x;

Qnp nBn0

the existence of the first integral follows from the continuity of Zt .x/, for the second integral we use the bound obtained in (i).  Lemma 12 Z.x; t/  0, for x 2 Qnp and t > 0. Proof Since etAw .kkp / is radial, by using Qnp X f0g D we have Z.x; t/ D

1 X iD1

D

1 X

etAw . p / i

F j2Z

pj S0n and formula (2.8),

Z p .x  /dn  kkp Dpi

h i   i iC1 pni etAw . p /  etAw . p / .pi xp /  0

iD1

since Aw is increasing function of i, cf. Lemma 5.



Theorem 13 The function Z.x; t/ has the following properties: (i) Z.x; R t/  0 for any t > 0; (ii) Qn Z.x; t/dn x D 1 for any t > 0; p

(iii) Zt .x/ 2 C.Qnp ; R/ \ L1 .Qnp / \ L2 .Qnp / for any t > 0; (iv) Zt .x/  Zt0 .x/ D ZtCt0 .x/ for any t, t0 > 0; (v) lim Z.x; t/ D ı.x/ in D0 .Qnp /, where ı denotes the Dirac distribution. t!0C

Proof (i) It follows from Lemma 12. (ii) Since Zt .x/, Fx! .Zt .x// D etAw .kkp / 2 C \ L1 , for any t > 0, cf. Lemma 10 and Lemma 11 (ii), the result follows from the inversion formula for the Fourier transform. (iii) It follows from Lemma 10 and Lemma 11 (ii). (iv) By the previous property Zt .x/ 2 L1 for any t > 0, then

0 1 etAw .kkp / et Aw .kkp / Zt .x/  Zt0 .x/ D F!x

0 1 e.tCt /Aw .kkp / D ZtCt0 .x/: D F!x

22

2 Parabolic-Type Equations and Markov Processes

(v) Since we have etAw .kkp / 2 C.Qnp ; R/ \ L1 for t > 0, cf. Lemma 10, the inner product D e

tAw .kkp /

E

Z

; D

etAw .kkp / ./dn 

Qnp

defines a distribution on Qnp , then, by the dominated convergence theorem, D E lim etAw .kkp / ; D h1; i

t!0C

and thus D E ˝ ˛ lim hZ .x; t/ ; i D lim etAw .kkp / ; F 1 D 1; F 1 D .ı; / :

t!0C

t!0C



2.2.5 Markov Processes Over Qnp   Along this section we consider Qnp ; kkp as complete non-Archimedean metric space and use the terminology  and results  of [39, Chapters 2, 3]. Let B denote the Borel  -algebra of Qnp . Thus Qnp ; B; dn x is a measure space. We set p.t; x; y/ WD Z.x  y; t/ for t > 0, x; y 2 Qnp ; and (R P.t; x; B/ D

B

p.t; y; x/dn y

1B .x/

for t > 0;

x 2 Qnp ;

B2B

for t D 0:

Lemma 14 With the above notation the following assertions hold: (i) p.t; x; y/ is a normal transition density; (ii) P.t; x; B/ is a normal transition function. Proof The result follows from Theorem 13, see [39, Section 2.1] for further details. 

2.2 Operators W, Parabolic-Type Equations and Markov Processes

23

Lemma 15 The transition function P.t; x; B/ satisfies the following two conditions: (i) for each u  0 and compact B lim supP.t; x; B/ D 0 [Condition L(B)];

x!1 tu

(ii) for each  > 0 and compact B lim supP.t; x; Qnp n Bn .x// D 0 [Condition M(B)].

t!0C x2B

Proof (i) By Lemma 11 and the fact that kkp is an ultranorm, we have Z P.t; x; B/  Ct

1 n 1 d y D tC kxk˛ vol .B/ for x 2 Qnp n B: kx  yk˛ p p

B

Therefore lim supP.t; x; B/ D 0. x!1 tu

(ii) Again, by Lemma 11, the fact that kkp is an ultranorm, and ˛ 1 > n, we have Z P.t; x; Qnp n Bn .x//  Ct

1 n d y D Ct kx  yk˛ p

kxykp >

Z

1 n d z kzk˛ p

kzkp >

D C0 .˛ 1 ; ; n/ t: Therefore lim supP.t; x; Qnp n Bn .x//  lim supC0 .˛ 1 ; ; n/ t D 0:

t!0C x2B

t!0C x2B

 Theorem 16 Z.x; t/ is the transition density of a time and space homogeneous Markov process which is bounded, right-continuous and has no discontinuities other than jumps. Proof The result follows from [39, Theorem 3.6] by using that .Qnp ; kxkp / is semicompact space, i.e. a locally compact Hausdorff space with a countable base, and P.t; x; B/ is a normal transition function satisfying conditions L.B/ and M.B/, cf. Lemmas 14, 15. 

24

2 Parabolic-Type Equations and Markov Processes

2.2.6 The Cauchy Problem Consider the following Cauchy problem: 8 @u n < @t .x; t/  Wu.x; t/ D 0; x 2 Qp ; t 2 Œ0; 1/ ; :

u .x; 0/ D u0 .x/;

(2.13)

u0 .x/ 2 Dom.W/;

    1 Aw kkp Fx! for 2 Dom.W/, see (2.10), and where .W / .x/ D F!x u W Qnp  Œ0; 1/ ! C is an unknown function. We say that a function u.x; t/ is   a solution of (2.13), if u.x; t/ 2 C .Œ0; 1/ ; Dom.W// \ C1 Œ0; 1/ ; L2 .Qnp / and u satisfies (2.13) for all t  0. In this section, we understand the notions of continuity in t, differentiability in t and equalities in the L2 .Qnp / sense, as it is customary in the semigroup theory. We know from Proposition 7 that the operator W generates a C0 semigroup .T .t//t0 , then Cauchy problem (2.13) is well-posed, i.e. it is uniquely solvable with the solution continuously dependent on the initial datum, and its solution is given by u.x; t/ D T .t/u0 .x/, for t  0, see e.g. [24, Theorem 3.1.1]. However the general theory does not give an explicit formula for the semigroup .T .t//t0 . We show that the operator T .t/ for t > 0 coincides with the operator of convolution with the heat kernel Zt  . In order to prove this, we first construct a solution of Cauchy problem (2.13) with the initial value from D without using the semigroup theory. Then we extend the result to all initial values from Dom.W/, see Propositions 18– 20. 2.2.6.1 Homogeneous Equations with Initial Values in D To simplify the notation, set Z0  u0 D .Zt .x/  u0 .x// jtD0 WD u0 . We define the function u .x; t/ D Zt .x/  u0 .x/; for t  0:

(2.14)

Since Zt .x/ 2 L1 for t > 0 and u0 2 D.Qnp /  L1 .Qnp /, the convolution exists and is a continuous function, see e.g. [100, Theorem 1.1.6]. Lemma 17 Take u0 2 D with the support of ub0 contained in BnR , and u .x; t/, t  0 defined as in (2.14). Then the following assertions hold: (i) u .x; t/ is continuously differentiable in time for t  0 and the derivative is given by

@u.x; t/ 1 etAw .kkp / Aw .kkp /1BnR ./  u0 .x/I D F!x @t

2.2 Operators W, Parabolic-Type Equations and Markov Processes

25

(ii) u.x; t/ 2 Dom.W/ for any t  0 and

1 etAw .kkp / Aw .kkp /1BnR ./  u0 .x/: .Wu/.x; t/ D F!x Proof (i) The proof is similar to the one given for Lemma 110 in Chap. 4. (ii) Note that etAw .kkp / ub0 ./, Aw .kkp /etAw .kkp / ub0 ./ 2 C \ L2 \ L1 for t  0, i.e. u.x; t/ 2 Dom.W/ for t  0. Now   1 Aw .kkp /F!x .u.x; t// .Wu/.x; t/ D F!x

1 Aw .kkp /etAw .kkp / ub0 ./ D F!x

1 Aw .kkp /etAw .kkp / 1BnR ./b u0 ./ D F!x

1 etAw .kkp / Aw .kkp /1BnR ./  u0 .x/: D F!x  As a direct consequence of Lemma 17 we obtain the following result. Proposition 18 Assume that u0 2 D. Then function u .x; t/ defined in (2.14) is a solution of Cauchy problem (2.13). 2.2.6.2 Homogeneous Equations with Initial Values in L2 We define T.t/u D

8 < Zt  u; t > 0 :

(2.15) u;

t D 0;

for u 2 L2 . Lemma 19 The operator T.t/ W L2 .Qnp / ! L2 .Qnp / is bounded for any fixed t  0. Proof For t > 0, the result follows from the Young inequality by using the fact that Zt 2 L1 , cf. Theorem 13 (iii).  Proposition 20 The following assertions hold. (i) The operator W generates a C0 semigroup .T .t//t0 . The operator T .t/ coincides for each t  0 with the operator T.t/ given by (2.15). (ii) Cauchy problem (2.13) is well-posed and its solution is given by u.x; t/ D Zt  u0 , t  0.

26

2 Parabolic-Type Equations and Markov Processes

Proof (i) By Proposition 7 (iii) the operator W generates a C0 semigroup .T .t//t0 . Hence Cauchy problem (2.13) is well-posed, see e.g. [24, Theorem 3.1.1]. By Proposition 18, T .t/jD D T.t/jD and both operators T .t/ and T.t/ are defined on the whole L2 and bounded, cf. Lemma 19. By the continuity we conclude that T .t/j D T.t/ on L2 . Now the statements follow from well-known results of the semigroup theory, see e.g. [24, Theorem 3.1.1], [41, Chap. 2, Proposition 6.2]. 

2.2.6.3 Non-homogeneous Equations Consider the following Cauchy problem: 8 @u n < @t .x; t/  Wu.x; t/ D g.x; t/; x 2 Qp ; t 2 Œ0; T ; T > 0; :

u .x; 0/ D u0 .x/;

(2.16)

u0 .x/ 2 Dom.W/:

We say that a function u.x; t/ is a solution of (2.16), if u.x; t/ belongs to  C .Œ0; T/; Dom.W// \ C1 Œ0; T; L2 .Qnp / and if u.x; t/ satisfies equation (2.16) for t 2 Œ0; T.   Theorem 21 Assume that u0 2 Dom.W/ and g 2 C Œ0; 1/; L2 .Qnp / \ L1 ..0; 1/; Dom.W//. Then Cauchy problem (2.16 ) has a unique solution given by Z

Z tZ

u.x; t/ D

Z.x  ; t/u0 ./d  C n

Qnp

0

Qnp

Z.x  ; t  /g.; /dn d :

Proof The result follows from Proposition 20 by using some well-known results of the semigroup theory, see e.g. [24, Proposition 4.1.6]. 

2.2.7 The Taibleson Operator and the p-Adic Heat Equation We set p.n/ .˛/ WD

1  p˛n , for ˛ 2 Rn f0g : 1  p˛

This function is called the p-adic Gamma function. The function k˛ .x/ D

jjxjj˛n p .n/

p .˛/

;

˛ 2 Rn f0; ng ;

x 2 Qnp ;

2.2 Operators W, Parabolic-Type Equations and Markov Processes

27

is called the multi-dimensional Riesz Kernel; it determines a distribution on D.Qnp / as follows. If ˛ ¤ 0, n, and ' 2 D.Qnp /, then Z 1  pn 1  p˛ n '.0/ C jjxjj˛n .k˛ .x/; '.x// D p '.x/ d x 1  p˛n 1  p˛n jjxjjp >1 Z 1  p˛ n jjxjj˛n C p .'.x/  '.0// d x: 1  p˛n jjxjjp 1

(2.17)

Then k˛ 2 D0 .Qnp /, for Rn f0; ng. In the case ˛ D 0, by passing to the limit in (2.17), we obtain .k0 .x/; '.x// WD lim .k˛ .x/; '.x// D '.0/; ˛!0

i.e., k0 .x/ D ı .x/, the Dirac delta function, and therefore k˛ 2 D0 .Qnp /, for Rn fng. It follows from (2.17) that for ˛ > 0, Z 1  p˛ .k˛ .x/; '.x// D jjxjjp˛n .'.x/  '.0// dn x: (2.18) 1  p˛n Qnp Definition 22 The Taibleson pseudodifferential operator D˛T , ˛ > 0, is defined as   1 jjjj˛p Fx! ' , for ' 2 D.Qnp /. .D˛T '/.x/ D F!x By using (2.18) and the fact that .F k˛ / .x/ equals jjxjj˛p , ˛ ¤ n, in D0 .Qnp /, we have  ˛  DT ' .x/ D .k˛  '/ .x/ Z 1  p˛ D jjyjj˛n .'.x  y/  '.x// dn y: p 1  p˛n Qnp

(2.19)

Then the Taibleson operator belongs to the class of operators W introduced before. The right-hand side of (2.19) makes sense for a wider class of functions, for example, for locally constant functions '.x/ satisfying Z jjxjjp 1

jjxjjp˛n j'.x/j dn x < 1:

A similar observation is valid in general for operators of W type. The equation @u.x; t/ C .D˛T u/.x; t/ D 0; @t

x 2 Qnp ;

t  0;

28

2 Parabolic-Type Equations and Markov Processes

where  is a positive constant, is a multi-dimensional analog of the p-adic heat equation introduced in [111].

2.3 Elliptic Pseudodifferential Operators, Parabolic-Type Equations and Markov Processes In this section we consider following Cauchy problem: 8 @u.x;t/ < @t C . f .@; ˇ/ u/ .x; t/ D 0, x 2 Qnp , n  1, t  0 :

(2.20)

u .x; 0/ D ' .x/ ;

where f .@; ˇ/ is an elliptic pseudodifferential operator of the form

1 . f .@; ˇ/ / .x; t/ D F!x j f ./jˇp Fx! .x; t/ : Here ˇ is a positive real number, and f ./ 2 Qp Œ 1 ; : : : ;  n  is a homogeneous polynomial of degree d satisfying the property f ./ D 0 ,  D 0. We establish the existence of a unique solution to Cuachy problem (2.20) in the case in which ' .x/ is a continuous and an integrable function. Under these hypotheses we show the existence of a solution u .x; t/ that is continuous in x, for a fixed t 2 Œ0; T, bounded, and integrable function. In addition the solution can be presented in the form u .x; t/ D Z .x; t/  ' .x/ where Z .x; t/ is the fundamental solution (also called the heat kernel) to Cauchy’s Problem 2.20: Z .x; t; f ; ˇ/ WD Z .x; t/ D

R Qnp

ˇ

p .x  / etj f ./jp dn ;  2 Qnp ; t > 0:

(2.21)

The fundamental solution is a transition density of a Markov process with space state Qnp .

2.3.1 Elliptic Operators Let h ./ 2 Qp Π1 ; : : : ;  n  be a non-constant polynomial.

In this sectionwe work with operators of the form h .@; ˇ/ D F 1 jhjˇp F ; ˇ > 0, 2 D Qnp . We

2.3 Elliptic Pseudodifferential Operators, Parabolic-Type Equations and. . .

29

will say that h .@; ˇ/ is a pseudodifferential operator with symbol jhjˇp D jh ./jˇp . The operator h .@; ˇ/ has a self-adjoint extension with dense domain in L2 . Definition 23 Let f ./ 2 Qp Œ 1 ; : : : ;  n  be a non-constant polynomial. We say that f ./ is an elliptic polynomial of degree d, if it satisfies: (i) f ./ is a homogeneous polynomial of degree d, and (ii) f ./ D 0 ,  D 0. Lemma 24 (i) There are infinitely many elliptic polynomials. (ii) For any n 2 N X f0g and p ¤ 2, there exists an elliptic polynomial h . 1 ; : : : ;  n / with coefficients in Z p and degree 2d.n/ WD 2d such that jh . 1 ; : : : ;  n /jp D k. 1 ; : : : ;  n /k2d p .

(2.22)

Proof (i) Assume that h . 1 ; : : : ;  n / is an elliptic polynomial of degree d. Take 2 2   2 Q p such that the equation x D  has no solutions in Qp , then h . 1 ; : : : ;  n /   2d nC1 is an elliptic polynomial of degree 2d. Since there are elliptic quadratic forms for 1  n  4, see e.g. [22, Chapter 1], one concludes the existence of infinitely many elliptic polynomials. (ii) By choosing  2 Z p , it follows from (i) that if h . 1 ; : : : ;  n / is an elliptic polynomials of degree d with coefficients in Z p , then  h . 1 ; : : : ;  n /2   2d is elliptic with coefficients in Z . We pick d such that p nC1 p 2 2d does not divide d. We prove by induction on n that h . 1 ; : : : ;  n /   nC1 satisfies If ˇ (2.22). Assume, ˇ ˇ ˇas induction ˇhypothesis, that h . 1 ; : :ˇ : ;  n/ satisfies (2.22). 2d ˇ ˇ 2d ˇ ˇ 2ˇ 2 2d ˇ   ˇh . 1 ; : : : ;  n / ˇ ¤ ˇ nC1 ˇ , then ˇh . 1 ; : : : ;  n /    nC1 ˇ D  1 ; : : : ;  nC1 p . pˇ p ˇ ˇ pˇ ˇ ˇ ˇ ˇ m If ˇh . 1 ; : : : ;  n /2 ˇ D ˇ 2d , taking  D p u , with unC1 2 Z ˇ nC1 nC1 nC1 p , we have p

p

ˇ ˇ ˇ ˇ ˇ ˇ 2 2md ˇ m m 2d ˇ . p D p  ; : : : ; p  /  u ˇ ˇh ˇh . 1 ; : : : ;  n /2   2d 1 n nC1 nC1 ˇ : p

p

2   We note that h p1  1 ; : : : ; p1  n  u2d nC1 2 Zp , otherwise h . pm  1 ; : : : ; pm  n /2   u2d nC1 0 mod p and by using that p does not divide 2d, i.e. p ¤ 2 and the Hensel lemma, there 2 2d is impossible. exists a nontrivialˇ solution of h . ˇ 1 ; : : : ;  n /   nC1 D 0,ˇ which ˇ2d ˇ 2d 2ˇ ˇ ˇ Finally, by using ˇh . 1 ; : : : ;  n / ˇ D k. 1 ; : : : ;  n /kp D  nC1 p D p2md , we p ˇ ˇ  2d ˇ 2 2d ˇ  have ˇh . 1 ; : : : ;  n /   nC1 ˇ D   1 ; : : : ;  nC1 p . p

Lemma 25 Let f ./ 2 Qp Π,  D . 1 ; : : : ;  n /, be an elliptic polynomial of degree d. Then there exist positive constants C0 D C0 . f /, C1 D C1 . f / such that C0 kkdp  j f ./jp  C1 kkdp , for every  2 Qnp :

(2.23)

30

2 Parabolic-Type Equations and Markov Processes

Proof Without loss of generality we may assume that  ¤ 0. Let Q 2 Q p be an ˇ ˇ ˇQˇ element such that ˇ ˇ D kk ¤ 0. We first note that p

p

ˇ ˇd ˇ 1 ˇ ˇ ˇ ˇ ˇ j f ./jp D ˇQ ˇ ˇ f Q  ˇ , p

(2.24)

p

˚  1 with Q  2 S0n D z 2 Znp I kzkp D 1 . Now j f jp is continuous on S0n , that is a compact subset of Znp , then infz2Sn0 j f .z/jp , and supz2Sn0 j f .z/jp are attained on S0n , and since j f jp > 0 on S0n , we have supz2Sn0 j f .z/jp  infz2Sn0 j f .z/jp > 0. Therefore from (2.24) we have

!

ˇ ˇ ˇ ˇd ˇ Q ˇd ˇ ˇ infn j f .z/jp ˇ ˇ  j f ./jp  sup j f .z/jp ˇQ ˇ : p

z2S0

p

z2Sn0

 Along this section f ./ will denote an elliptic polynomial of degree d. Now, since cf ./ is elliptic for any c 2 Q p when f ./ is elliptic, we will assume that all the elliptic polynomials have coefficients in Zp . Lemma 26 Let A  Qnp be an open compact subset such that 0 … A. There exist a finite number of points Q i 2 A, i D 1;    ; L0 , and a constant M WD M .A; f / 2 NX f0g such that AD

ˇ ˇ  M n FL0 Q ˇ ˇ and j f ./jp jQ C. pM Zp /n D ˇ f Q i ˇ ; i D 1;    ; L0 : iD1  i C p Zp i p

Proof By (2.23), for  2 A, 0

j f ./jp  C0 kkdp  C0 inf kkdp  pM .A;f / ; 2A

where M 0 WD M 0 .A; f / is a positive integer constant. Now for Q i 2 A and y 2 Znp ,





f Q i C pM y D f Q i C pM T Q i ; y ;

where T Q i ; y is a polynomial function in Q i , y, with sup Q i 2A;y2Znp

ˇ

ˇ ˇ Q ˇ ˇT  i ; y ˇ  pı : p

2.3 Elliptic Pseudodifferential Operators, Parabolic-Type Equations and. . .

31

We set M D M 0 C ı C 1. Then ˇ

ˇ ˇ ˇ

ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ Q ˇ f  i C pM y ˇ D ˇ f Q i C pM T Q i ; y ˇ D ˇ f Q i ˇ . p

p

p

Now, since A is open compact, there exist a finite number of points Q i 2 A, such that F n  A D i Q i C pM Zp . Remark 27 Lemma 26 is valid for arbitrary polynomials satisfying only f ./ D 0 ,  D 0. Indeed, by using that A is compact and that j f ./jp is continuous, 0 there exists a constant M 0 such that j f ./jp  pM for  2 A. Definition 28 If f ./ 2 Zp Œ is an elliptic polynomial of degree d, then we say that j f jˇp is an elliptic symbol, and that f .@; ˇ/ is an elliptic pseudodifferential operator of order d. By Lemma 24, the Taibleson operator is elliptic for p ¤ 2. However, there ˇˇ ˇ ˇ ˇ are elliptic symbols which are not radial functions. For instance, ˇ 21  p 22 ˇ D p h n oiˇ 2 2 max j 1 jp ; p1 j 2 jp . Then, there are two different generalizations of the Taibleson operator (or Vladimirov operator): the W operators which are pseudodifferential operators with radial symbols, and the elliptic operators which include pseudodifferential operators with non-radial symbols.

2.3.2 Decaying of the Fundamental Solution at Infinity n

Lemma 29 For every t > 0, jZ .x; t/j  Ct dˇ , where C is a positive constant. ˇ Furthermore, etj f ./jp 2 L1 as a function of , for every t > 0. 1

Proof Let an integer m be such that pm1  .Ct/ dˇ  pm . By applying (2.23), jZ .x; t/j 

D

R Qnp

R Qnp

dˇ

eCtkkp dn   kp.m1/  kp

dˇ

e

R

.m1/dˇ

ep

kkdˇ n p

Qnp

d  p C n

n

dˇ

n  dˇ

d 

R Qnp

! e

n

d z t dˇ :

kzkdˇ n p

The result follows from the fact that ekzkp is an integrable function.



32

2 Parabolic-Type Equations and Markov Processes

Define ZL .x; t; f ; ˇ/ WD ZL .x; t/ D

R

ˇ

. pL Zp /

n

p .x  / etj f ./jp dn ; L 2 N;

where ˇ > 0, t > 0, and f ./ 2 Zp Œ 1 ; : : : ;  n  is an elliptic polynomial of degree d. Lemma 30 If kxkp  pMC1 and tpMdˇ kxkpdˇ  1, where M is the constant defined in Lemma 26, then there exists a positive constant C such that jZ0 .x; t/j  Ct kxkpdˇn : Proof By applying Fubini’s Theorem, Z0 .x; t/ D

1 .1/l R P n tl p .x  / j f ./jˇl p d : lŠ n lD0 Zp

By using the fact that kxkp > pMC1 > 1, be rewritten as Z0 .x; t/ D

R

Znp p

(2.25)

.x  / d D 0, and thus (2.25) can

1 .1/l R P n tl  .x  / j f ./jˇl p d : lŠ Znp p lD1

(2.26)

We set I .j; l/ WD I .x; f ; ˇ; j; l/ D

R Znp

  n p pj x   j f ./jˇl p d ; for j  0, l  1;

and     R   n IQ j; l; S0n WD IQ x; f ; ˇ; j; l; S0n D p pj x   j f ./jˇl p d ; Sn0

n  for j  0, l  1. By decomposing Znp as the disjoint union of pZp and S0n , I .0; l/ D D

R Znp

p .x  / j f ./jˇl p d R

. pZp /

n

R ˇl n p .x  / j f ./jˇl p d C p .x  / j f ./jp d 

  D pnˇdl I .1; l/ C IQ 0; l; S0n :

Sn0

2.3 Elliptic Pseudodifferential Operators, Parabolic-Type Equations and. . .

33

By iterating this formula k-times, we obtain I .0; l/ D

k P jD0

  pj.nCˇdl/ IQ j; l; S0n C p.kC1/.nCˇdl/ I .k C 1; l/ :

Hence I .0; l/ admits the expansion I .0; l/ D

1 P jD0

  pj.nCˇdl/ IQ j; l; S0n :

(2.27)

On the other hand, since S0n is open compact and f is elliptic, by applying Lemma 26, we obtain IQ .j; l; A/ D

L0 P



ˇ ˇˇl R   ˇ ˇ pMn p pj x  Q i ˇ f Q i ˇ p pjCM x  y d n y; p Zn p

iD1

(2.28)

Now by using 8 < 0 if j < M  ord .x/  jCM  n p p x  y d y D : Znp 1 if j  M  ord .x/ ; R

with ord.x/ D mini ord .xi /, we can rewrite IQ .j; l; A/ as 8

ˇ ˇˇl L0 P ˇ ˇ ˆ ˆ p pj x  Q i ˇ f Q i ˇ if j  M  ord .x/ < pMn ˆ ˆ :

p

iD1

0

(2.29) otherwise.

We set ˛ WD ˛ .x/ D M  ord .x/  1 because kxkp D pord.x/  pMC1 . With this notation, by combining (2.27)–(2.29) and using that f ./ has coefficients in Zp and Q i 2 Znp , i D 1; : : : ; l, jI .0; l/j  pMn 

L ˇ ˇ 1 P0 ˇ Q ˇˇl P j.nCˇdl/ p ˇ f i ˇ iD1

L0 1  p.nCˇd/

p

jD˛

˛ˇld : kxkn p p

34

2 Parabolic-Type Equations and Markov Processes

By using this estimation for jI .0; l/j in (2.26), jZ0 .x; t/j 

L0 1  p.nCˇd/

Mdˇ dˇ

p tkxk e  1 ; kxkn p

finally, by using the hypothesis tpMdˇ kxkpdˇ  1, we have jZ0 .x; t/j  Ct kxkpdˇn :  Proposition 31 If p t > 0.

Mdˇ

t kxkdˇ p

 1, then jZ .x; t/j 

Ct kxkpdˇn ,

for x 2

Qnp

and

ˇ

Proof By Lemma 29, p .x  / etj f ./jp 2 L1 as a function of , for x 2 Qnp and t > 0 fixed. Then, by using the dominated convergence theorem, Z .x; t/ D lim ZL .x; t/ D lim L!1

R

L!1

. pL Zp /

ˇ

n

p .x  / etj f ./jp dn :

By a change of variables we have R     ˇ Lˇd ZL .x; t/ D pLn p pL x   ep tj f ./jp d D pLn Z0 pL x; pLˇd t : Znp

Now by applying the Lemma 30, jZL .x; t/j  Cp

Ln

tpLˇd

!

nCˇd kxpL kp

C

t kxknCˇd p

;

where C is a constant independent of L. Therefore jZ .x; t/j D lim jZL .x; t/j  Ct kxkpnˇd ; L!1

if pMdˇ t kxkpdˇ  1.



Theorem 32 For any x 2 Qnp and any t > 0,

dˇn 1 ; jZ .x; t/j  At kxkp C t ˇd where A is a positive constant.

2.3 Elliptic Pseudodifferential Operators, Parabolic-Type Equations and. . . 1

35

1

Proof If t ˇd  pM kxkp , then t ˇd  kxkp because M  1, and by applying Proposition 31, jZ .x; t/j 

Ct kxkdˇn p



1 1 1  Ct kxkp C t ˇd 2 2

dˇn

2dˇCn Ct 

dˇCn : 1 kxkp C t ˇd 1

Now if kxkp < t ˇd , by applying Lemma 29,

dˇn 1 n 2dˇCn Ct kxkp C t ˇd  Ct ˇd  jZ .x; t/j :  When considering Z.x; t/ as a function of x for t fixed we will write Zt .x/ as before. Corollary 33  With  the hypothesis of Theorem 32, the following assertions hold: (i) Zt .x/ 2 L Qnp , for 1   1, for t > 0; (ii) Zt .x/ is a continuous function in x, for t > 0 fixed. Proof (i) The first part follows directly from the estimation given in Theorem 32. (ii) ˇ The continuity follows from the fact that Zt .x/ is the Fourier transform of etj f ./jp , t > 0, that is an integrable function by Lemma 25. 

2.3.3 Positivity of the Fundamental Solution Theorem 34 Z.x; t/  0 for every x 2 Qnp and every t > 0. Proof We start by making the following observation about the fiber of f W Qnp ! Qp at  2 Qp . (Claim A) f 1 ./ is a compact subset of Qnp . Since f is continuous f 1 ./ is a closed subset of Qnp . By applying (2.23), ( f

1

./   2

Qnp I kkp



jj C0

1d )

;

and thus f 1 ./ is a bounded subset of Qnp . ˚  (Claim B) The critical set Cf D  2 Qnp I rf ./ D 0 of the mapping f is reduced to the origin of Qnp .

36

2 Parabolic-Type Equations and Markov Processes

This claim follows from the Euler identity 1 X @f ./  D f ./ ; d iD1 i @ i n

and the fact that f is an elliptic polynomial.

  On the other hand, since p .x  / 1f 1 ./ ./ 2 D Qnp , as a function of , for  ¤ 0, by applying integration on fibers to Z .x; t/, see Chap. 1, formula (1.3), with t > 0 fixed, 1 0 Z Z ˇ B C Z .x; t/ D etjjp @ p .x  / j GL jA d; f ./D

Qp nf0g

where j GL j is the measure induced by the Gel’fand-Leray form along the fiber f 1 ./. Hence in order to prove the theorem, it is sufficient to show that 0 B F .; x/ WD @

1

Z

C p .x  / j GL jA  0, for every x 2 Qnp n f0g .

f ./D

Let Q be a fixed point of f 1 ./,  2 Qp n f0g. By Claim B we may assume, after renaming the variables if necessary, that @f Q ¤ 0. We set y D ./ with @ n

( yj WD

 j

j D 1; : : : ; n  1 f Q C pe   f Q j D n:

By applying the non-Archimedean implicit function theorem (see Chap. 1, Theorem 1) there exist e, l 2 N such that y D ./ is a bianalytic mapping from Znp onto  l n p Zp . Then 0  D 1 .y/ D @y1 ; : : : ; yn1 ;

1 X

1 Gj .y/A ;

jD1

where Gj .y/ is a form of degree j, and G1 .y/ ¤ 0. By shrinking the neighborhoods around Q and the origin, i.e., by taking e and l big enough, we may assume that the following conditions hold: ˇ ˇ ˇ ˇ (C) the Jacobian J 1 of 1 satisfies ˇJ 1 .y/ˇp D ˇJ 1 .0/ˇp , for every y 2  l n p Zp ;

2.3 Elliptic Pseudodifferential Operators, Parabolic-Type Equations and. . .

0 (D) ord @xn

1 X

37

1

n  Gj .y/A  0, for any y 2 pl Zp .

jD1

Since f 1 ./,  2 Qp n f0g, is a compact subset by Claim A, F .; x/ can expressed as a finite sum of integrals of the form Z p .x  / j GL j : Q e Znp \f 1 ./ Cp

Now by changing variables  D 1 .y/, and using (C), (D), we obtain Z p .x  / j GL j Q e Znp \f 1 ./ Cp

ˇ D ˇJ



ˇ 1 .0/ˇ

Z p pl Zn1 p

ˇ D ˇJ



ˇ 1 .0/ˇ

Z p

0  p @

n1 X

xj  j  xn

jD1

0  p @

pl Zn1 p

n1 X

1 X

1 Gj .y/A dn1 y

lD1

1 xj  j A dn1 y

jD1

ˇ ˇ

D pl.n1/ ˇJ 1 .0/ˇp 1pl Zn1 .x/  0; p where 1pl Zn1 .x/ denotes the characteristic function of pl Zpn1 . Therefore p F .; x/  0. 

2.3.4 Some Additional Results   We denote by Cb WD Cb Qnp ; R the R-vector space of all functions ' W Qnp ! R which are continuous and satisfy k'kL1 D supx2Qnp j' .x/j < 1. Proposition 35 The fundamental solution has the following properties: R (i) Qn Z .x; t/ dn x D 1, for any t > 0; p R (ii) if '2Cb , then lim.x;t/!.x0 ;0/ Qn Z .x  y; t/ ' .y/ dn y D ' .x0 /; p R (iii) Z .x; t C t0 / D Qn Z .x  y; t/ Z .y; t0 / dn y, for t, t0 > 0. p

38

2 Parabolic-Type Equations and Markov Processes

Proof (i) It follows from Corollary 33 and the Fourier inversion formula. R (ii) We set u .x; t/ D Qn Z .x  y; t/ ' .y/ dn y. We have to show that p

lim

.x;t/!.x0 ;0/

u .x; t/ D ' .x0 / ,

for any fixed x0 2 Qnp . Since ' is continuous at x0 there exists a ball  ˚ Bne .x0 / D y 2 Qnp I ky  x0 kp  pe , such that j' .y/  ' .x0 /j < 2 , for every y 2 Bne .x0 /. Then ju .x; t/  ' .x0 /j  jI1 j C jI2 j, where ˇ ˇ ˇ ˇ R ˇ n ˇ Z .x  y; t/ Œ' .y/  ' .x0 / d yˇ ; jI1 j W D ˇ ˇ ˇkyx0 k pe p ˇ ˇ ˇ ˇ R ˇ ˇ Z .x  y; t/ Œ' .y/  ' .x0 / dn yˇ : jI2 j W D ˇ ˇ ˇkyx0 k >pe p

By using the continuity of ' and (i), jI1 j
pe

dn z  C1 t k'kL1 , kzkdˇn p

for t > 0, where C1 is a positive constant. Note that k'kL1 D 0, implies ' 0, since ' is a continuous function. In this case the theorem is valid. For this reason we assume that k'kL1 > 0 . Hence jI2 j
0: We have obtained a contradiction, thus u.x; t/  0. Finally taking u.x; t/ instead of u.x; t/, we conclude that u.x; t/ 0. 

3.6 Markov Processes In this section we show that the fundamental solution ƒ.x; t; ;  / of Cauchy problem (3.15) is the transition density of a Markov process. We need some preliminary results. Lemma 63 If the coefficients ak .x; t/ and b.x; t/ are nonnegative, then ƒ.x; t; ; /  0. R Proof It is sufficient to show that u.x; t/ D Qn ƒ.x; t; ; /'./dn   0, where p u.x; t/ is the solution of Cauchy problem (3.15) with f .x; t/ 0, and initial condition u.x; 0/ D '.x/  0 with ' 2 D.Qnp /. From (3.28), (3.29), and Lemma 44 (iii), it follows that u.x; t/ ! 0 as kxkp ! 1:

(3.41)

74

3 Non-Archimedean Parabolic-Type Equations with Variable Coefficients

Now, if u.x; t/ < 0, then there exist x0 2 Qnp and t0 2 .0; T such that inf

0tT; x2Qnp

u .x; t/ D u.x0 ; t0 / < 0:

This implies that .W˛ u/.x0 ; t0 /  0, .W˛ k u/.x0 ; t0 /  0 for all k, and On the other hand,

(3.42) @u .x ; t / @t 0 0

 0.

N X @u .x; t/  a0 .x; t/.W˛ u/.x; t/  ak .x; t/.W˛ k u/.x; t/ D 0: @t kD1

By using the uniform parabolicity condition a0 .x; t/  > 0, we get .W˛ u/.x0 ; t0 / D 0, then by (2.3), u.x; t0 / is constant, and by (3.41), u.x; t0 / 0, which contradicts (3.42).  R Lemma 64 If b.x; t/ 0, then Qn ƒ.x; t; ;  /dn  D 1. p

Proof By integrating (3.30) in the variable  over whole the space Qnp ; and by using R

R Lemma 50 (iii), we have @t@ Qn ƒ.x; t; ; /dn  D 0, thus Qn ƒ.x; t; ; /dn  is p p independent of t. Now, by integrating (3.28) over whole space Qnp in variable  and by using Lemma 44 (iv), we have Z

Zt Z Z ƒ.x; t; ; /d  D 1 C

Z.x  ; t  ; ; / .; ; ; /dn d n d :

n

 Qnp Qnp

Qnp

The result is obtained by taking t D  in the above formula.



Lemma 65 If b.x; t/ 0 and f .x; t/ 0, then the function ƒ.x; t; ; / satisfies the following property: Z ƒ.x; t; ;  / D

ƒ.x; t; y;  /ƒ.y;  ; ;  /dn y:

(3.43)

Qnp

Proof Consider the following initial value problem: (

Q  a0 .x; t/.W˛ u/.x; t/  .Wu/.x; t/ D 0 n u .x; / D '.x/; x 2 Qp and t 2 .;  ; @u .x; t/ @t

(3.44)

3.7 The Cauchy Problem Is Well-Posed

by Theorem 59, u.x; / D

R Qnp

75

ƒ.x;  ; ;  /'./dn . Now consider

8 @u Q .x; t/  a0 .x; t/.W˛ u/.x; t/  .Wu/.x; t/ D 0 ˆ ˆ < @t Z u .x;  / D ˆ ˆ :

ƒ.x;  ; ;  /'./dn ; x 2 Qnp , t 2 . ; T; with  <  < T; Qnp

(3.45) by Theorem 59 and Fubini’s theorem, the solution of (3.45) is given by Z u.x; t/ D Qnp

0

1

Z

B n C n @ ƒ.x; t; y;  /ƒ.y;  ; ; /d yA './d : Qnp

On the other hand, (3.45) is equivalent to (

Q  a0 .x; t/.W˛ u/.x; t/  .Wu/.x; t/ D 0 n u .x; / D '.x/; x 2 Qp ; t 2 .; T ; @u @t .x; t/

(3.46)

R which has solution given by u .x; t/ D Qn ƒ.x; t; ;  /'./dn . Now, by Theop rem 62, 0 1 Z Z Z B C ƒ.x; t; ; /'./dn  D @ ƒ.x; t; y;  /ƒ.y;  ; ; /dn yA './dn ; Qnp

Qnp

Qnp

for any test function ', which implies (3.43).



Theorem 66 If the coefficients ak .x; t/, k D 1;    ; N, are nonnegative bounded continuous functions, b.x; t/ 0, ˛ > n C 1, 0   < ˛ 1  n (if a1 .x; t/ D    ak .x; t/ 0; we shall suppose that 0   < ˛  n), and f .x; t/ 0, then the fundamental solution ƒ.x; t; ; / is the transition density of a bounded rightcontinuous Markov process without second kind discontinuities. Proof The result follows from [39, Theorem 3.6] by using Lemmas 63, 64, 65, and (3.28)–(3.29), and Lemma 44 (iii). 

3.7 The Cauchy Problem Is Well-Posed In this section, we study the continuity of the solution of Cauchy problem (3.15) with respect to ' .x/ and f .x; t/. We assume that the coefficients ak .x; t/; k D 0; 1; : : : ; N are nonnegative bounded continuous functions, b.x; t/ is a bounded

76

3 Non-Archimedean Parabolic-Type Equations with Variable Coefficients

continuous function, 0   < ˛ 1  n (if a1 .x; t/ D : : : D ak .x; t/ 0, we shall suppose that 0   < ˛  n), ' .x/ 2 M and f .x; t/ 2 M , uniformly in t, with 0   < ˛ 1  n. We identify M with the R-vector space of all the functions “ .x; t/ 2 M , uniformly in t,” and introduce on M the following norm: k kM

ˇ ˇ ˇ .x; t/ ˇ ˇ ˇ WD sup sup ˇ ˇ:  t2Œ0;T x2Qnp ˇ 1 C kxkp ˇ

From now on, we consider M as topological vector space with the topology induced by kkM . We also consider M  M as topological vector space with the topology induced by the norm kkM C k?kM . Theorem 67 With the above hypotheses, consider the following operator: M  M

L M !  .' .x/ ; f .x; t// ! u .x; t/ ; where u .x; t/ is given by (3.27). Then   ku .x; t/kM  C k' .x/kM C kf .x; t/kM ; i.e. L is a continuous operator. Proof We write u.x; t/ D u1 .x; t/ C u2 .x; t/ where Zt Z u1 .x; t/ D

ƒ.x; t; ; /f .; /dn d and 0 Qnp

Z u2 .x; t/ D

ƒ.x; t; ; 0/'./dn ; Qnp

as before. Now Zt Z jƒ.x; t;  ; /j jf .; /j dn d

ju1 .x; t/j  0 Qnp

 kf .x; t/kM

8 ˆ

= > ;

;

3.7 The Cauchy Problem Is Well-Posed

77

by (3.28)–(3.29), (3.17) and Proposition 45, ju1 .x; t/j  C0 kf .x; t/kM

C

t NC1 XZ

8 t 0 and 0  xp;i  p  1

iDk

otherwise,

as before. We recall that a function fp W Qp ! C which is locally constant with compact support is called a Bruhat-Schwartz function. The space of such functions  is denoted as D Qp , as before.   For fp 2 D Qp , its Fourier transform b fp is defined by   b fp  p D

Z Qp

    p xp  p fp xp dxp :

    The Fourier transform induces a linear isomorphism of D Qp onto D Qp   b  satisfying b fp  p D fp  p .

4.2 Preliminaries

83

In the case p D 1 the additive character is defined by 1 .x1 / WD exp .2ix1 /. Let D.R/ denote the Schwartz space. The Fourier transform of f1 2 D.R/, denoted fc 1 , is defined by fc 1 . 1 / D

Z R

1 .x1  1 / f1 .x1 / dx1 :

The Fourier transform induces a linear isomorphism of D .R/ onto D .R/. By choosing a suitable multiple of the Lebesgue measure, the Fourier transform satisfies c fc 1 . 1 / D f1 . 1 /. The additive adelic character  W A ! C is defined by  .x/ D

Y

   p xp

for x D .x1 ; x2 ; x3 ; : : :/ :

p

Remark 69 By abuse of notation we will denote also by , the additive character on Af , defined by  .x/ D

Y

   p xp

for x D .x2 ; x3 ; : : :/ :

p 0. If p ¤ q, then logq p j > 0 and logq p j ¤ 0, where fg denotes the       fractional part function. Hence logq p j D logq p j , and since logq pj D   ˚       logq p j C logq p j , we have logq pj D  logq p j . Now ˆ. pj /ˆ. p j / D

Y

qŒŒlogq p  j

q

Y

j qŒŒlogq p 

q

j D pŒŒlogp  pŒŒlogp p 

pj

Dp

Y

Y

qŒŒlogq p  j

q¤p

qŒŒlogq CŒŒlogq pj

pj

Y

j qŒŒlogq p 

q¤p

 D p

q¤p

Y

j j qŒlogq p Œlogq p  D p:

q¤p

(ii) The formulas follow from definitions (4.12)–(4.13). We notice p that the formulas are not valid if n ¤ p j .p For instance, if we take n D 5, then n D 3, nC D 5, and .n /C D 5 ¤ 5. (iii) In the case j  0, the announced formula follows from: Claim A 

 j

logq p D

8 < j1 :

if q D p

 logq p j if q ¤ p:

Indeed, ˆ. p j / D

Y

j qŒŒlogq p  D

Y

q

D ppj1

qŒlogq p  D p j j

q

Y q¤p

qŒ

j logq p

Dp

Y

Y

j qŒlogq p 

q¤p

qŒ

j logq p

 D pˆ. p j /: 

q

j1 Claim A is verified  pj < p j implies that  as follows. We first notice that p j j  1  logq p < j, now by using the left continuity of the function Œ we get the first case of the formula. We now set pj D qˇ with  q a prime number different from p. From qˇ < p j < qˇC1 , one gets ˇ D logq p j , which establish the second case in the formula for q ¤ p. We now consider a prime k satisfying k ¤ q ¤ p.

90

4 Parabolic-Type Equations and Markov Processes on Adeles

    Take ˛ D logk pj and  D logk p j . If ˛ <  , then k˛ < qˇ < k < p j , which contradicts the fact that pj D qˇ . Since ˛   , we conclude that ˛ D  . Finally, we show the formula in the case pj , with j > 0. We first notice that if ; p˛k k ; p j are all the prime powers  p j , i.e. if ˆ. p j / D p˛1 1 : : : p˛k k p j , then

p˛1 1 ;   

j

pC D

8 1C˛t for some t 2 f1;    ; kg < pt :

for some prime q ¤ pl , l 2 f1;    ; kg ;

q

and j

ˆ. pC / D

By using that ˆ. pj / D

ˆ. pj /

8 ˛1 j 1C˛ ˛ 1C˛ < p1 : : : pt t : : : pk k p j if pC D pt t :

p˛1 1 : : : p˛k k p j q

p ˆ. p j /

j

if pC D q:

1 j and pj D pC we have

1 j D D ˆ pC

8 j t pt if p D p1C˛ ˆ t ˆ j C ˆ ˆ p < C ˆ q ˆ ˆ : ˆ p j

j

if pC D q;



C

which implies that ˆ. pj / D   ˆ.pj / ˆ pj , i.e. ˆ. pj / D p .

1 ˛ ˛ p1 1 :::pk k p j

D

1 , ˆ. p j /

and thus pˆ. pj / D

p ˆ. p j /

D 

Lemma 76 (i) The adelic ball Br WD Br .0/ is a compact subset and its volume is given by vol .Br / D ˆ.r/: (ii) The adelic sphere Sr WD Sr .0/ is a compact subset and its volume is given by vol .Sr / D ˆ.r/  ˆ.r /: Proof The compactness of Br .0/ was established in Corollary 72. Since Sr .0/ is a closed subset of Af and Sr .0/  Br .0/ we conclude that Sr .0/ is compact. The formulas for volumes follows immediately from (4.9), (4.11) and (4.7). 

4.3 Metric Structures, Distributions and Pseudodifferential Operators on Af

91

4.3.2 The Fourier Transform of Radial Functions Definition 77 A function f W Af ! C is said to be radial if its restriction to any sphere Sr , r > 0, is a constant function, i.e. f jSr D fr 2 C, r > 0. If f is a radial function, then there exists a function h W R ! C such that f ./ D h.kxk/. By abuse of notation, we will denote a radial function f in the form f D f .kxk/. Lemma 78 Let f W Af ! C be an integrable function. Then the following assertions hold: (i) Z f .x/ dxAf D

Af

Z

X

f .x/ dxAf :

pm ; m2Znf0g

Spm

In the particular case in which f is a radial function this formula takes the form Z Af

(ii) Take A.i/ D

F m2J

f .x/ dxAf D

X

  f .pm / vol Spm :

pm ; m2Znf0g

Spm  Af , where J is a (countable) subset of Z n f0g, then

Z Af

X Z

f .x/ 1A.i/ .x/ dxAf D

pm ; m2J

f .x/ dxAf :

Spm

In the particular case in which f is a radial function this formula takes the form Z Af

(iii) Assume that Af D

f .x/ 1A.i/ .x/ dxAf D

X

  f .pm / vol Spm :

pm ; m2J

F i2N

A.i/ with each A.i/ is a disjoint union of spheres, then

Z Af

f .x/ dxAf D

XZ i2N

A.i/

f .x/ dxAf :

Proof The proof follows by general techniques in measure theory, the compactness of the adelic balls and spheres, see Lemma 76, and the characterization of the adelic integrals for positive functions given in [50, p. 21].  To simplify notations, throughout this subsection the expressions k0k1 and j0j1 p in the inequalities mean 1. The following theorem describes the Fourier transform of a radial function.

92

4 Parabolic-Type Equations and Markov Processes on Adeles

Theorem 79 Let f D f .kxk/ W Af ! C be a radial function in L1 .Af /. Then the following formula holds: fO ./ D

X

  j  ˆ q j f .q j /  f .qC /

for any  2 Af ;

(4.16)

q j 0

Note that on the sets A.0;q/Pwe have kxk P D kxk0 and on the sets A.1;q/ we have .0;q/ kxk D kxk1 . Then fO ./ D q fO ./ C q fO .1;q/ ./, where fO .k;q/ ./ WD

Z A.k;q/

 .x/ f .kxk/ dxAf ;

k D 0; 1; q is a prime:

We set ˇ q WD ˇ q ./ D Œlogq kk

(4.17)

with convention that ˇ q .0/ D C1. We also set ı .t/ D 1 if t D 0 and ı .t/ D 0 otherwise. To simplify the proof, we first present the final formulas for the functions fO .k;q/ ./, the proofs are given later. Claim 81 X q

fO .1;q/ ./ D 0

if kk > 1;

(A)

4.3 Metric Structures, Distributions and Pseudodifferential Operators on Af

X

fO .1;q/ ./ D

q

.x/1

ˇqX  X    1 1 f .q j /ˆ q j q 1 jD1

q 1:  q q

(C)

(D)

Combining (A), (B), (C), (D) we obtain fO ./ D

X 1 1 q q

X

  f .q /ˆ q j



j

jˇ q ./1; j¤0

X1       f kk1 ˆ kk1 ı q j  kk :  q q; j

(4.18)

Note that the last sum over q and j involving the function ı means that we take the only term corresponding to the prime number q such that kk D q j for some j 2 Z n f0g. Now the proof of the theorem may be finished as follows. Since f 2 L1 .Af / j and vol.SpP D ˆ. p j /  ˆ. pj /, see Lemma 76, j / D vol fx 2 Af I kxk D p g ˇ ˇ  the series q j ˆ.q j /  ˆ.qj / ˇ f .q j /ˇ is convergent. Because of the inequality ˇ  ˇ P ˆ.q j /  ˆ.qj /  12 ˆ.q j / the series q j ˆ q j ˇ f .q j /ˇ converges as well, hence we may arbitrary reorder the terms in (4.18). By the properties of the entire part function, the following inequalities hold for the function ˇ q (see (4.17)): q j  qŒlogq kk1 < q logq kk D kk1 ;

j < ˇ q ; j 2 Z;

(4.19)

q j  qŒlogq kk  q logq kk D kk1 ;

j  ˇ q ; j 2 Z;

(4.20)

94

4 Parabolic-Type Equations and Markov Processes on Adeles

where the equality in the second inequality is possible only when kk is a power of q. Suppose in (4.18), that kk D pk for some prime number p and integer k ¤ 0. It follows from inequalities (4.19), (4.20) that the formula (4.18) may be written as

X0    1  1 f .q j /ˆ.q j /  f kk1 ˆ kk1 1 q p j 1

fO ./ D

q 0 (let us denote it by D again) with domain n    o (4.35) Dom .D / WD f 2 L2 Af I kk fO 2 L2 Af is a self-adjoint operator. Moreover, the following assertions hold: (i) (ii) (iii) (iv)

D is a positive operator;  D is m-accretive, i.e. D ˚ j is an m-dissipative operator;   the spectrum  .D / D p I p is a prime; j 2 Z n f0g [ f0g;   D is the infinitesimal generator of a contraction C0 semigroup T .t/ t0 .   Moreover, the semigroup T .t/ t0 is bounded holomorphic (or analytic) with angle =2.

Proof (i) It follows from the Steklov–Parseval equality that for any f 2 L2 .Af / 



Z

.D f ; f / D .kk F f ; F f / D

Af

ˇ ˇ2 kk ˇF f ˇ d Af  0:

(ii), (iv) The result follows from the well-known corollary from the Lumer-Phillips Theorem, see e.g. [41, Chapter 2, Section 3] or [24]. For the property of the semigroup of being holomorphic, see e.g. [7, 3.7] or [41, Chapter 2, Section 4.7]. (iii) Since D is self-adjoint and positive, .D /  Œ0; 1/. Consider the eigenvalue problem D f D f , f 2 D.D /,  > 0. By applying the Fourier transform we obtain the equivalent equation .kk  /fO D 0:

(4.36)

If  D pj for some prime p and j 2 Z n f0g then the inverse Fourier j transform of the characteristic ˚ j function of Sp j D f 2Af I kk D p g is a solution of (4.36) . If  62 p I p is a prime; j 2 Znf0g , then the functions ˇ ˇ ˇ 1 ˇ ˇ  ˇ and ˇ kk  ˇ are bounded, hence the equation D f  f D h is kk  kk  uniquely solvable for any h 2 L2 .Af / and  2 .D /. The point 0 belongs to .D / as a limit point.    The representation of the generated semigroup T .t/ t0 in the case  > 1 is presented in detail in Theorem 114.

104

4 Parabolic-Type Equations and Markov Processes on Adeles

4.4 Metric Structures, Distributions and Pseudodifferential Operators on A 4.4.1 A Structure of Complete Metric Space for the Adeles  that A D R  Af . Then any x 2 A can be written uniquely as x D We recall x1 ; xf 2 R  Af D A. Set for x; y 2 A   A .x; y/ WD jx1  y1 j1 C xf ; yf ; where .x; y/ was defined in (4.6). Then .A; A / is a complete metric space, see Proposition 71. Note that A .x; y/ is topologically equivalent to  ˚ Q .x; y/ WD max jx1  y1 j1 ; .x; y/ ;

x; y 2 A;

which induces on A the product topology. The topology of the restricted product on A is equal to the product topology on R  Af , where R is equipped with the usual topology and Af with the restricted product topology. Hence the following result holds. Proposition 95 The restricted product topology on A is metrizable, a metric is given by A . Furthermore, .A; A / is a complete metric space and .A; A / as a topological space is homeomorphic to .R; j  j1 /  Af ; .  .i/ Remark 96 .A; A / is a second-countable topological space. More precisely, B1       .j/ .i/ Bf i;j2N is a countable base, where B1 i2N is a countable base of R; j  j1 and  .j/    Bf j2N is a countable base of Af ; , see Remark 73 (ii). Therefore .A; A / is a semi-compact space.

4.4.2 Distributions on A The space of Bruhat-Schwartz functions, denoted D.A/, consists of finite linear  combinations of functions of type h .x/ D h1 .x1 / hf xf with h1 2 D.R/, Schwartz space on R, and hf 2 D.Af /. The space D.A/ is dense in L% .A; dxA / for 1  % < C1, see e.g. [37, Theorem 2.9]. The space of distributions on D.A/ is the strong dual space of D.A/.

4.4 Metric Structures, Distributions and Pseudodifferential Operators on A

105

4.4.3 Pseudodifferential Operators and the Lizorkin Space on A ˇ

We consider the pseudodifferential operator DR DW Dˇ , ˇ > 0 on D.R/ defined by

  Dˇ h .x1 / D F1 j 1 jˇ1 Fx1 ! 1 h ; 1 !x1

h 2 D.R/:

(4.37)

Recall that the operator Dˇ is the real Riesz fractional operator and represents a fractional power of the Laplacian, see e.g. [101, §8], [102, §25]. ˛;ˇ We introduce the pseudodifferential operator DA DW D˛;ˇ , ˛; ˇ > 0 on D.A/ defined by

 ˛    1 D˛;ˇ h .x/ D F!x j 1 jˇ1 C  f  Fx! h ;

h 2 D.A/:

(4.38)

Lemma 97 With the above notation D˛;ˇ W D.A/ ! C .A; C/ \ L2 .A/ Proof It is sufficient to show the result for a factorizable function h D h1 hf , h1 2   bf  f , D .R/, hf 2 D.Af /. Since hO ./ D hc 1 . 1 / h



     D˛;ˇ h .x/ D hf xf Dˇ h1 .x1 / C h1 .x1 / D˛ hf xf :

(4.39)

  Note that Dˇ h1 2 L2 .R/ \ C .R; C/, D˛ hf 2 L2 .Af / \ C Af ; C , cf. Lemma 92, and since dxA D dx1 dxAf we conclude hf Dˇ h1 , h1 D˛ hf 2 C .A; C/ \ L2 .A/.  The space D.A/ is not invariant under the action of the operator D˛;ˇ . To overcome such an inconvenience, we introduce an adelic version of the Lizorkin space of the second kind. First we recall that the real Lizorkin space of test functions, see e.g. [101, §2] or [102, §25], is defined by  L0 .R/ D f1 2 D.R/I



Z R

xn1 f1 .x1 / dx1

D 0; for n 2 N :

The real Lizorkin space can be equipped with the topology of the space D.R/, which makes L0 .R/ a complete space. The real Lizorkin space is invariant with respect to Dˇ , is dense in Lp .R/, 1 < p < 1, and admits the following characterization: f1 2 L0 .R/ if and only if f1 2 D.R/ and ˇ ˇ dn ˇ F f . / D 0; 1 ˇ n d 1  1 D0

for n 2 N:

106

4 Parabolic-Type Equations and Markov Processes on Adeles

We introduce an adelic Lizorkin space of the second kind L0 WD L0 .A/ as L0 .A/ D L0 .R/ ˝ L0 .Af /: The space L0 .A/ consists of finite linear combinations of factorizable functions h.x/ D h1 .x1 /hf .xf / with h1 2 L0 .R/, hf 2 L0 .Af /. Note that L0 .A/ is a subspace of D.A/ and it may be equipped with the topology of D.A/. Lemma 98 With the above notation the following assertions hold: (i) D˛;ˇ L0 D L0 for ˛; ˇ > 0; (ii) L0 is dense in L2 .A/. Proof (i) It is sufficient to consider a factorizable function h D h1 hf , h1 2 L0 .R/, hf 2 L0 .Af /. Since D˛ hf 2 L0 .Af / and Dˇ h1 2 L0 .R/, see Lemma 93 and [101, (9.1)], we conclude from (4.39) that D˛;ˇ L0 .A/  L0 .A/. Conversely, take h 2 L0 .A/. We want to show that the equation D˛;ˇ g D h has a solution g 2 L0 .A/. We may assume without loss of generality that h D h1 hf , h1 2 L0 .R/, hf 2 L0 .Af /. Applying the Fourier transform we obtain gO ./ D

hO 1 . 1 /hO f . f / ˇ

j 1 j1 C k f k˛

:

(4.40)

Since hf 2 L0 .Af /, it follows from (4.32) that hO f . f / D

N X

ci 1BR . i / . f /;

iD1

where the balls BR . i / are disjoint and 0 62 BR . i / for i D 1; : : : ; N. It follows from non-Archimedean property that the function k f k is constant on each of the balls BR . i /, hence we may rewrite (4.40) as gO ./ D

N X ci hO 1 . 1 / ˇ

iD1

j 1 j1 C di

1BR . i / . f /;

where di WD k i k˛ > 0 are constants. It may be easily checked that the functions ci hO 1 . 1 / ˇ

j 1 j1 Cdi

are Fourier transforms of real Lizorkin functions and the functions

1BR . i / . f / are Fourier transforms of Lizorkin functions on Af , thus g 2 L0 .A/. (ii) Since L0 .R/ is dense in L2 .R/, see [101, Thm. 3.2], and L0 .Af / is dense in L2 .Af / by Lemma 93, the tensor product L0 .A/ D L0 .R/ ˝ L0 .Af / is dense in the tensor product L2 .R/ ˝ L2 .Af / which is isomorphic to the space L2 .A/, see e.g. [95, Theorem II.10]. 

4.4 Metric Structures, Distributions and Pseudodifferential Operators on A

107

Similarly to Sect. 4.3.4 we may consider the operator D˛;ˇ as an operator acting on L2 .A/ . It is easy to see that the operator D˛;ˇ with the domain Dom.D˛;ˇ / D D.A/ is symmetric. Moreover, similarly to the proof of Lemma 98 (i) we may check that .D˛;ˇ ˙ i/L0 .A/ D L0 .A/, i.e. the ranges of the operators D˛;ˇ ˙ i are dense in L2 .Af /, hence the operator D˛;ˇ is essentially self-adjoint. The following description of the closure holds. Lemma 99 The closure of the operator D˛;ˇ , ˛; ˇ > 0 (let us denote it by D˛;ˇ again) with domain

n o   Dom D˛;ˇ WD f 2 L2 .A/ I j 1 jˇ1 C kk˛ fO 2 L2 .A/

(4.41)

is a self-adjoint operator. Moreover, the following assertions hold: D˛;ˇ  0; D˛;ˇ is m-accretive, i.e. D˛;ˇ is an m-dissipative operator; the spectrum  .D˛;ˇ / D Œ0; 1/; ˛;ˇ D  is the infinitesimal generator of a  contraction C0 semigroup T˛;ˇ .t/ t0 . Moreover, the semigroup T˛;ˇ .t/ t0 is bounded holomorphic (or analytic) with angle =2.

(i) (ii) (iii) (iv)

Proof The proofs of (i), (ii) and (iv) are similar to the corresponding proofs in Lemma 94. We outline only the proof of (iii). Suppose  > 0 is given. We show that the equation .D˛;ˇ  /g D h cannot be solved for some h 2 L2 .A/. We pick a factorizable h and by applying the Fourier transform we get gO ./ D

hO 1 . 1 /hO f . f / ˇ

j 1 j1 C k f k˛  

:

We pick hO f . f / D 1BR . 0 / ./, where k 0 k˛ <  and 0 62 BR . 0 /, then k f k D k 0 k for any  f 2 BR . 0 / and gO ./ D

hO 1 . 1 / ˇ

j 1 j1 C k 0 k˛  

1BR . 0 / ./:

  If we take h1 2 D.R/ such that hO 1 .  k 0 k˛ /1=ˇ ¤ 0, then the function hO 1 . 1 / ˇ

j 1 j1 Ck 0 k˛ 

62 L2 .R/, and a fortiori gO 62 L2 .A/, hence  2  .D˛;ˇ /.



  The representation of the generated semigroup T˛;ˇ .t/ t0 , in the case ˛ > 1, 0 < ˇ  2, is presented in detail in Theorem 127.

108

4 Parabolic-Type Equations and Markov Processes on Adeles

4.5 The Adelic Heat Kernel on Af In this section we introduce the adelic heat kernel on Af as the inverse Fourier ˛ transform of etkyk with y 2 Af , kyk defined by (4.5), ˛ > 1 and t > 0. In Sects. 4.5, 4.6 and 4.7 we work only with finite adeles, for this reason in the variables we omit the subindex ‘f ’. ˛

Proposition 100 Consider the function kykˇ etkyk for fixed t > 0, ˇ  0 and ˛ > 1. Then   ˛ kykˇ etkyk 2 L% Af ; dyAf for any 1  % < C1. Proof It is sufficient to show that for any t > 0 and ˇ  0 Z

˛

I.t/ WD Af

kykˇ etkyk dyAf < C1:

According to Lemmas 78 and 76 Z

X

˛

Af

kykˇ etkyk dyAf D

pmˇ etp

m˛

  ˆ. pm /  ˆ. pm / ;

pm ; m¤0

thus we have to prove the convergence of the latter series. We consider two cases: m < 0 and m > 0. m˛ If m < 0, then pmˇ etp  1 and S .t/ WD

X

pmˇ etp

m˛



ˆ. pm /  ˆ. pm /



pm ; m0

pm ; m>0

We recall that the Prime Number Theorem is equivalent to ln ˆ.x/ D

.x/ x;

x ! 1;

4.5 The Adelic Heat Kernel on Af

109

see e.g. [33], hence there exists a constant C such that 1 XX

SC .t/ 

pmˇ etp

m˛ CCpm

:

p mD1

We want to show the existence of a positive constant M D M .ˇ/ such that pmˇ etp

m˛ CCpm

 Mp1m

for all m  1 and prime p;

or equivalently that p1Cm.ˇC1/ etp CCp  M. Since p1Cm.ˇC1/  e.ˇC1/p for all m ˛m m  1 and p  2, consider e.CCˇC1/p tp . This expression is less than or equal to 1

˛1 and hence there exist only a finite number of pairs .p; m/ 1 when pm  CCˇC1 t for which it can be greater than 1, so the announced constant exists. Therefore m˛

SC .t/  M

1 XX

m

m

p1m  2M

X

p mD1

p2 < C1:



p

Definition 101 We define the adelic heat kernel on Af as Z Z .x; tI ˛/ WD Z .x; t/ D

˛

Af

 .x/ etkk d Af ;

x 2 Af ; t > 0; ˛ > 1: (4.42)

By Proposition 100 the integral is convergent. When considering Z .x; t/ as a function of x for t fixed we will write Zt .x/. By applying Theorem 79 to the function ˛ etkk we obtain the following result. Proposition 102 The following representation holds for the heat kernel: Z .x; t/ D

j ˛   j˛ ˆ q j etq  et.qC /

X

for t > 0; x 2 Af ;

(4.43)

q j 0:

Proof From the inequality 1  ex  x valid for x  0 we obtain j

˛

j

˛

etq  et.qC /  1  et.qC /  t.qC /˛ : j˛

j

(4.44)

110

4 Parabolic-Type Equations and Markov Processes on Adeles

Then with the use of the inequality 12 ˆ.q j /  ˆ.q j /  ˆ.qj / we have Z.x; t/ D

X q j 0 Z kyk>

Zt .y/ dyAf  Ct < C1:

(4.45)

Proof The statements (i) and (ii) immediately follows from formulas (4.43) and (4.44), respectively. (iii) By (4.44) and Lemma 78 we have Z kyk>

Zt .y/dyAf 

X

ˆ. pk /  2tpk˛ ˆ. pk  / D 2t 

pk >

X

pk˛ < C1;

pk >

where we have used (4.14) and (4.15).



Theorem 105 The adelic heat kernel on Af satisfies the following: (i) (ii) (iii) (iv) (v) (vi) (vii)

Z R .x; t/  0 for any t > 0; Af Zt .x/ dxAf D 1 for any t > 0; Zt .x/ 2 L1 .Af / for any t > 0; Zt .x/  Zt0 .x/ D ZtCt0 .x/ for any t, t0 > 0; limt!0C Zt .x/ D ı .x/ in D0 Af ; Zt .x/ is a uniformly continuous function for any fixed t > 0; Z.x; t/ is uniformly continuous in t, i.e. Z.x; t/ 2 C..0; 1/; C.Af // or limt0 !t maxx2Af jZ.x; t/  Z.x; t0 /j D 0 for any t > 0.

Proof (i) It follows from Corollary 104. ˛ (ii) For any t > 0 the function etkk  is continuous   at  D 0 and by Proposi ˛ tkk 1 2 A \ L Af . Then Zt .x/ 2 C Af ; R \ tion 100 we have e 2 L f  L2 Af . Now the statement follows from the inversion formula for the Fourier transform on Af . (iii) The statement follows from (i) and (ii).

4.6 Markov Processes on Af

111

  (iv) By the previous property Zt .x/ 2 L1 Af for any t > 0. Then



˛ ˛ ˛ 0 0 1 1 Zt .x/  Zt0 .x/ D F!x etkk et kk D F!x e.tCt /kk D ZtCt0 .x/ :     ˛ (v) Since etkk 2 C Af ; R \ L1 Af , cf. Proposition 100, the scalar product  tkk˛  e ; f ./ D

Z

˛

etkk f ./ d Af

Af

  with f 2 D Af

defines a distribution on Af . Since the support of f is compact, cf. Lemma 87,   ˛ and etkk 2 L1 Af , the dominated convergence theorem with the characteristic function of the support of f as a dominant function implies lim

t!0C

˛ etkk ; f ./ D .1; f /

     and then, as F D Af D D Af , we have



   tkk˛  1 1 e ; f .x/ D 1; F!x f D .ı; f / : lim Z.x; t/; f D lim F!x t!0C

t!0C



  ˛ ˛ 1 etkk and etkk 2 L1 Af for t > 0, Zt .x/ is (vi) Since Zt .x/ D F!x uniformly continuous in x for any fixed t > 0. ˛ ˛ 0 (vii) Suppose that t < t0 . By the mean value theorem etkk  et kk D .t0  ˛ t/kk˛ et.kk/kk , where t < t.kk/ < t0 . Hence ˇ ˇZ

ˇ ˇ 0 tkk˛ t0 kk˛ ˇ d Af ˇˇ jZ.x; t/  Z.x; t /j D ˇ  .  x/ e e Af

ˇ ˇZ ˇ ˇ ˛ t.kk/kk˛ ˇ D jt  t jˇ  .  x/ kk e d Af ˇˇ Af Z ˛ kk˛ et0 kk d Af ;  jt  t0 j 0

Af

for some 0 < t0 < t; t0 . Now the statement follows from Proposition 100.



4.6 Markov Processes on Af   Along this section we consider Af ; as the complete non-Archimedean metric space and use the terminology, notation and results of [39,  ChaptersTwo, Three]. Let B denote the -algebra of the Borel sets of Af . Then Af ; B; dxAf is a measure space. Let 1B .x/ denote the characteristic function of a set B 2 B.

112

4 Parabolic-Type Equations and Markov Processes on Adeles

We assume along this section that ˛ > 1 and set p .t; x; y/ WD Z .x  y; t/

for t > 0; x; y 2 Af ;

and (R P .t; x; B/ WD

B

p .t; x; y/ dyAf

1B .x/

for t > 0; x 2 Af ; B 2 B for t D 0:

Lemma 106 With the above notation the following assertions hold: (i) p .t; x; y/ is a normal transition density; (ii) P .t; x; B/ is a normal transition function. Proof The result follows from Theorem 105, see [39, Sec. 2.1] for further details. Lemma 107 The transition function P .t; y; B/ satisfies the following two conditions: (i) for each u  0 and a compact B lim sup P .t; x; B/ D 0I

x!1 tu

ΠCondition L.B/

(ii) for each  > 0 and a compact B   lim sup P t; x; Af n B .x/ D 0:

t!0C x2B

ŒCondition M.B/

Proof Since B is a compact, dist.x; B/ DW d.x/ ! 1 as x ! 1. Since the function we obtain from (4.44) that Z.x  y; t/ ˛ ˆ.x/ is1non-decreasing,  ˛ 2u d.x/ ˆ for any y 2 B and t  u. Hence P.t; x; B/  2u d.x/ .d.x//   ˆ .d.x//1  vol.B/ ! 0 as x ! 1. To verify Condition M.B/ we proceed as follows: for y 2 Af n B .x/ we have kx  yk > . The statement follows from (4.45):   P t; x; Af n B .x/  C./t ! 0;

t !0C: 

Theorem 108 Z.x; t/ is the transition density of a time- and space homogenous Markov process which is bounded, right-continuous and has no discontinuities other than jumps. Proof The result follows from [39, Theorem 3.6], Remark 73 (ii) and Lemmas 106, 107. 

4.7 Cauchy Problem for Parabolic Type Equations on Af

113

Remark 109 The more strict version of Condition M.B/ which is sufficient for the continuity of a Markov process, namely, that for each  > 0 and a compact B   1 sup P t; x; Af n B .x/ D 0; t!0C t x2B

ΠCondition N.B/

lim

does not hold for the function Z.x; t/. This may be easily seen if we take  D 1=4. In such case by Proposition 102 and Lemma 78 we have Z

Z Af nB1=4 .x/

Zt .x  y/ dyAf 

S1=3 .x/

Zt .x  y/ dyAf

D vol S1=3 .x/ 

˛ j  j  tqj˛ t qC e ˆ q e

X q j 1, D˛ is the pseudodifferential operator defined by (4.33) with the domain given by (4.35) and u W Af  Œ0; 1/ ! C is an unknown  function.  We say that a function u.x; t/ is a solution of (4.46) if u 2 C Œ0; 1/; Dom.D˛ / \   C1 Œ0; 1/; L2 .Af / and u satisfies equation (4.46) for all t  0. We understand the notions of continuity in t, differentiability in t and equalities in the L2 .Af / sense, as it is customary in the semigroup theory. More precisely, we say that a function u.x; t/ is continuous in t at t0 if limt!t0 ku.x; t/  u.x; t0 /kL2 .Af / D 0; the function u0t .x; t/ is the time derivative of function u.x; t/ at   0/  u0t .x; t0 /L2 .Af / D 0; two functions f .x; t/ and g.x; t/ are t0 if limt!t0  u.x;t/u.x;t tt0 equal at t0 if kf .x; t0 /  g.x; t0 /kL2 .Af / D 0. We know from Lemma 94 that the operator D˛ generates a C0 semigroup. Therefore Cauchy problem (4.46) is well-posed, i.e. it is uniquely solvable with

114

4 Parabolic-Type Equations and Markov Processes on Adeles

the solution continuously dependent on the initial data, and its solution is given by u.x; t/ D T .t/u0 .x/, t  0, see e.g. [7, 24, 41]. the general theory does  However  not give an explicit formula for the semigroup T .t/ t0 . We show that the operator T .t/ for t > 0 coincides with the operator of convolution with the heat kernel Zt  . In order to prove this, we first construct a solution of Cauchy problem 4.46 with the initial value from D.Af / without using the semigroup theory. Then we extend the result to all initial values from Dom.D˛ /, see Proposition 112 and Theorem 114. We show in Theorem 117 that in the case u0 2 L0 .Af /, the function u.x; t/ is the  solution of Cauchy problem 4.46 in a stricter sense, i.e. u.x; t/ 2 C1 Œ0; 1/; L0 .Af / and all limits and equalities are understood pointwise.

4.7.1 Homogeneous Equations with Initial Values in D .Af / We first consider Cauchy problem 4.46 ˇ initial value from the space D.Af /.  with the To simplify notations, set Z0  u0 D Zt  u0 ˇtD0 WD u0 . Note that such definition is consistent with Theorem 105 (v). We define the function u.x; t/ D Zt .x/  u0 .x/;

t  0:

(4.47)

    Since Zt .x/ 2 L1 Af for t > 0 and u0 .x/ 2 D.Af /  L1 Af , the convolution exists and is a continuous function, see [100, Theorem 1.1.6]. Lemma 110 Let u0 2 D.Af / and u.x; t/, t  0 is defined by (4.47). Then u.x; t/ is continuously differentiable in time for t  0 and the derivative is given by  ˛ tkk˛  @u 1 .x; t/ D F!x kk e  1BR ./  u0 .x/; @t

(4.48)

where 1BR ./ is the characteristic function of the ball BR , R D .1=`/ and ` is the parameter of constancy of the function u0 , see (4.31) and Proposition 89. Proof Let ht .x/ be a function defined by the right-hand side of (4.48 ). Since ˛ kk˛ etkk  1BR ./ 2 L1 .Af / \ L2 .Af / for any t  0, the function ht .x/ is welldefined and belongs to C.Af / \ L2 .Af /. Let t0  0. Consider a limit  u.x; t/  u.x; t /    0 lim   ht .x; t0 / 2 t!t0 L .Af / t  t0  etkk˛  et0 kk˛  ˛   D lim  uO 0 ./ C kk˛ et0 kk 1BR ./  uO 0 ./ 2 t!t0 L .Af / t  t0   etkk˛  et0 kk˛

˛   D lim  C kk˛ et0 kk 1BR ./  uO 0 ./ 2 ; t!t0 L .Af / t  t0

4.7 Cauchy Problem for Parabolic Type Equations on Af

115

where we have applied Steklov-Parseval formula and the fact that supp uO 0  BR which follows from Proposition 89. By applying the mean value theorem twice we obtain ˛

˛

etkk  et0 kk ˛ 0 ˛ ˛ C kk˛ et0 kk D kk˛ et kk C kk˛ et0 kk t  t0 D .t0  t0 /kk2˛ et

00 kk˛

;

where t0 D t0 .kk/ is a point between t0 and t and t00 D t00 .kk/ is a point between t0 and t0 (and thus between t0 and t). Hence   etkk˛  et0 kk˛

˛   C kk˛ et0 kk 1BR ./  uO 0 ./ 2  L .Af / t  t0  jt  t0 jR2˛ kOu0 ./kL2 .Af / ! 0;

t ! t0 ;

i.e. ht .x/ is the time derivative of the function u.x; t/ for any t  0. The proof of the continuous differentiability in time of u.x; t/ follows from the time continuity of ht .x/ which can be checked similarly.  Lemma 111 Let u0 2 D.Af / and u.x; t/, t  0 is defined by (4.47). Then u.x; t/ 2 Dom.D˛ / for any t  0 and  ˛ tkk˛  1 kk e D˛ u.x; t/ D F!x  1BR ./  u0 .x/;

(4.49)

where 1BR ./ is the characteristic function of the ball BR , R D .1=`/ and ` is the parameter of constancy of the function u0 , see (4.31) and Proposition 89.     ˛ Proof Note that uO 0 2 D.Af / which implies that etkk  uO 0 ./ 2 L1 Af \ L2 Af     ˛ and kk˛ etkk  uO 0 ./ 2 L1 Af \ L2 Af for any t  0. Hence we may calculate D˛ u.x; t/ by formula (4.33). For t > 0 we obtain    ˛  ˛ 1 1 D˛ u.x; t/ D F!x Z t ./  uO 0 ./ kk  uO .; t/ D F!x kk b    ˛ tk k˛  ˛ tkk˛ 1 1 D F!x 1BR ./  uO 0 ./ D F!x  1BR ./  u0 .x/; kk e kk e

where we have used the fact that supp uO 0  BR . For t D 0 we obtain   ˛ 1 D˛ u.x; 0/ D D˛ u0 .x/ D F!x kk  uO 0 ./    ˛ 0k k˛  ˛ 0kk˛ 1 1 D F!x 1BR ./  uO 0 ./ D F!x  1BR ./  u0 .x/: kk e kk e



116

4 Parabolic-Type Equations and Markov Processes on Adeles

As an immediate consequence from Lemmas 110 and 111 we obtain: Proposition 112 Let the function u0 2 D.Af /. Then the function u.x; t/ defined by (4.47) is a solution of Cauchy problem (4.46).

4.7.2 Homogeneous Equations with Initial Values in L2 .Af / Consider the operator T.t/, t  0 of convolution with the heat kernel, i.e. T.t/u D Zt  u:

(4.50)

Since Zt 2 L2 .Af /, the convolution Zt  u is a continuous function of x for t > 0 and any u 2 L2 .Af /, see [100, Theorem 1.1.6]. Lemma 113 The operator T.t/ W L2 .Af / ! L2 .Af / is bounded. Proof Consider a function u 2 L2 .Af /. Since Zt 2 L1 .Af /, see Theorem 105 (iii), by the Young inequality and Theorem 105 (ii) kZt  ukL2  kZt kL1  kukL2 D kukL2 : Hence T.t/u D Zt  u 2 L2 .Af / and kT.t/k  1.



Theorem 114 Let ˛ > 1. Then the following assertions hold.   (i) The operator D˛ generates a C0 semigroup T .t/ t0 . The operator T .t/ coincides for each t  0 with the operator T.t/ given by (4.50). (ii) Cauchy problem 4.46 is well-posed and its solution is given by u.x; t/ D Zt  u0 , t  0. ˛ Proof According to Lemma 94, the operator D generates a C0 semigroup T .t/ t0 . Hence Cauchy problem 4.46 is well-posed, see e.g. [24, Theorem 3.1.1]. By Proposition 112, T .t/jD.Af / D T.t/jD.Af / and both operators T .t/ and T.t/ 2 are defined on the whole  L .Af / and bounded. By the continuity we conclude that 2 T .t/ D T.t/ on L Af . Now the statements follow from well-known results of the semigroup theory, see e.g. [7, Proposition 3.1.9.], [24, Theorem 3.1.1], [41, Ch. 2, Proposition 6.2].    Remark 115 Since the semigroup T .t/ t0 is holomorphic, Cauchy problem 4.46 possesses smoothing effect, see e.g. [7, Corollary 3.7.21]. More precisely, consider Cauchy problem 4.46 with weaker requirement on the initial value, namely, let u0 2 L2 .Af /. Then there exists a unique function

      u.x; t/ 2 C Œ0; 1/; L2 .Af / \ C .0; 1/; Dom.D˛ / \ C1 .0; 1/; L2 .Af /

4.7 Cauchy Problem for Parabolic Type Equations on Af

117

satisfying the equation for t > 0 and satisfying the initial condition. That is, this weaker Cauchy problem is solvable for arbitrary initial data and the solution is infinitely differentiable in t for t > 0.

4.7.3 Homogeneous Equations with Initial Values in L0 .Af / We now consider the Cauchy problem 8 ˆ < @u.x; t/ C D˛ u.x; t/ D 0; x 2 Af ; t 2 Œ0; C1/ @t ˆ : u.x; 0/ D u .x/; u0 .x/ 2 L0 .Af /; 0

(4.51)

with the initial value from the space L0 .Af / and the pseudodifferential operator D˛ with the smaller domain Dom.D˛ / D L0 .Af /. We say that a function u.x; t/ is a classical solution of (4.51), if u 2 C1 Œ0; 1/; L0 .Af / and u satisfies equation (4.51) for all t  0, with the understanding that all the involved limits are taken in the topology of L0 .Af /. Lemma 116 Let u0 2 L0 .Af / and the function u.x; t/ is defined by (4.47). Then u.x; t/ 2 L0 .Af / for any t > 0. ˛

Proof Since etkk is locally constant outside of the origin, the function ht ./ D ˛ ˛ 1 etkk  uO 0 ./ D Zt .x/u0 .x/ 2 etkk  uO 0 ./ 2 D.Af / with ht .0/ D 0. Then F!x L0 .Af / for t > 0.  Theorem 117 Let the function u0 2 L0 .Af /. Then the function u.x; t/ defined by (4.47) is the classical solution of Cauchy problem 4.51. Proof By Lemma 116 the function u.x; t/ is correctly defined and u.x; t/ 2 Dom.D˛ / D L0 .Af / for all t  0. We assert that there exist constants ` and R not dependent on t such that u.x; t/ 2 ˛ DR` and D˛ u.x; t/ 2 DR` for all t  0. Consider the function ht ./ D etkk  uO 0 ./, tkk˛ t  0. Since u0 2 L0 , the function e is locally constant on the support of ˛ uO 0 . Moreover, the parameter of constancy of etkk on the support of uO 0 does not 0 0 0 depend on t. Hence there exist parameters ` and R such that ht 2 DR` 0 for any t  0.  1  0 .1=R / By Proposition 89 we have F ht .x/ D u.x; t/ 2 D.1=`0 / for any t  0. Similar ˛ ˛ proof works for the function gt ./ WD F .D u.x; t// D kk˛ etkk  uO 0 ./. We recall that for finite dimensional spaces, the uniform convergence is equivalent to the L2 -convergence. Since DR` is a finite dimensional space, cf. Proposition 89, by applying Lemmas 110 and 111, we have @u .x; t/ 2 L0 .Af /, u.x; t/ is a @t  1 solution of Cauchy problem (4.51) and u belongs to C Œ0; 1/; L0 .Af / . 

118

4 Parabolic-Type Equations and Markov Processes on Adeles

4.7.4 Non Homogeneous Equations Consider the following Cauchy problem 8 ˆ < @u.x; t/ C D˛ u.x; t/ D f .x; t/; @t ˆ : u.x; 0/ D u .x/; 0

x 2 Af ; t 2 Œ0; T; T > 0

(4.52)

u0 .x/ 2 Dom.D˛ /:

We  say that a function   u.x; t/ is a solution of (4.52 ), if u belongs to C Œ0; T; Dom.D˛ / \ C1 Œ0; T; L2 .Af / and if u satisfies equation (4.52) for t 2 Œ0; T.   Theorem 118 Let ˛ > 1 and let f 2 C Œ0; T; L2 .Af / . Assume that at least one of the following conditions is satisfied:   T/; Dom.D˛ / ; (i) f 2 L1 .0;  (ii) f 2 W 1;1 .0; T/; L2 .Af / . Then Cauchy problem (4.52) has a unique solution given by Z u.x; t/ D Af

Z .x  y; t/ u0 .y/ dyAf

Z tZ

C 0

Af

 Z .x  y; t  / f .y; / dyAf d:

Proof With the use of Theorem 114 the proof follows from well-known results of the semigroup theory, see e.g. [7, Proposition 3.1.16], [24, Proposition 4.1.6]. 

4.8 The Adelic Heat Kernel on A We recall that the Archimedean heat kernel is defined as Z ˇ Z.x1 ; tI ˇ/ D 1 . 1 x1 / etj 1 j1 d 1 ; t > 0; ˇ 2 .0; 2 : R

This heat kernel is a solution of the pseudodifferential equation

@u .x1 ; t/ ˇ 1 C F1 F u .x ; t/ D 0: j j 1 1 1 x1 ! 1 1 !x1 @t For a more detailed discussion of the Archimedean heat kernel and its properties the reader may consult [38, Section 2] and references therein. From now on we will denote heat kernel (4.42) as Z.xf ; tI ˛/.

4.9 Markov Processes on A

119

Definition 119 For fixed ˛ > 1, ˇ 2 .0; 2 we define the heat kernel on A as Z Z .x; tI ˛; ˇ/ WD

A

 .  x/ e



ˇ ˛ t j 1 j1 Ck f k

d A ;

x 2 A; t > 0:



ˇ ˛ Since etj 1 j1 2 L1 .R; d 1 /, etk f k 2 L1 Af ; d Af , cf. [38, Property 2.2] and Proposition 100 , and d A D d 1 d Af , we have ˇ  ˛     Z .x; tI ˛; ˇ/ D F 1 etj 1 j1 F 1 etk f k D Z.x1 ; tI ˇ/Z xf ; tI ˛ : (4.53)

For t > 0 fixed, we use the notation Zt .xI ˛; ˇ/ instead of Z .x; tI ˛; ˇ/. Theorem 120 The adelic heat kernel on A possesses the following properties (i) (ii) (iii) (iv) (v) (vi) (vii)

Z R .x; tI ˛; ˇ/  0 for any t > 0; A Z .x; tI ˛; ˇ/ dxA D 1 for any t > 0; Zt .xI ˛; ˇ/ 2 L1 .A/ for any t > 0; Zt .xI ˛; ˇ/  Zt0 .xI ˛; ˇ/ D ZtCt0 .xI˛;ˇ/ for any t; t0 > 0; limt!0C Z .x; tI ˛; ˇ/ D ı .x1 /  ı xf D ı .x/ in D0 .A/; Zt .xI ˛; ˇ/ is a uniformly continuous function for any fixed t > 0; Z.x; tI ˛; ˇ/ is uniformly continuous in t, i.e. Z.x; tI ˛; ˇ/ belongs to C..0; 1/; C.A// or limt0 !t maxx2A jZ.x; tI ˛; ˇ/  Z.x; t0 I ˛; ˇ/j D 0 for any t > 0.

Proof The statement from (4.53) and the corresponding properties for  follows   Z.x1 ; tI ˇ/ and Z xf ; tI ˛ , see [38, Section 2] and Theorem 105.

4.9 Markov Processes on A Let B.A/ denote the -algebra of the Borel sets of .A; A /. Along this section we suppose that ˛ > 1 and ˇ 2 .0; 2 are fixed parameters. We set p .t; x; yI ˛; ˇ/ WD Z.x  y; tI ˛; ˇ/

for t > 0; x; y 2 A:

Note that   p .t; x; yI ˛; ˇ/ D Z.x1  y1 ; tI ˇ/Z xf  yf ; tI ˛   DW p .t; x1 ; y1 I ˇ/ p t; xf ; yf I ˛ ;

120

4 Parabolic-Type Equations and Markov Processes on Adeles

    where p .t; x1 ; y1 I ˇ/ D Z.x1  y1 ; tI ˇ/ and p t; xf ; yf I ˛ WD p t; xf ; yf D Z.xf  yf ; tI ˛/. We also define for x1 ; y1 2 R and B1 2 B .R/ (R P .t; x1 ; B1 I ˇ/ WD

B1

p .t; x1 ; y1 I ˇ/ dy1

1B1 .x1 /

for t > 0 for t D 0

and for x; y 2 A and B 2 B .A/ (R P .t; x; BI ˛; ˇ/ WD

B

  p .t; x1 ; y1 I ˇ/ p t; xf ; yf I ˛ dy1 dyAf

1B .x/

for t > 0 for t D 0:

Lemma 121 With the above notation the following assertions hold: (i) p .t; x; yI ˛; ˇ/ is a normal transition density; (ii) P .t; x; BI ˛; ˇ/ is a normal transition function. Proof The statement follows from the corresponding properties for the functions   p .t; x1 ; y1 I ˇ/ and p t; xf ; yf , see Lemma 106. Lemma 122 The transition function P .t; x; BI ˛; ˇ/ satisfies the following two conditions: (i) for each u  0 and a compact B lim sup P .t; x; BI ˛; ˇ/ D 0I

x!1 tu

ŒCondition L.B/

(ii) for each  > 0 and a compact B

ı lim sup P t; x; A n B .x/ I ˛; ˇ D 0;

t!0C x2B

ŒCondition M.B/

ı

where B .x/ WD fy 2 AI A .x; y/ < g. Proof (i) Note that there exist compact subsets K1  R and Kf  Af such that B  K1  Kf . Then P.t; x; BI ˛; ˇ/  P.t; x1 ; K1 I ˇ/P.t; xf ; Kf /:   Since A .0; x/ ! 1 we have either 0; xf ! 1 or jx1 j1 ! 1. Therefore it is sufficient to show that lim sup P.t; xf ; Kf / D 0

xf !1 tu

(4.54)

4.9 Markov Processes on A

121

and lim sup P.t; x1 ; K1 I ˇ/ D 0:

(4.55)

x1 !1 tu

The equality (4.54) follows from Lemma 107. By [38, (2.2)] 1

Z.t; x1 I ˇ/ 

Ct ˇ

for t > 0; x1 2 R:

2

t ˇ C x21

(4.56)

Then Z P.t; x1 ; K1 I ˇ/ D

Z.t; x1  y1 I ˇ/dy1 K1 1

Z

1

 Ct ˇ K1

2 ˇ

t C .x1  y1 /2

dy1 :

As x1 ! 1 we have dist .x1 ; K1 / ! 1 and jx1  y1 j  dist.x1 I K1 / for any y1 2 K1 and 1 2 ˇ

t C .x1  y1 /

2



1 2

dist .x1 I K1 /

Hence 1

Z

1

lim sup P.t; x1 ; K1 I ˇ/  lim Cu ˇ

x1 !1 tu

x1 !1

:

K1

2

dist .x1 I K1 /

dy1 D 0:

ı ı   (ii) Since B .x/ B 2 .x1 /  B 2 xf , where

n ı o B 2 .x1 / D y1 2 RI jx1  y1 j < 2 ı    ı  and B 2 xf is given by (4.8), we have A n B .x/  R  Af n B 2 .x1 / 

ı        

 and with the use of B 2 xf  R n B 2 .x1 /  Af [ R  Af n B 2 xf [38, (2.1)] and Theorem 105 (ii) we obtain ı   P t; x; A n B .x/ I ˛; ˇ 

Z jx1 y1 j1  2

Z C

.xf ;yf /> 2

p .t; x1 ; y1 I ˇ/ dy1

  p t; xf ; yf dyAf

ı       P t; x1 ; R n B 2 .x/ I ˇ C P t; xf ; Af n B 2 xf :

122

4 Parabolic-Type Equations and Markov Processes on Adeles

Now the result follows from Lemma 107 and the inequality ı   P t; x1 ; R n B 2 .x/ D

Z jy1 j 2

Z.t; y1 I ˇ/ dy1

Z C

jy1 j 2

Z DC

1

tˇ 2

t ˇ C y21

jz1 j 2 t1=ˇ

dy1

1 dz1 ! 0 1 C z21

as t ! C0:



Theorem 123 Z .x; tI ˛; ˇ/ with ˛ > 1 and ˇ 2 .0; 2 is the transition density of a time- and space homogenous Markov process which is bounded, right-continuous and has no discontinuities other than jumps. Proof The result follows from [39, Theorem 3.6], Remark 96 (ii) and Lemmas 121, 122. 

4.10 Cauchy Problem for Parabolic Type Equations on A In this section we study Cauchy problems for parabolic type equations on A and present analogues of the results of Sects. 4.7.1, 4.7.2 and 4.7.4.

4.10.1 Homogeneous Equations Consider the following Cauchy problem 8 @u .x; t/ ˆ C D˛;ˇ u .x; t/ D 0; x 2 A; t 2 Œ0; C1/ < @t

ˆ : u.x; 0/ D u .x/; u0 .x/ 2 Dom D˛;ˇ ; 0

(4.57)

where ˛ > 1, ˇ 2 .0; 2, D˛;ˇ is the pseudodifferential operator defined by (4.38) with the domain given by (4.41) and u W A  Œ0; 1/ ! C is an unknown function.  We say that a function u.x; t/ is a solution of (4.57), if u 2 C Œ0; 1/; Dom.D˛;ˇ / \  C1 Œ0; 1/; L2 .A/ and if u satisfies equation (4.57) for all t  0. As in Sect. 4.7 we understand the notions of continuity, differentiability and equalities in the sense of L2 .A/.

4.10 Cauchy Problem for Parabolic Type Equations on A

123

We first consider Cauchy problem (4.57) with the initial value from D.A/. We define the function u.x; t/ WD u.x; tI ˛; ˇ/ D Zt .xI ˛; ˇ/  u0 .x/ D Zt .x/  u0 .x/;

t  0;

(4.58)

ˇ  where Z0  u0 D Zt  u0 ˇtD0 WD u0 . Note that such definition is consistent with Theorem 120 (v). Since Zt .x/ 2 L1 .A/ for t > 0 and u0 .x/ 2 D.A/  L1 .A/, the convolution exists and is a continuous function, see Theorem 120 (ii), [100, Theorem 1.1.6]. Lemma 124 Let u0 2 D.A/ and u is defined by (4.58). Then u belongs to  C Œ0; 1/; Dom.D˛;ˇ / and 1 D˛;ˇ u D F!x



 ˛  t j 1 jˇ1 Ck f k˛

 j 1 jˇ1 C  f  e Fx! u0

(4.59)

for t  0. Proof We first verify that u.; t/ 2 Dom.D˛;ˇ / for t  0. Without loss of generality we may assume that u0 .x/ D u1 .x1 /uf .xf / with u1 2 D.R/ and uf 2 D.Af /. Since Fx! u.x; t/ D e



˛ ˇ t j 1 j1 Ck f k

  uO 1 . 1 / uO f  f ;

we have       j jˇ C  ˛ Fx! u 2 1 1 f L .A/

˛   ˛     ˇ    f  etk f k uO f  f L2 .Af /  etj 1 j1 uO 1 . 1 / L2 .R/ ˛      ˇ C etk f k uO f  f L2 .Af /  j 1 jˇ1 etj 1 j1 uO 1 . 1 / L2 .R/   ˛             f  uO f  f L2 .Af /  uO 1 L2 .R/ C uO f L2 .Af /  j 1 jˇ1 uO 1 . 1 / L2 .R/         D D˛ uf L2 .Af / ku1 kL2 .R/ C uf L2 .Af / Dˇ u1  2 ;

L .R/

where we used the equality d A D d Af d 1 and the Parseval-Steklov formula. Therefore u.x; t/ 2 Dom.D˛;ˇ / for t  0 and formula (4.59) holds. To verify the continuity, assume again that u0 .x/ D u1 .x1 /uf .xf / with u1 2 D.R/, uf 2 D.Af /. With the use of the Parseval-Steklov equality and the mean value theorem we obtain    u .x; t/  u x; t0  2 lim L .A/ 0

t !t

 t.j jˇ Ck k˛ / ˇ    t0 .j 1 j1 Ck f k˛ /  e 1 1 f u O  f L2 .A/ D lim  e . / u O 1 f 1 0 t !t

124

4 Parabolic-Type Equations and Markov Processes on Adeles ˛      ˇ    t  t0 j 1 jˇ C  f ˛ eQt.j 1 j1 Ck f k / uO 1 . 1 / uO f  f  2 D lim 1 L .A/ 0

t !t

  ˛     lim jt  t0 j   j 1 jˇ1 C  f  uO 1 . 1 / uO f  f L2 .A/ 0 t !t

    ˇ ˇ    ˇt  t0 ˇ D 0;  D˛ uf L2 .Af / ku1 kL2 .R/ C uf L2 .Af / Dˇ u1 L2 .R/ lim 0 t !t

 ˛   ˇ where Qt D Qt j 1 j1 C  f  is a point between t and t0 .



Lemma 125 Let u0 2 D.A/ and u.x; t/, t  0 is defined by (4.58). Then u.x; t/ is continuously differentiable in time for t  0 and the derivative is given by ˛  ˛   ˇ  @u 1 .x; t/ D F!x j 1 jˇ1 C  f  et.j 1 j1 Ck f k / Fx! u0 : @t

(4.60)

Proof Assume that u0 .x/ D u1 .x1 /uf .xf / with u1 2 D.R/, uf 2 D.Af /. By reasoning as in the proofs of Lemmas 110 and 124, we have  uO .; t/  uO .; t /    ˛  t j 1 jˇ1 Ck f k˛

  0 lim  C j 1 jˇ1 C  f  e Fx! u0  2 t!t0 L .A/ t  t0

˛ ˇ      ˛ 2 Qt j 1 j1 Ck f k D lim jt  t0 j   j 1 jˇ1 C  f  e Fx! u0 L2 .A/ t!t0

  ˛   lim jt  t0 j  .j 1 jˇ1 C  f  /2 Fx! u0 L2 .A/ t!t0

ˇ ˇ   ˇt  t0 ˇ D 0;  kD2˛ uf k  ku1 k C 2kD˛ uf k  kDˇ u1 k C kuf k  kD2ˇ u1 k lim 0 t !t

where we have used the fact that D.R/  Dom.Dˇ / for any ˇ > 0 and D.Af /  Dom.D˛ / for any ˛ > 0. To verify the continuity of @u .x; t/, we proceed similarly: @t  @u  @u   lim  .x; t/  .x; t0 / 2 t!t0 @t L .A/ @t   ˛ 2 Qt j 1 jˇ1 Ck f k˛

  ˇ  D lim jt0  tj .j 1 j1 C  f  / e uO 1 . 1 / uO f  f L2 .A/ t!t0

  ˛    lim jt0  tj .j 1 jˇ1 C  f  /2 uO 1 . 1 / uO f  f L2 .A/ D 0: t!t0

where we used the mean value theorem with a point Qt between t and t0 .



As an immediate consequence from Lemmas 124 and 125 we obtain Proposition 126 Let the function u0 2 D.A/. Then the function u.x; t/ defined by (4.58) is a solution of Cauchy problem (4.57).

4.10 Cauchy Problem for Parabolic Type Equations on A

125

Consider the operator T.tI ˛; ˇ/, t  0 of convolution with the adelic heat kernel T.tI ˛; ˇ/u D Zt  u:

(4.61)

As in Sect. 4.7, the convolution Zt  u is a continuous function of x for t > 0 and any u 2 L2 .A/ and the operator T.tI ˛; ˇ/ W L2 .A/ ! L2 .A/ is bounded. By reasoning as in the proof of Theorem 114, we obtain Theorem 127 Let ˛ > 1 and ˇ 2 .0; 2. Then the following assertions hold.   (i) The operator D˛;ˇ generates a C0 semigroup T .tI ˛; ˇ/ t0 . The operator T .tI ˛; ˇ/ coincides for each t  0 with the operator T.tI ˛; ˇ/ given by (4.61). (ii) Cauchy problem (4.57) is well-posed and its solution is given by u.x; t/ D Zt  u0 , t  0.

4.10.2 Non Homogeneous Equations Consider the following Cauchy problem 8 @u .x; t/ ˆ C D˛;ˇ u .x; t/ D f .x; t/ ; x 2 A; t 2 Œ0; T ; T > 0 < @t

ˆ : u.x; 0/ D u .x/; u0 .x/ 2 Dom D˛;ˇ : 0

(4.62)

We u.x; t/ is a solution of (4.62 ), if u belongs to  say that a function  C Œ0; T; Dom.D˛;ˇ / \ C1 Œ0; T; L2 .A/ and if u satisfies equation (4.62) for t 2 Œ0; T.   Theorem 128 Let ˛ > 1, ˇ 2 .0; 2 and let f 2 C Œ0; T; L2 .A/ . Assume that at least one of the following conditions is satisfied:   (i) f 2 L1 .0; T/; Dom.D˛;ˇ / ;   (ii) f 2 W 1;1 .0; T/; L2 .A/ . Then Cauchy problem (4.62) has a unique solution given by Z u.x; t/ D A

Z .x  y; tI ˛; ˇ/ u0 .y/ dyA

Z t Z

C 0

A

 Z .x  y; t  I ˛; ˇ/ f .y;  / dyA d:

Proof With the use of Theorem 127 the proof follows from well-known results of the semigroup theory, see e.g. [7, Proposition 3.1.16], [24, Proposition 4.1.6]. 

Chapter 5

Fundamental Solutions for Pseudodifferential Operators, and Equations of Schrödinger Type

5.1 Introduction This chapter aims to explore the connections between local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields. In the 50s Gel’fand and Shilov showed that fundamental solutions for certain types of partial differential operators with constant coefficients can be obtained by using local zeta functions [49]. The existence of fundamental solutions for general differential operators with constant coefficients was established by Atiyah [15] and Bernstein [18] using local zeta functions. A similar program can be carried out in the p-adic setting. In this chapter, we give a detailed proof of the existence of a fundamental solution for a pseudo-differential operator with a symbol of the form j f j˛p , with f an arbitrary polynomial, see Theorem 134. This result was proved in [123], see also [120], by Zúñiga-Galindo, here we present a complete proof, including a review of the method of analytic continuation of Gel’fand-Shilov, see [49]. This chapter is organized as follows. In Sects. 5.2–5.3, we review the method of analytic continuation of Gel’fand-Shilov, and the basic aspects of the local zeta functions. In Sect. 5.4, we prove the existence of a fundamental solution Eˇ for a pseudodifferential operator f .@; ˇ/, see Theorem 134. For small ˇ, there exists a simple formula for Eˇ , see Theorem 137. In Sect. 5.5, we compute fundamental solutions for quasielliptic operators, see Theorem 142 and Corollary 143. In Sect. 5.6, we compute fundamental solutions for Schrödinger-type pseudodifferential operators, see Theorem 146. We study certain homogeneous equations attached to these operators, see Theorem 149, and finally we study some initial value problems for Schrödinger-type pseudodifferential equations, see Theorem 150.

© Springer International Publishing AG 2016 W.A. ZúQniga-Galindo, Pseudodifferential Equations Over Non-Archimedean Spaces, Lecture Notes in Mathematics 2174, DOI 10.1007/978-3-319-46738-2_5

127

128

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of. . .

5.2 The Method of Analytic Continuation of Gel’fand-Shilov In this section, we review the method of analytic continuation of Gel’fand-Shilov, see [49, Appendix B] or [57, pp. 65–67]. Let U be a non-empty, open subset of C. Assume that U ! D0 s ! Ts is an D0 -valued function on U. We say that Ts is continuous, respectively holomorphic, on U if .Ts ; / is continuous, respectively holomorphic, on U for every in D. Let X be a continuous curve of finite length in U (i.e. continuous on X \ U), then .R; / WD

R

.Ts ; / ds

X

is defined for every in D and defines an element of D0 . We denote this distribution as R R D Ts ds: X

Suppose that Ts is holomorphic on U, and that the puncture disc fs 2 CI 0 < js  s0 j  rg is contained in U for some s0 2 C (not necessarily in U) and some r > 0. Then .Ts ; / can be expanded in its Laurent series: .Ts ; / D

P

.ak ; / .s  s0 /k ;

k2Z

where ak D

R 1 Ts ds p 2 1 jss0 jr .s  s0 /kC1

(5.1)

is in D0 . Now, since D0 is complete Ts D

P

ak .s  s0 /k

k2Z

gives an element of D0 , the Laurent expansion of Ts at s0 . We say that s0 is a pole of Ts if ak ¤ 0 (in D0 ) only for a finite number of k < 0.

5.3 Igusa’s Local Zeta Functions

129

5.3 Igusa’s Local Zeta Functions We set for a > 0 and s 2 C, as WD es ln a . Let f ./ 2 Qp Œ 1 ; : : : ;  n  be a nonconstant polynomial. The p-adic complex power j f jsp associated to f (also called the Igusa zeta function of f ) is the distribution 

 j f jsp ; WD

R Qnp Xf 1 .0/

./ j f ./jsp dn , s 2 C, Re .s/ > 0.

The Igusa local zeta functions are connected with the number of solutions of polynomial congruences mod pm and with exponential sums mod pm . There are many intriguing conjectures connecting the poles of local zeta functions with topology of complex singularities, see e.g. [30, 57]. The local zeta functions can be defined on any locally compact field K, i.e. for R, C, or finite extensions of Qp or of Fp ..t//, with Fp the finite field with p elements. In the Archimedean case K D R or C, the study of local zeta functions was initiated by I.M. Gel’fand and G.E. Shilov [49]. The main motivation was the construction of fundamental solutions for partial differential operators with constant coefficients. Indeed, the meromorphic continuation of the local zeta functions imply the existence of fundamental solutions, this fact was established, independently, by Atiyah [15] and Bernstein [18], see also [57, Theorem 5.5.1 and Corollary 5.5.1]. On the other hand, in the middle 60s, A. Weil initiated the study of local zeta functions, in the Archimedean and non Archimedean settings, in connection with the Poisson-Siegel formula. In the 70s, Igusa developed a uniform theory for local zeta functions over local fields of characteristic zero [57]. Theorem 129 (Igusa [57, Theorem 8.2.1]) For Re.s/ > 0, j f jsp defines a D0 -valued holomorphic function, and  it has  a meromorphic continuation to the whole complex plane such that j f jsp ; is a rational function of ps . More precisely, there exists a finite collection of pairs of no-negative integers f.NE ; vE / 2 N  NX f0g I E 2 T g depending only on f , such that Q 

 1  pvE NE s  j f jsp

E2T

becomes a D0 -valued holomorphic function on the whole complex plane. Remark 130 (i) Notice that the poles of j f jsp have the form s D  NvEE C NE2i ln p Z. (ii) Let f W Qnp ! Qp be a polynomial mapping satisfying f .0/ D 0. Let f.NE ; vE / 2 N  NX f0g I E 2 T g be as in Theorem 129. Set  WD  . f / D minE NvEE .   Then  is the real part of a pole of j f jsp ; for some 2 D, cf. [110, Theorem 2.7] or [58]. Notice that  < 0. This result implies that R

./

Qnp Xf 1 .0/

j f ./jˇp

dn  < 1 for any ˇ satisfying 0 < ˇ <  . f / :

130

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of. . .

From these observations and by applying Theorem 129, we get the following result: Corollary 131 With the above notation, 

 j f jsp ; D

R Qnp Xf 1 .0/

./ j f ./jsp dn , for Re.s/ >  . f / ;

defines a D0 -value holomorphic function. Remark 132 We call a D0 -valued holomorphic function a holomorphic distribution on D.

5.4 Fundamental Solutions for Pseudodifferential Operators Let f ./ 2 Qp Œ 1 ; : : : ;  n  be an arbitrary non-constant polynomial. A pseudodifferential operator with symbol j f ./jˇp , ˇ > 0, is an extension of an operator of the form

1 . f .@; ˇ/ / .x/ D F!x j f ./jˇp Fx! , for 2 D: Definition 133 Let Eˇ 2 D0 , with ˇ > 0. We say that Eˇ is a fundamental solution for f .@; ˇ/ u D , with 2 D;

(5.2)

if u D Eˇ  is a solution of (5.2) in D0 . At this point, it is important to mention that we cannot use the standard definition of fundamental solution, i.e. f .@; ˇ/ Eˇ D ı, because D is not invariant under the action of f .@; ˇ/. Theorem 134 There exists a fundamental solution for f .@; ˇ/ u D , with 2 D. Proof We first note that the existence of a fundamental solution for f .@; ˇ/ u D , with 2 D, is equivalent to the existence of a distribution Tˇ D F Eˇ satisfying j f ./jˇp Tˇ D 1 in D0 ,

(5.3)

R where .1; / D dn x. Indeed, assume that (5.3) holds. Then for any F 2 D, j f ./jˇp Tˇ F D F in D0 . The product j f ./jˇp Tˇ F is well-defined as a commutative and associative product, cf. [105, Theorem 3.19], thus,     j f ./jˇp Tˇ F D j f ./jˇp Tˇ F D j f ./jˇp F Eˇ  D j f ./jˇp F u D F

5.4 Fundamental Solutions for Pseudodifferential Operators

131

in D0 , i.e. f .@; ˇ/ u D in D0 . The other implication is established in a similar form. D j f jˇp j f jsp in D0 , for any s 2 C. Claim A j f jsCˇ p We show the existence of a solution for the division problem (5.3). By Theorem 129, j f jsp has a meromorphic continuation to the complex plane, then let j f jsp D

P

ak .s C ˇ/k

k2Z

be the Laurent expansion of j f jsp at ˇ, with ak 2 D0 . Since the real parts of the D j f jsp j f jˇp is a poles of j f jsp are negative rational numbers, by Claim A, j f jsCˇ p ˇ holomorphic distribution at s D ˇ. Therefore j f jp ak D 0 for all k < 0 (notice that by (5.1) and Claim A, j f jˇp ak 2 D0 ), and D j f jˇp a0 C j f jsCˇ p

1 P

j f jˇp ak .s C ˇ/k :

kD1

(5.4)

By applying the dominated convergence theorem in (5.4): lim

s!ˇ



; D .1; / D j f jˇp a0 ; j f jsCˇ p

hence Tˇ D a0 , and Eˇ D F 1 a0 . Proof of Claim A We recall that the product j f jsp j f jˇp is defined by

n o

j f jsp j f jˇp ; ' D lim j f jsp ; j f jˇp  ı j ' for Re.s/ > 0, j!C1

  where ı j .x/ D pnj  p j kxkp , if the limit exists for all ' 2 D. By using the fact ˇ  ˇ that if x 2 Qnp X f 1 .0/, then ˇ f x C p j yQ ˇp D j f .x/jp for j big enough (depending on x) and for any yQ 2 Znp , we have

lim

j!C1

D

0

R Qnp Xf 1 .0/

R Qnp Xf 1 .0/

' .x/ j f .x/jsp @pnj

1

R

kyxkp

pj

j f .y/jˇp dn yA dn x

' .x/ j f .x/jsCˇ dn x; p

for j big enough and Re.s/ > 0. The result follows by using the meromorphic continuation of j f .x/jsp to the whole complex plane. 

132

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of. . .

Remark 135 Notice that if ˇ is not a pole of j f jsp , then Tˇ D j f jˇ p and Eˇ D 1 F Tˇ . From this observation, by applying Theorem 134 and Corollary 131, we get the following result. Corollary 136 Assume that 0 < ˇ <  . f /. Then Z

 .x/ D u .x/ D F 1 j f jˇ p

p .x  / .F / ./ jf

Qnp Xf 1 .0/

./jˇp

dn 

is a solution of f .@; ˇ/ u D in D0 , for any 2 D. n o Set Dom. f .@; ˇ// WD 2 L2 I j f jˇp F 2 L2 . Then f .@; ˇ/ W Dom.f .@; ˇ// ! L2 is a well-defined linear operator. Theorem 137 If 0 < 2ˇ <  . f /, then the following assertions hold: (i) the mapping v ! Eˇ  v is continuous operator from L2 into L2 ; (ii) f .@; ˇ/ u D v, with v 2 L2 , has a solution in L2 given by Z u .x/ D

p .x  / .F v/ ./ jf

Qnp Xf 1 .0/

./jˇp

dn :

Furthermore, if v 2 L1 \ L2 , then u is a continuous function. In this case u is the unique continuous solution of the equation. Proof (i) By Corollary 136,   Eˇ  v .x/ D

Z

p .x  / .F v/ ./ Qnp Xf 1 .0/

jf

./jˇp

dn , v 2 D:

  By using the density of D Qnp in L2 and the Cauchy-Schwarz inequality, ˇ ˇ ˇE ˇ  v ˇ 

sZ

dn  Qnp Xf 1 .0/

j f ./j2ˇ p

! kvkL2  C kvkL2 ,

we get that v ! Eˇ  v gives rise to a continuous operator from L2 into L2 .

5.5 Fundamental Solutions for Quasielliptic Pseudodifferential Operators

133

2 1 1 (ii) Now, since F u D j f jˇ p F v, with v 2 L , j f jˇ 2 L , by (i), we have u D p F 1 1 ˇ  v D Eˇ  v 2 L2 . Finally, if v 2 L1 , the continuity of u follows j f jp

from the fact that F v 2 L1 , by using the dominated convergence theorem. 

5.5 Fundamental Solutions for Quasielliptic Pseudodifferential Operators In this section we compute fundamental solutions for quasielliptic pseudodifferential operators, these operators were introduced by Galeano-Peñaloza and Zúñiga-Galindo in [48] as a generalization of the elliptic pseudodifferential operators introduced in [122], see also [52]. Definition 138 Let f ./ 2 Zp Œ 1 ; : : : ;  n  be a non-constant polynomial, and let ! D .! 1 ; : : : ; ! n / 2 .NX f0g/n . We say that f is a quasielliptic polynomial of degree d with respect to ! if the two following conditions hold: (1) f ./ D 0 ,  D 0, and (2) f .! 1  1 ; : : : ; ! n  n / D d f ./, for any  2 Q p. For ! WD .! 1 ; : : : ; ! n / 2 .NX f0g/n , we set j!j WD ! 1 C : : : C ! n . Proposition 139 Let f ./ 2 Zp Œ 1 ; : : : ;  n  be a quasielliptic polynomial of degree d with respect to !. Then j f jsp has a meromorphic continuation to the whole complex plane of the form: 

 R j f jsp ; D . ./  .0// j f ./jsp dn  C .0/ Znp

C

R

Qnp XZnp

L . ps / 1  pj!jds

./ j f ./jsp dn ;

s s where with rational coefficients such that j!j L .p / is a polynomial in p L p d ¤ 0.

Proof The result follows from the meromorphic continuation of the integral Z.s; f / WD Z.s/ D

R Znp Xf0g

j f ./jsp dn :

More precisely, from the following fact: Claim Z.s/ D

L . ps / ; 1  pj!jds

(5.5)

134

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of. . .

where L .ps / is a polynomial in ps with rational coefficients. In addition, Z.s/ has a pole satisfying Re .s/ D j!j d . The proof of this Claim is as follows. Set  ˚ A WD . 1 ; : : : ;  n / 2 Znp I ord. i /  wi for i D 1; : : : ; n ; and Ac WD Znp X A. Then Z.s/ D

R

j f ./jsp d C

A

R Ac

j f ./jsp d D pj!jds Z .s/ C

R Ac

j f ./jsp d;

i.e. Z.s/ D

R

1 pj!jds

1

Ac

j f ./jsp dn :

Let I be a proper subset of f1; : : : ; ng. Set m D .m1 ; : : : ; mn / 2 Nn such that 0  mi  wi  1 , i … I. Define ˚  A .I; m/ WD . 1 ; : : : ;  n / 2 Znp I ord. i /  wi , i 2 I and ord. i / D mi , i … I : Then A .I; m/ is an open and compact subset of Znp , and Ac is a open and compact subset of Znp because it is a finite union of subsets of the form A .I; m/. We now show that JI;m .s/ WD

R

j f ./jsp d D LI;m .ps / ;

A.I;m/

where LI;m .ps / is a polynomial in ps with rational coefficients. Since the origin does not belong to A .I; m/, there exists a finite covering of A .I; m/ byˇ balls Bi WD ˇ s   Q C pM Zp n , with M WD M.I; m/ and Q 2 A .I; m/, such that j f js jB D ˇˇ f Q ˇˇ for i i i p i p

any i, cf. Lemma 26 and Remark 27 in Chap. 2. Therefore JI;m .s/ is a polynomial in ps with rational coefficients, and Z.s/ is a rational function of the form Z.s/ D

8 0:

(5.10)

QnC1 Xh1 .0/ p

  ˚  Notice that the hypersurface Vh Qp WD Vh D . ; / 2 QnC1 W h .; / D 0 does p not have singular points, i.e. the system of equations h .; / D rh .; / D 0 has no solutions in QnC1 p . Technically speaking, Vh is a closed submanifold of codimension one of QnC1 p .

138

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of. . .

˚  Set Vh; D .; / 2 QnC1 W   h0 ./ D  , for  2 Qp fixed. Then Vh; is a p   , we closed submanifold of codimension one for every . For .;  / 2 D QnC1 p take F ./ WD

R

. ; / j GL j ;

(5.11)

Vh;

where  GL is a Gel’fand-Leray form along Vh; . On other hand, by using integration on the fibers, see Chap. 1, formula (1.3) or [57, Section 7.6], one gets from (5.10)– (5.11): 

 R jhjsp ; D jjsp F ./ d, for Re.s/ > 0:

(5.12)

Qp

Now by using the meromorphic continuation of the distribution jjsp , the fact that   F ./ 2 D Qp , and (5.11), we get a meromorphic continuation for jhjsp . The distribution jhjsp , Re.s/ > 0, has the following meromorphic continuation to the whole complex plane: 

 R  ˚ jhjsp ; D jjsp F ./  F .0/ d C



Zp

C

R

Qp XZp

1  p1 1  p1s

F .0/

jjsp F ./ d:

We now observe that 1 F!z

  R F ./ D p .z/ Qp

D

R

(

R

) . ; / j GL j d

Vh;

p .z C zh0 .// .; / ddn d;

(5.13)

QnC1 p

for further details the reader may consult [57, Theorem 8.3.1]. Hence   R 1 F!z F ./ jzD0 D . ; / ddn : QnC1 p

Definition 144 An extension of an operator of the form

  ˇ .h .@; ˇ/ / .x; t/ D F 1 ./j F , for 2 D QnC1  h t! j !t 0 p p !x

x!

is called a Schrödinger-type pseudodifferential operator.

(5.14)

5.6 Schrödinger-Type Pseudodifferential Equations

139

These operators were introduced by Kochubei, see [80] and references therein, and also [120]. By Proposition 5.6 and Theorem 137 we get the following result: R p .xt /.F v/./ n Corollary 145 If 0 < 2ˇ < 1, then u .x; t/ D QnC1 d d is the is ˇ p the unique solution in L2 of h .@; ˇ/ u D v, for v 2 L2 .

j h0 ./jp

From Proposition 5.6 by using the technique presented in the proof of Theorem 134, and the formulas (5.13)–(5.14), we get the following result. Theorem 146 With the above notation. Let Eˇ be fundamental solution for h .@; ˇ/ u D , with 2 D. (1) If ˇ ¤ 1, then  R R  Eˇ ; D

˚

 p . C h0 .//  1 .; / jjˇp

Zp QnC1 p



C

1  p1 1  p1Cˇ

R

.; / ddn 

QnC1 p

R p . C h0 .// .; /

R

C

ddn d

jjˇp

Qp XZp QnC1 p

ddn d:

(2) If ˇ D 1, then .E1 ; / D

R R

˚  p . C h0 .//  1 .; / jjp

Zp QnC1 p



1  p1 C 2 C

R

R

.; / ddn 

QnC1 p

R p . C h0 .// . ; /

Qp XZp QnC1 p

ddn d

jjp

ddn d:

  R Corollary 147 Take ˇ > 1 and v 2 D QnC1 satisfying QnC1 v .t; x/ dtdn x D 0. p p Then ˚   R R p . C h0 .//  1 v .t   ; x  /  u1 .x; t/ D ddn d 2 D QnC1 p ˇ jjp Qp QnC1 p is a solution of h .@; ˇ/ u1 D v.

140

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of. . .

5.6.1 The Homogeneous Equation Theorem 148 Consider n    nC1 o   ˇ 2 I v O  h Qp ! L2 QnC1 : h .@; ˇ/ W v 2 L2 QnC1 ./j 2 L j 0 p p p Then, the initial value problem: 8 n ˆ < .h .@; ˇ/ u0 / .x; t/ D 0, x 2 Qp , t 2 Qp (5.15)

ˆ : u .x; 0/ D 2 D Qn  ; 0 p   has a unique solution in D0 QnC1 given by p u0 .x; t/ D

R Qnp

p .th0 ./    x/ .F / ./ dn ;

(5.16)

this function is locally constant in .x; t/ and square-integrable in x for any t 2 Qp .  If the datum in (5.15) can be taken in L2 Qnp , then (5.16) is the unique solution   in D0 QnC1 of (5.15). p   Proof We first established the theorem in the case 2 D Qnp . Any non-trivial solution u0 2 L2 , i.e. u0 is not zero almost everywhere, of h .@; ˇ/ u0 D 0 necessarily satisfies ess supp F u0  Vh , where ess supp./ denotes the essential support of a measurable function. Let 1Vh .; / denote the characteristic function of Vh , and set   .F u0 / .; / WD .F / ./  1Vh .; /, for 2 D Qnp . From the fact that 1Vh .; /  1suppF u0 ./ has compact support in QnC1 p , it follows that .F u0 / .; / is a bounded function with compact essential support contained in Vh that satisfies j  h0 ./jˇp .F u0 / .;  / D 0:

(5.17)

  Consider for  2 Qp X f0g fixed, the distribution on D Qnp : ..F / ./  1Vh .;  / ; .// D

R fh0 ./D g

.F / ./ ./ j GL j ;

where  GL is a Gel’fand-Leray differential form on fh0 ./ D g, we note  GL ¤ d 1 ^    d n because  is fixed, the existence of the form  GL is a consequence of the fact fh0 ./ D g,  fixed in Qp X f0g, is a non-singular hypersurface because rh0 ./ D 0 ,  D 0. Now, since .F / ./  1Vh .; /,  2 Qp X f0g fixed, is a

5.6 Schrödinger-Type Pseudodifferential Equations

141

distribution with compact support, we have 1 ..F / ./  1Vh .; // D F!x

R fh0 ./D g

p .  x/ .F / ./ j GL j

  in D0 Qnp . We now notice that the function !

R fh0 ./Dg

p .  x/ .F / ./ j GL j

has compact support because h0 .suppF / is compact in Qp . Then 1 u0 .x; t/ WD F1 !t F!x ..F / ./  1Vh .;  // R R D p .t    x/ .F / ./ j GL j d Qp Xf0g fh0 ./D g

  ˚  in D0 QnC1 . We now recall that  2 Qnp I h0 ./ D  is a non-singular hypersurp face for  ¤ 0, then R Qnp

./ dn  D

R Qp Xf0g

(

R fh0 ./Dg

)

./ j GL j d

see [57, Section 7.6], hence u0 .x; t/ D

R Qnp

  : p .th0 ./    x/ .F / ./ dn  in D0 QnC1 p

We note that u0 .x; t/ is a locally constant function since it is the inverse Fourier transform  of a distribution with compact2 support. Furthermore, since u0 .x; t/ D 1 p .th0 .// .F / ./ , u0 .; t/ 2 L for any t. The uniqueness of the solution F!x follows from the fact that any non-trivial solution has the form (5.16) for some 2 D.   We now take 2 L2 Qnp . Notice that (5.17) is still valid in this case. To establish the formula (5.16) we take replace .F / by k .F /, where k is the characteristic .k/ function of the ball Bnk , an set F u0 .; / WD k ./ .F / ./  1Vh .; /. Then .k/

u0 .x; t/ D .k/

R kkp pk

  : p .th0 ./    x/ .F / ./ dn  in D0 QnC1 p

Then u0 .; t/ L2 u0 .; t/. ! 



142

5 Fundamental Solutions for Pseudodifferential Operators, and Equations of. . .

Theorem 149 (i) The operator v ! Tt v, from L2 into L2 , where t 2 Qp and Z .Tt v/ .x/ WD

Qnp

p .th0 ./    x/ .F v/ ./ dn 

is linear continuous, and satisfies: (1) T0 D I is the identity operator on L2 ; (2) TtCs D Tt Ts for any t; s 2 Qp ; (3) Tt v ! v as t ! 0 in L2 ; (4) Dt Tt v D h0 .@; ˇ/ Tt v, for any v 2 L2 , t 2 Qp , where operator and

Dt is the Vladimirov  n ˇ 1 .h0 .@; ˇ/ / .x/ D F!x jh0 ./jp Fx! , for 2 D Qp . (ii) The function u0 .x; t/ D .Tt v/ .x/ is the unique solution of the following initial value problem: 8 < .h .@; ˇ/ u0 / .x; t/ D 0 :

(5.18) v 2 L2 :

u0 .x; 0/ D v;

Proof

  1 p .th0 .// Fx! v . From this identity one gets (i) Notice that .Tt v/ .x/ D F!x (1), (2) and (3). (4) By using the fact that D is dense in L2 , it is sufficient to established the announced formula in the case of test functions. Take 2 D, since )! ( R 1 ; .Tt / .x/ D F !t p .  x/ .F / ./ j GL j h0 ./D0

  in D0 Qp , and the Vladimirov operator extends to distributions having Fourier transform with compact support, we have Dt .Tt / .x/ D D

R Qp

R Qnp

( p .t / jjp

R  h0 ./D0

) p .  x/ .F / ./ j GL j d

jh0 ./jp p .th0 ./    x/ .F / ./ dn :

  1 p .th0 .// Fx! , we have On the other hand, since .Tt / .x/ D F!x .h0 .@; ˇ/ Tt / .x/ D

R Qnp

jh0 ./jp p .th0 ./    x/ .F / ./ dn ;

therefore Dt .Tt / .x/ D .h0 .@; ˇ/ Tt / .x/. (ii) It follows from Theorem 148.



5.6 Schrödinger-Type Pseudodifferential Equations

143

5.6.2 The Inhomogeneous Equation   R Theorem 150 Let ˇ > 1 and v 2 D QnC1 v .t; x/ dtdn x D 0, and satisfying QnC1 p p  n let u0 .x/ 2 D Qp . Consider the following initial value problem: 8 < .h .@; ˇ/ u/ .x; t/ D v .x; t/ ; x 2 Qnp ; t 2 Qp :

(5.19) u .x; 0/ D u0 .x/ :

Set u00 .x/

R R

WD

˚

 p . C h0 .//  1 v .; x  / jjˇp

Qp QnC1 p

ddn d

  Then u00 .x/ 2 D Qnp and u .x; t/ D

R

   p .th0 ./    x/ F u0  u00 ./ d n 

Qnp

C

R R

Qp QnC1 p

˚

 p . C h0 .//  1 v .t  ; x  / jjˇp

ddn d

  is the unique solution in D0 QnC1 of (5.19). p Proof From Theorem 148 and Corollary 147 imply that u .x; t/ is a solution of (5.19). The uniqueness of the solution is a consequence of Theorem 148. 

Chapter 6

Pseudodifferential Equations of Klein-Gordon Type

6.1 Introduction In the 1980s I. Volovich proposed that the world geometry in regimes smaller than the Planck scale might be non-Archimedean [112, 113]. This hypothesis conducts naturally to consider models involving geometry and analysis over Qp . Since then, a big number of articles have appeared exploring these and related themes, see e.g. [36], [107, Chapter 6] and the references therein. In this chapter, which is based on [119], we introduce a new class of nonArchimedean pseudodifferential equations of Klein-Gordon  type. We work on the p-adic Minkowski space which is the quadratic space Q4p ; Q where Q.k/ D k02  k12  k22  k32 . Our starting point is a result of Rallis-Schiffmann that asserts ˚  the existence of a unique measure on Vt D k 2 Q4p I Q .k/ D t which is invariant under the orthogonal group O.Q/ of Q, see Proposition 153 or [92]. By using Gel’fand-Leray differential forms, we reformulate this results in terms of Dirac distributions ı .Q .k/  t/ invariant under O.Q/, see Remark 154 and Lemma 156. We introduce the positive and negative mass shells VmC2 and Vm2 , here m is the ‘mass parameter’ which is taken to be a nonzero p-adic number. The restriction of ı .Q .k/  t/ to Vm˙2 gives two distributions ı ˙ .Q .k/  t/ which are invariant under L"C , the ‘Lorentz proper group’, see Definition 162, and that satisfy ı .Q .k/  t/ D ı C .Q .k/  t/ C ı  .Q .k/  t/, see Lemma 163. The p-adic Klein-Gordon type pseudodifferential operators introduced here have the form hˇ i ˇ ˇQ.k/  m2 ˇ˛ Mx!k ' ; ˛ > 0, m 2 Qp X f0g , .˛;m '/ .x/ D M1 k!x p where M denotes the Fourier-Minkowski transform. We solve the Cauchy problem for these operators, see Theorem 174.

© Springer International Publishing AG 2016 W.A. ZúQniga-Galindo, Pseudodifferential Equations Over Non-Archimedean Spaces, Lecture Notes in Mathematics 2174, DOI 10.1007/978-3-319-46738-2_6

145

146

6 Pseudodifferential Equations of Klein-Gordon Type

The equations .˛;m / .t; x/ D 0 have many similar properties to the classical Klein-Gordon equations, see e.g. [31, 32, 103]. These equations  admit  plane waves  1 as weak solutions, see Lemma 177; the distributions aM ı Q .k/  m2 C C   bM1 ı  Q .k/  m2 , a; b 2 C, are weak solutions of these equations, see Proposition 171. The locally constant functions Z .t; x/ D

˚  p .x  k/ p .t! .k// C .k/ C p .t! .k//  .k/ d 3 k;

(6.1)

UQ;m

where p ./ denotes the standard additive character of Qp and ˙ are locally constant functions with support in UQ;m , are weak solutions of these equations. At this point, it is relevant to mention that the operators, equations and techniques introduced here are new. In [5] a very general theory for pseudodifferential operators and equations involving symbols that vanish only at the ˇ origin was ˇ˛ developed. This theory cannot be applied here because our symbols (ˇQ.k/  m2 ˇp ) have infinitely many zeros. Finally, we want to mention that another type of p-adic hyperbolic equations was introduced by Kochubei in [79].

6.2 Preliminaries Remark 151 Along this chapter p will denote a prime number different from 2. We recall that any non-zero p-adic number x can be written uniquely as pord.x/ ac .x/,  where ac .x/ 2 Z p is the angular component of x. For x D x0 C x1 p C    2 Zp , we denote by x the digit x0 considered as an element of Fp , the finite field with p elements.

6.2.1 Fourier Transform on Finite Dimensional Vector Spaces Let E be a finite dimensional vector space over Qp and p a non-trivial additive character of Qp as before. Let Œx; y be a symmetric non-degenerate Qp - bilinear form on E  E. Thus Q.e/ WD Œe; e, e 2 E is a non-degenerate quadratic form on E. We identify E with its algebraic dual E by means of Œ; . We now identify the dual group (i.e. the Pontryagin dual) of .E; C/ with E by taking he; e i D p .Œe; e / where Œe; e  is the algebraic duality. The Fourier transform takes the form Z 'O .y/ D

' .x/ p .Œx; y/ dx for ' 2 L1 .E/ ,

E

where dx is a Haar measure on E.

6.2 Preliminaries

147

Let L .E/ be the space of continuous functions ' in L1 .E/ whose Fourier transform 'O is in L1 .E/.The measure dx can be normalized uniquely in such manner c that .b ' / .x/ D ' .x/ for every ' belonging to L .E/. We say that dx is a self-dual measure relative to p .Œ; /. For further details about the material presented in this section the reader may consult [115].

6.2.2 The p-Adic Minkowski Space We take E to be the Qp -vector space of dimension 4. By fixing a basis we identify E with Q4p considered as a Qp -vector space. For x D .x0 ; x1 ; x2 ; x3 / WD .x0 ; x/ and y D .y0 ; y1 ; y2 ; y3 / WD .y0 ; y/ in Q4p we set Œx; y WD x0 y0  x1 y1  x2 y2  x3 y3 WD x0 y0  x  y;

(6.2)

which is a symmetric non-degenerate bilinear form. From now on, we use Œx; y to   mean bilinear form (6.2 ). Then Q4p ; Q , with Q.x/ D Œx; x is a quadratic vector   space and Q is a non-degenerate quadratic form on Q4p . We will call Q4p ; Q the p-adic Minkowski space.  On Q4p ; Q , the Fourier-Minkowski transform takes the form Z M Œ' .k/ D

  p .Œx; k/ ' .x/ d4 x for ' 2 L1 Q4p ,

(6.3)

Q4p

where d4 x is a self-dual measure for p .Œ; /, i.e. M ŒM Œ' .x/ D ' .x/ for   every ' belonging to L Q4p . Notice that d4 x is equal to a positive multiple of the normalized Haar measure on Q4p , i.e. d4 x D Cd4 .x/, with C > 0. Remark 152 (i) We recall that the usual Fourier transform F on Q4p has the form Z F Œ' Œk D

  p .x0 k0 C x1 k1 C x2 k2 C x3 k3 / ' .x/ d4 .x/ for ' 2 L1 Q4p ,

Q4p

where d4 .x/ is the normalized Haar measure of Q4p . The connection between M and F is given by the formula F ŒM Œ' .x0 ; x1 ; x2 ; x3 / D C' .x0 ; x1 ; x2 ; x3 / ; which is equally valid for integrable functions as well as distributions.

148

6 Pseudodifferential Equations of Klein-Gordon Type

(ii) Note that C D 1, i.e. d4 x D d 4 .x/. Indeed, take ' .x/ to be the characteristic function of Z4p , now ' .x/ D M ŒM Œ' .x/ D CMk!x ŒF Œ' .k0 ; k/ D CMk!x Œ' .k0 ; k/ D C2 F Œ' .x0 ; x/ D C2 ' .x/ , therefore C D 1.

6.2.3 Invariant Measures Under the Orthogonal Group O.Q/ We set Q.x/ D Œx; x as before. We also set 2

1 0 6 0 1 G WD 6 40 0 0 0

3 0 0 0 0 7 7: 1 0 5 0 1

Then Q.x/ D xT Gx, where T denotes the transpose of a matrix. The orthogonal group of Q.x/ is defined as    ˚ O.Q/ D ƒ 2 GL4 Qp I Œƒx; ƒy D Œx; y    ˚ D ƒ 2 GL4 Qp I ƒT Gƒ D G : Notice that any ƒ 2 O.Q/ satisfies det ƒ D ˙1. We consider O.Q/ as a p-adic Lie  subgroup of GL4 Qp , which is also a p-adic Lie group. For t 2 Q p , we set ˚  Vt WD k 2 Q4p I Q .k/ D t : Proposition 153 (Rallis-Schiffman [92, Proposition 2–2]) The orthogonal group O .Q/ acts transitively on Vt . On each orbit Vt there is a measure which is invariant under O .Q/ and unique up to multiplication by a positive constant. For each t 2 Q p , let d t be a measure on Vt invariant under O .Q/. Since Vt is closed in Q4p , it is possible to consider d t as a measure on Q4p supported on Vt .

6.2 Preliminaries

149

6.2.4 Some Remarks About p-Adic Analytic Manifolds We need some basic results about p-adic manifolds in the sense of Serre, the reader may consult Chap. 2 or [57, 104]. Since rQ.k/ ¤ 0 for any k 2 Vt , by using the non-Archimedean implicit function theorem, one verifies that Vt is a p-adic closed submanifold of codimension 1, i.e a p-adic analytic hypersurface without singularities. The condition rQ.k/ ¤ 0 for any k 2 Vt , implies the existence of Gel’fand-Leray differential form t on Vt , i.e. dk0 ^ dk1 ^ dk2 ^ dk3 D dQ .k/ ^ t :

(6.4)

We denote the corresponding measure as jt j .A/ for an open compact subset A of Vt . O The notation k0 ; : : : ; kl.j/ ; : : : ; k3 means omit the l .j/th coordinate. We now describe this measure in a suitable chart. We may assume that Vt is a countable disjoint union of submanifolds of the form .j/

Vt

8

9 < .k0 ; : : : ; k3 / 2 Q4 I kl.j/ D hj k0 ; : : : ; kO l.j/ ; : : : ; k3 = p

WD ; : with k0 ; : : : ; kO l.j/ ; : : : ; k3 2 Vj ;

(6.5)



where hj k0 ; : : : ; kO l.j/ ; : : : ; k3 is a p-adic analytic function on some open compact subset Vj of Q3p , and .j/

@Q @kl.j/

.j/

.z/ ¤ 0 for any z 2 Vt . If A is a compact open subset

contained in Vt , then Z jt j .A/ D h1 j .A/

dk0 : : : dkO l.j/ : : : dk3 ˇ ˇ ; ˇ @Q ˇ ˇ @kl.j/ .k/ˇ

(6.6)

p

.j/

where we are identifying the set A  Vt with the set of all the coordinates of the points of A, which is a subset of Q3p , and h1 the subset of Q4p j .A/ denotes

consisting of the points .k0 ; k1 ; k2 ; k3 / such that kl.j/ D hj k0 ; : : : ; kO l.j/ ; : : : ; k3 for

k0 ; : : : ; kO l.j/ ; : : : ; k3 2 A. Remark 154 (i) Let T .Vt / denote the family of all compact open subsets of Vt . Then t is a additive function on T .Vt / such that t .A/  0 for every A in T .Vt /. By Carath éodory’s extension theorem t has a unique extension to the  -algebra generated by T .Vt /. We also note that the measure t is supported on Vt . (ii) Let D .Vt / denote the C-vector space generated by the characteristic functions of the elements of T .Vt /. The fact that t is a positive additive function on

150

6 Pseudodifferential Equations of Klein-Gordon Type

T .Vt / is equivalent to say that D .Vt / ! '

Z

!

C ' jt j

Vt

is a positive distribution. We can identify the measure jt j with a distribution on Q4p supported on Vt . (iv) Some authors use ı .Q .k/  t/ or ı .Q .k/  t/ d4 k to denote the measure jt j. We will use ı .Q .k/  t/.   Remark 155 Let G0 be a subgroup of GL4 .Qp /. Let ' 2 D Q4p and let ƒ 2 G0 . We define the action of ƒ on ' by putting   .ƒ'/ .x/ D ' ƒ1 x ;   and the action of ƒ on a distribution T 2 D0 Q4p by putting   .ƒT; '/ D T; ƒ1 ' : We say that T is invariant under G0 if ƒT D T for any ƒ 2 G0 . Lemma 156 With the above notation, we have d t D C jt j for some positive constant C. Proof By Remark 154 and Proposition 153, it is sufficient to show that the distribution ı .Q .k/  t/ is invariant under O .Q/, i.e. Z

Z ' .ƒk/ jt .k/j D

Vt

' .k/ jt .k/j Vt

  for any ƒ 2 O.Q/ and ' 2 D Q4p . Now, since Vt is invariant under ƒ, it is sufficient to show that jt .k/j D jt .y/j under k D ƒ1 y, for any ƒ 2 O.Q/. To verify this fact we note that   dk0 ^ dk1 ^ dk2 ^ dk3 D det ƒ1 dy0 ^ dy1 ^ dy2 ^ dy3 and dQ .k/ D dQ.y/ under k D ƒ1 y. Now by (6.4) and the fact that the restriction of t to Vt is unique we have t .k/ D .det ƒ/ t .y/ on Vt , i.e. jt .k/j D jt .y/j under k D ƒ1 y on Vt . 

6.2 Preliminaries

151

6.2.5 Some Additional Results on ı .Q .k/  t/ We now take t D m2 with m 2 Q p . Notice that Vm2 has infinitely many points and that .k0 ; k/ 2 Vm2 if and only if .k0 ; k/ 2 Vm2 . In order to exploit this symmetry we need a ‘notion of positivity’ on Qp . To motivate our definitions consider a D  pn an C pnC1 anC1 C    D pn ac .a/ 2 Q p , with ac .a/ 2 Zp , i.e. an ¤ 0, then a D .p  an / pn C .p  1  anC1 / pnC1 C    C .p  1  a0 / C .p  1  a1 / p C    D pn ac.a/: Thus, changing the sign of a is equivalent to changing the sign of its angular component. On the other hand, the x2 D a has two solutions if and only if equation

an  n is even and p D 1 , here p denotes the Legendre symbol. The condition

an D 1 means that the equation z2 an mod p has two solutions, say ˙z0 , p n o n o pC1 and z 2 ; : : : ; p  1 . because p ¤ 2, with z0 2 1; : : : ; p1 0 2 2 We define     p1 pC1    F ; : : : ; p  1  F D 1; : : : ; and F D FC p p p p. 2 2 Motivated by the above discussion we introduce the following notion of ‘positivity’. C Definition 157 We say that a 2 Q p is positive if ac.a/ 2 Fp , otherwise we declare a to be negative. We will use the notation a > 0, in the first case, and a < 0 in the second case.

The reader must be aware that this notion of positivity is not compatible with the arithmetic operations on Fp neither on Q p because these fields cannot be ordered. We now define the mass shells as follows: VmC2 D f.k0 ; k/ 2 Vm2 I k0 > 0g and Vm2 D f.k0 ; k/ 2 Vm2 I k0 < 0g : Hence Vm2 D VmC2

F

Vm2

F

f.k0 ; k/ 2 Vm2 I k0 D 0g :

Notice that VmC2 !

Vm2

.k0 ; k/ ! .k0 ; k/

(6.7)

152

6 Pseudodifferential Equations of Klein-Gordon Type

is a bijection. We define … W Q4p

! Q3p

.k0 ; k/ ! k;     and … VmC2 D … Vm2 WD UQ;m . Given k 2 UQ;m , there are two p-adic numbers, k0 > 0 and k0 < 0, such that .k0 ; k/, .k0 ; k/ 2 Vm2 , thus we can define the following two functions: UQ;m ! k

Q p

Q p

UQ;m !

p ! k  k C m2 DW k0

, k

p !  k  k C m2 DW k0 :

Furthermore, we obtain the following description of the sets Vm˙2 : o n p Vm˙2 D .k0 ; k/ 2 Q4p I k0 D ˙ k  k C m2 , for k 2 UQ;m :

(6.8)

Lemma 158 With the above notation the following assertions hold: (i) UQ;m is an open subset of Q3p ; p (ii) the functions ˙ k  k C m2 are p-adic analytic on UQ;m ; (iii) UQ;m is p-adic bianalytic equivalent to each Vm˙2 , and Vm˙2 are open subsets of Q3p ; (iv) m2 .k0 ; k/ jV ˙ D m2

dk1 ^ dk2 ^ dk3 jUQ;m and p ˙2 k  k C m2

d3 k ˇ jUQ;m ; jm2 .k0 ; k/j jV ˙ D ˇˇp ˇ m2 ˇ k  k C m2 ˇ p

where d3 k is the normalized Haar measure of Q3p ; Z   (v) ' .k0 ; k/ jm2 .k0 ; k/j D 0 for any ' .k0 ; k/ 2 D Q4p . f.k0 ;k/2Vm2 Ik0 D0g .j/

Proof Take a point .k0 ; k/ 2 VmC2 , then .k0 ; k/ 2 Vt for some j, see (6.5), thus 0 00 there exist an open compact subset UC D UC  UC containing .k0 ; k/ and a p-adic 00 0 analytic function hC W UC ! UC such that  ˚ 00 VmC2 \ U D .k0 ; k/ 2 UC I k0 D hC .k/ with k 2 UC :

(6.9)

6.2 Preliminaries

153

Now by (6.8), we have 00 D hC .k/ jUC

p 00 ; k  k C m 2 j UC

p which implies that k  k C m2 is a p -adic analytic function on UQ;m , which is an 00 open subset of Q3p since it is the union of all the UC which are open. In this way we establish (i)–(ii). We now prove (iii). By (ii) Vm˙2

UQ;m ! k

p

! ˙ k  k C m2 ; k DW i˙ .k/

are p-adic bianalytic mappings, and by (i) Vm˙2 are open subsets of Q3p . The formulas (iv)–(v) follow from (6.4) by a direct calculation.



Remark 159 Let X be a locally compact and totally disconnected topological space. We denote by D.X/ the set of all complex-valued and locally constant functions on X. Any such function is a linear combination of characteristic functions of compact open sets. The strong dual space D0 .X/ of D.X/ agrees with the algebraic dual of D.X/. For further details the reader may consult [57, Chapter 7]. Lemma 160 (i) Each of the spaces D.Vm˙2 / is isomorphic to D.UQ;m / as C-vector space. (ii) If W UQ;m ! C is a function with compact support, then Z UQ;m

d3 k ˇ D .k/ ˇp ˇ ˇ ˇ k  k C m2 ˇ p

Z

  ı i1 ˙ .k/ jm2 .k0 ; k/j .

V ˙2 m

4 (iii) If m 2 Q p and ' W Qp ! C is a function with compact support, then

Z

Z ' .k/ jm2 .k/j D

V 2 m

' UQ;m

Z C UQ;m

p

d3k ˇ k  k C m 2 ; k ˇp ˇ ˇ ˇ k  k C m2 ˇ

p

p

d3 k ˇ : '  k  k C m 2 ; k ˇp ˇ ˇ ˇ k  k C m2 ˇ

(6.10)

p

Proof (i) Let ˙ be a function in D.Vm˙2 /, by applying Lemma 158 (iii), we have ˙ ıi˙ 2 D.UQ;m /. Conversely, if ' 2 D.UQ;m /, then, by Lemma 158 (iii), ' ıi1 ˙ 2 D.Vm˙2 /. (ii) The formula follows from (i) by applying Lemma 158 (iv). (iii) The formula follows from (6.7) by applying (ii) and Lemma 158 (v). 

154

6 Pseudodifferential Equations of Klein-Gordon Type

Remark 161 We set 

   ı ˙ Q .k/  m2 ; ' WD

Z

' .k/ jm2 .k/j ; ' 2 D.Q4p /:

V ˙2 m

  Then ı ˙ Q .k/  m2 2 D0 .Q4p / and by(6.10),       ı Q .k/  m2 D ı C Q .k/  m2 C ı  Q .k/  m2 : 

1 0 Now, if we take ƒ0 WD 0 I33

 D ƒ1 0 2 O .Q/, then

    ı  Q .k/  m2 D ƒ0 ı C Q .k/  m2 :   Notice that instead of ƒ0 we can use any ƒ satisfying ƒ VmC2 D Vm2 .

6.2.6 The p-Adic Restricted Lorentz Group Definition 162 We define the p-adic restricted Lorentz group L"C to be the largest   subgroup of SO.Q/ such that L"C Vm˙2 D Vm˙2 . "

Notice that LC is a non-trivial subgroup of SO.Q/. Indeed, take ƒ in   ˚  SO .3/ D R 2 GL3 Qp I RT D R1 , det R D 1 ; and define   1 0 Q ƒD : 0ƒ Then    T  T 1 0 Q Q Gƒ Q D G, i.e. ƒ Q 2 SO.Q/, ƒ D and ƒ 0 ƒ1   Q  ƒk Q D k  k we have ƒ Q V ˙2 D V ˙2 . and since ƒk m m

6.3 A p-Adic Analog of the Klein-Gordon Equation

155

"

At the moment, we do not know if LC D f1g  SO.3/. It seems that this depends on Qp , which can be replaced for any locally compact field of characteristic different from 2.   Lemma 163 The distributions ı ˙ Q .k/  m2 are invariant under L"C . Proof Take ƒ 2 L"C , then 

Z

   ƒı ˙ Q .k/  m2 ; ' D

Z ' .ƒk/ jm2 .k/j D

V ˙2 m

' .k/ jm2 .k/j V ˙2 m

  because ƒ Vm˙2 D Vm˙2 and dm2 is invariant under any element of O.Q/, see proof of Lemma 156. 

6.3 A p-Adic Analog of the Klein-Gordon Equation Given a positive real number ˛ and a nonzero p-adic number m, we define the pseudodifferential operator       D Q4p ! C Q4p \ L2 Q4p '

!

˛;m ';

hˇ i ˇ 2 ˇ˛ ˇ where .˛;m '/ .x/ WD M1 k!x Œk; k  m p Mx!k ' .     We set EQ;m Q4p WD EQ;m to be the subspace of D0 Q4p consisting of the ˇ˛ ˇ   distributions T such that the product ˇŒk; k  m2 ˇp MT exists in D0 Q4p , here ˇ ˇ ˇ R ˇ ˇŒk; k  m2 ˇ˛ denotes the distribution ' ! 4 ˇŒk; k  m2 ˇ˛ ' .k/ d4 k. Notice that Qp p p   E Q4p , the space of locally constant functions, is contained in EQ;m . We consider   EQ;m as topological space with the topology inherited from D0 Q4p . Definition 164 A weak solution of   ˛;m T D S, with S 2 D0 Q4p ,   is a distribution T 2 EQ;m Q4p satisfying (6.11). For a subset U of Q4p we denote by 1U its characteristic function.

(6.11)

156

6 Pseudodifferential Equations of Klein-Gordon Type

  Lemma 165 Let T, S 2 D0 Q4p . The following assertions are equivalent:   (i) there exists W 2 D0 Q4p such that TS D WI (ii) for each x 2 Q4p , there exists an open compact subset U containing x so that for each each k 2 Q4p : Z M Œ1U W .k/ WD

M Œ1U T .l/ M Œ1U S .k  l/ d 4 l

Q4p

exists. Proof Any distribution is uniquely determined by its restrictions to any countable open covering of Qnp , see e.g. [111, p. 89]. On the other hand, the product TS exists if and only if M ŒT  M ŒS exists, and in this case M ŒTS D M ŒT  M ŒS, see e.g. [111, p. 115]. Assume that TS D W exists and take a countable covering fUi gi2N of Q4p by open and compact subsets, then TS jUi D W jUi i.e. 1Ui TS D 1Ui W. We recall that the product of a finite number of distributions involving at least one distribution with compact support is associative and commutative, see e.g. [105, Theorem 3.19], then W jUi D 1Ui TS D 1Ui .1Ui TS/ D .1Ui T/ .1Ui S/ D T jUi S jUi : Now for each x 2 Q4p , there exists an open compact subset Ui containing x such that M ŒT jUi   M ŒS jUi  D M Œ1Ui T  M Œ1Ui S D M Œ1Ui W. Conversely, if for each x there exists an open compact subset Ui containing x (from this we get countable subcovering of Q4p also denoted as fUi gi2N ) such that M ŒT jUi   M ŒS jUi  D M ŒW jUi  i.e. T jUi S jUi D W jUi exists, then TS D W.  Corollary 166 If TS exists, then supp.TS/ supp .T/ \supp.S/. Proof Since x …supp.S/, there exists a compact open set U containing x such .S; '/ D 0 for any ' 2 D.U/, hence 1U S D 0, and M Œ1U T  M Œ1U S D 0 D M Œ1U W, i.e. W jU D 0, which means x …supp.W/.  Remark 167 Lemma 165 and Corollary 166 are valid in arbitrary dimension. These results are well-known in the Archimedean setting, see e.g. [96, Theorem IX.43], however, such results do not appear in the standard books of p-adic analysis, see e.g. [5, 80, 105, 111]. Remark 168

  (i) Let  denote the characteristic function of the interval Œ0; 1. Then  pj kxkp .n/

is the characteristic function of the ball Bj .0/. Let us recall definition of the   product of two distributions. Set ı j .x/ WD pnj  pj kxkp for j 2 N. Given

6.3 A p-Adic Analog of the Klein-Gordon Equation

157

  T; S 2 D0 Qnp , their product TS is defined by .TS; '/ D lim

j!C1

    S; T  ı j '

  if the limit exists for all ' 2 D Qnp . (ii) We assert that ˇ

ˇ ˇ ˇ

ˇŒk; k  m2 ˇ˛ MT; ' D MT; ˇŒk; k  m2 ˇ˛ ' p

p

    for any T 2 EQ;m Q4p and any ' 2 D Qnp . Indeed, by using the fact that Vm2 has d4 y-measure zero, ˇ

ˇ ˇ ˇ ˇŒk; k  m2 ˇ˛  ı j .k/ D ˇŒy; y  m2 ˇ˛ ; ı j .k  y/ p

p

Z

D p4j

ˇ ˇ ˇŒy; y  m2 ˇ˛ d4 y p

kC.

p/ Z

pj Z

D p4j

4

ˇ ˇ ˇ ˇ ˇŒy; y  m2 ˇ˛ d4 y D ˇŒk; k  m2 ˇ˛ p

p

4

kC.pj Zp / XVm2

for j big enough depending on k. Then ˇ

i

hˇ ˇ ˇ ˇŒk; k  m2 ˇ˛ MT; ' D lim MT .k/ ; ˇŒk; k  m2 ˇ˛  ı j .k/ ' .k/ p p j!C1



ˇ ˇ˛ D MT .k/ ; ˇŒk; k  m2 ˇp ' .k/ :   Lemma 169 A distribution T 2 EQ;m Q4p is a weak solution of ˛;m T D 0 if and only if suppMT  Vm2 . Proof Suppose that suppMT  Vm2 , then by Corollary 166, we have ˇ

ˇ ˇ˛ ˇ˛

supp ˇŒk; k  m2 ˇp MT  supp .MT/ \ supp ˇŒk; k  m2 ˇp D ; ˇ ˇ˛ ˇ ˇ˛ because ˇŒk; k m2 ˇp D ˇŒk; k m2 ˇp 1Q4p XVm2 in D0 .Q4p / (Vm2 has d4 k-measure zero)

ˇ ˇ ˇ˛ ˇ˛ and supp ˇŒk; k m2 ˇp 1Q4p XVm2  Q4p X Vm2 , therefore ˇŒk; k m2 ˇp MTD0.

158

6 Pseudodifferential Equations of Klein-Gordon Type

ˇ ˇ˛ Suppose now that ˇŒk; k  m2 ˇp MT D 0. By contradiction, assume that suppMT ª Vm2 . Then , there exists s0 2 Q4p X Vm2 and an open compact subset U  Q4p X Vm2 containing s0 such that .MT; 1U / ¤ 0. By using Remark 168-(ii) and by shrinking U if necessary, ˇ



ˇ ˇ ˇ ˇŒk; k  m2 ˇ˛ MT; 1U D MT; ˇŒk; k  m2 ˇ˛ 1U p

p

ˇ˛ ˇ D ˇŒs0 ; s0   m2 ˇ .MT; 1U / ¤ 0; p

ˇ ˇ˛ contradicting ˇŒk; k  m2 ˇp MT D 0.    Remark 170 Let ' 2 D Q4p and let ƒ 2 L"C , a Lorentz transformation. By using that Œk; x D 12 fQ .k C x/  Q .k/  Q .x/g, we have M Œ' .ƒx/ .k/ D M Œ' .ƒk/ and ƒM ŒT D M ŒƒT : Hence the Fourier transform preserves Lorentz invariance, or more generally, the Fourier transform preserves invariance under O .Q/. Proposition 171 The distributions     M ŒT .k/ D aıC Q .k/  m2 C bı  Q .k/  m2 , a; b 2 C, are weak solutions of ˛;m T D 0 invariant under L"C .

  Proof By Remark 170,ˇ it is sufficient to show that ı ˙ Q .k/  m2 are invariant ˇ ˛ solutions of ˇŒk; k  m2 ˇp M ŒT D 0, which follows from Lemmas 163–169. 

At this point we should mention that a similar results to Lemmas 163–169 and Proposition 171 are valid for the Archimedean Klein-Gordon equation, see e.g. [31] or [32, Chapter IV].

6.4 The Cauchy Problem for the Non-Archimedean Klein-Gordon Equation In this section we study the Cauchy problem for the p-adic Klein-Gordon equations.

6.4 The Cauchy Problem for the Non-Archimedean Klein-Gordon Equation

159

6.4.1 Twisted Vladimirov Pseudodifferential Operators Let C 1 denote the multiplicative group of complex numbers having modulus one.   Let  1 W Z p ! C1 be a non-trivial multiplicative character of Zp with positive conductor k, i.e. k is the smallest positive integer such that  1 j1Cpk Zp D 1. Some  authors call a such character a unitary character of Z p . We extend  1 to Qp  by putting  1 .x/ WD  1 .ac .x//. A quasicharacter of Qp (some authors use  multiplicative character) is a continuous homomorphism from Q p into C . Every quasicharacter has the form  s .x/ D  1 .x/ jxjs1 for some complex number s. p The distribution associated with  s .x/ has a meromorphic continuation to the whole complex plane given by Z . s .x/ ; ' .x// D

 1 .x/ jxjs1 p f' .x/  ' .0/g dx Zp

Z  1 .x/ jxjs1 p ' .x/ dx;

C

(6.12)

Qp XZp

see e.g. [111, p. 117]. On the other hand, s F Πs  ./ D p .s;  1 /  1 1 ./ jjp , for any s 2 C,

(6.13)

where p .s;  1 / D psk ap;k . 1 / , Z ˇ ˇ   k ap;k . 1 / D  1 .t/ p pk t dt and ˇap;k . 1 /ˇ D p 2 , Z p

see e.g. [111, p. 124]. Another useful formula is the following: Z . s .x/ ; ' .x// D

 1 .x/ f' .x/  ' .0/g jxjsC1 p

Qp

  dx, for Re.s/ > 0, and ' 2 D Qp . (6.14)

The formula follows from (6.12) by using that Z Qp XZp

0  1 .x/ jxjsC1 p

dx D @

1 X jD1

1 p

js A

Z

Z p

 1 .y/ dy D 0:

(6.15)

160

6 Pseudodifferential Equations of Klein-Gordon Type

Given ˛ > 0, we define the twisted Vladimirov operator by ˛

    Q ' .x/ D F 1  1 .k/ jkj˛ Fx!k .'/ for ' 2 D Qp . D x p k!x 1 Notice that     D Qp ! C Qp ; C \ L2 '

!



Q ˛' D x



is a well-defined linear operator.

  Lemma 172 For ˛ > 0 and ' 2 D Qp , the following formula holds:

Q ˛x ' .x/ D D

1 p .˛;  1 /

Z

Qp

 1 .y/ f' .x  y/  ' .x/g jyj˛C1 p

dy:

(6.16)

Proof By using (6.13), we have ˛

 ˛ Q x ' .x/ D F 1  1 D 1 .k/ jkjp .x/  ' .x/ D !x

1  ˛ .x/  ' .x/ p .˛;  1 /

1 . ˛ .y/ ; ' .x  y// p .˛;  1 / Z 1  1 .y/ f' .x  y/  ' .x/g D dy p .˛;  1 / jyj˛C1 p D

Zp

C

1 p .˛;  1 /

Z

Qp XZp

D

1 p .˛;  1 /

Z

jyj˛C1 p

1 p .˛;  1 /

Z

Qp XZp

where we used (6.15).

jyj˛C1 p

dx

 1 .y/ f' .x  y/  ' .x/g

Zp

C

 1 .y/ ' .x  y/

dy

 1 .y/ f' .x  y/  ' .x/g jyj˛C1 p

dy;



6.4 The Cauchy Problem for the Non-Archimedean Klein-Gordon Equation

161

Notice that the right-hand side of (6.16) makes sense for a wider class of functions. For instance, for any locally constant function u .x/ satisfying Z

ju .x/j jxj˛C1 p

Qp XZp

dx < 1.

Another useful formula is the following: Lemma 173 For ˛ > 0, we have  1 1

.x/ jxj˛p

1 D p .˛;  1 /

Z

˚   1 .y/ p .yx/  1 jyj˛C1 p

Qp

  dy in D0 Qp :

Proof The formula follows from (6.13) and (6.14). The proof is a simple variation of the proof given for the case in which  1 is the trivial character, see e.g. [80, Proposition 2.3].  From now on we put ac .x/  1 .x/ WD i p

! for x 2 Z p;



where z denotes the reduction mod p of z 2 Zp , p denotes the Legendre symbol. Notice that the conductor of  1 is 1 and  1 .x/ 2 f˙ig, furthermore,

1 p

D

8 < 1; if p  1 is divisible by 4 :

1 if p  3 is divisible by 4:

6.4.2 The Cauchy Problem for the p-Adic Klein-Gordon Equation In this section we take x0 D t and .x0 ; x/ D .t; x/ 2 Qp  Q3p . Our goal is to study the following Cauchy problem: 8 ˆ .˛;m u/ .t; x/ D 0; ˆ ˆ ˆ ˆ < u .t; x/ jtD0 D 0 .x/ ; ˆ ˆ ˆ ˆ ˆ : Q˛ Dx u .t; x/ jtD0 D 1 .x/ ;

(A) 0

2 D.Q3p / and F 1 Œ

0

1

2 D.Q3p / and F 1 Œ

1

  2 D UQ;m (B)   2 D UQ;m (C). (6.17)

162

6 Pseudodifferential Equations of Klein-Gordon Type

Theorem 174 Assume that p  3 is divisible by 4. Then Cauchy problem (6.17) has a weak solution given by Z u .t; x/ D UQ;m

p

d3k ˇ p t k  k C m2 C x  k uC .k/ ˇp ˇ ˇ ˇ k  k C m2 ˇ

Z C UQ;m

p

p

d3 k ˇ ; p t k  k C m2 C x  k u .k/ ˇp ˇ ˇ ˇ k  k C m2 ˇ

(6.18)

p

where 8 ˆ ˆ ˇ 1 > = > > ;

;

and 8 ˆ ˆ ˇ 1 > = > > ;

:

Proof We first show that u .t; x/ is a weak solution of (6.17)-(A). By applying Lemmas 158–160, we have Z UQ;m

p

d3 k ˇ p t k  k C m2 C x  k u˙ .k/ ˇp ˇ ˇ ˇ k  k C m2 ˇ Z D

p

  p .Œ.k0 ; k/ ; .t; x// u˙ i1 ˙ .k/ dm2 .k0 ; k/

V ˙2 m

D M1 .k0 ;k/!.t;x/



   2 u˙ ı i1 ; ˙ .k/ ı ˙ Q .k/  m

whence M.t;x/!.k0 ;k/ Œu .t; x/ D

    2 uC ı i1 C .k/ ı C Q .k/  m     2 : C u ı i1  .k/ ı  Q .k/  m

6.4 The Cauchy Problem for the Non-Archimedean Klein-Gordon Equation

163

By Lemma 169, u .t; x/ is a weak solution of (6.22), if     2  V m2 : supp u˙ ı i1 ˙ .k/ ı ˙ Q .k/  m This last condition is verified by applying Corollary 166 and the fact that      ˙ 2  Vm˙2  Vm2 : supp u˙ ı i1 ˙  Vm2  Vm2 and supp ı ˙ Q .k/  m The verification of (6.17)-(B) is straight forward. To verify (6.17)-(C) we proceed as follows. By using Lemma 172, Fubini’s theorem and Lemma 173, we get the following formula: 2 3 Z

3 p d k Q˛6 ˇ 7 p t k  k C m2 C x  k u˙ .k/ ˇp D t 4 ˇ ˇ 5 ˇ k  k C m2 ˇ U Q;m

p

p

Z  t k  k C m2 C x  k u˙ .k/ p ˇp ˇ D ˇ 2 ˇˇ k  k C m ˇ UQ;m p 8

o 9 n p ˆ > Z  1 .y/  ˙y k  k C m2  1 < = p 1 d3 k  dy ˆ > jyj˛C1 : p .˛;  1 / ; p Qp

D  1 1 .˙1/

Z

p p



p t k  k C m2 C x  k  1 k  k C m2 1

UQ;m

ˇp ˇ˛1 ˇ ˇ  ˇ k  k C m2 ˇ u˙ .k/ d3 k: p

Now Condition (6.17)-(C) follows from the previous formula.



Remark 175 (i) Note that the condition ‘p  3 is divisible by 4’ is required only to establish (6.17)-(C). At the moment, we do not know if this condition is necessary to have (6.17)-(C). (ii) The parameter ˛ does not have any influence on the solutions of (6.22). Theorem 176 The equation   .˛;m u/ .t; x/ D J .t; x/ ; J .t; x/ 2 D Q4p

(6.19)

164

6 Pseudodifferential Equations of Klein-Gordon Type

admits the following weak solution: u .t; x/ D E˛ .t; x/  J .t; x/ Z p

d3 k ˇ p t k  k C m2 C x  k uC .k/ ˇp C ˇ 2 ˇˇ k  k C m ˇ U Q;m

Z C UQ;m

p

p

d3 k ˇ ; p t k  k C m2 C x  k u .k/ ˇp ˇ ˇ ˇ k  k C m2 ˇ

(6.20)

p

  where E˛ .t; x/ is a distribution on S Q4p satisfying ˇ ˇ ˇŒk; k  m2 ˇ˛ M p

t ! k0 x!k

  ŒE˛ .t; x/ D 1 in D0 Q4p ,

(6.21)

  and uC .k/, u .k/ are arbitrary functions in D UQ;m . Proof Like in the classical case a solution of (6.19) is computed as u0 .t; x/Cu1 .t; x/ with u0 .t; x/ a particular solution of (6.19) and u1 .t; x/ a general solution of .˛;m u1 / .t; x/ D 0:

(6.22)

The existence of a fundamental solution for (6.19), i.e. a distribution E˛ .t; x/ is a  distribution on D Q4p such that E˛ .t; x/  J.t; x/ is a weak solution of (6.19), was established in Theorem 134 in Chap. 6. This fundamental solution satisfies (6.21). Finally, we verify that Z u1 .t; x/ D UQ;m

p

d3 k ˇ p t k  k C m2 C x  k uC .k/ ˇp ˇ ˇ ˇ k  k C m2 ˇ

Z C UQ;m

p

p

d3 k ˇ ; p t k  k C m2 C x  k u .k/ ˇp ˇ ˇ ˇ k  k C m2 ˇ

satisfies (6.22) as in the proof of Theorem 174.

p



Like in the Archimedean case the non-Archimedean Klein-Gordon equations admit plane waves. p Lemma 177 Let .E; p/ 2 Vm˙2 , i.e. E D ˙ p  p C m2 . Then u .t; x/ D p .Œ.t; x/ ; .E; p// is a weak solution of .˛;m u/ .t; x/ D 0. Proof (i) Since M.t;x/!.k0 ;k/ Œu .t; x/ D ı .k0  E; k  p/, the results follows from Lemma 169. 

6.5 Further Results on the p-Adic Klein-Gordon Equation

165

6.5 Further Results on the p-Adic Klein-Gordon Equation In this section we change the notation slightly, this facilitates the comparison with the classical results and constructions. Set p ! .k/ WD k  k C m2 for k 2 UQ;m : The function ! .k/ is a p-adic analytic function on UQ;m , cf. Lemma 158, and ! .k/ ¤ 0 for any k 2 UQ;m . Then, by Taylor formula, j! .k/jp is a locally constant     function on UQ;m , and if ˙ 2 D UQ;m , then j! .k/j˙1 p ˙ .k/ 2 D UQ;m . Then Z .t; x/ D

˚  p .x  k/ p .t! .k// C .k/ C p .t! .k//  .k/ d 3 k

UQ;m

(6.23)   is a weak solution of .˛;m / .t; x/ D 0 for any ˙ 2 D UQ;m . Note that if we replace p .x  k/ by p .x  k/ in (6.23) we get another weak solution. As in the classical case, for the quantum interpretation we specialize to ‘positive energy solutions’ which have  .k/ D 0. Then the solution is determined by a single complex valued initial condition. The solution (6.23) with initial condition ‰ 2 L2Q;m WD L2

˚

    ˆ 2 L2 Q3p I supp .F ˆ/  UQ;m ; d3 k

-where the condition ‘supp.F ˆ/  UQ;m ’ means that there exists a function ˆ0 in the equivalence class containing F ˆ such that supp.ˆ0 /  UQ;m - is given by Z ‰ .t; x/ D

p .t! .k/  x  k/ .F ‰/ .k/ d3 k:

(6.24)

UQ;m

Set   U .t/ W L2Q;m ! L2 Q3p ‰

  1 p .t! .k// Fx!k ‰ ; ! ‰ .t; x/ D Fk!x

for t 2 Qp . Lemma 178 (i) U .t/, t 2 Qp is a group of unitary operators on L2Q;m . (ii) st:  limt!0 U .t/ D I. Proof It is a straightforward calculation.



Chapter 7

Final Remarks and Some Open Problems

The theory pseudodifferential equations over p-adic fields is just beginning. There are several open problems connecting these equations with number theory, probability and physics. Here we want to pose some of them.

7.1 Zeta Functions, Adelic Riesz Kernels and Heat Equations We use all the notation introduced in Chap. 4. We propose to study the meromorphic continuation and the existence of functional equations, i.e. to compute explicitly the Fourier transform, of the following distributions: Z & Af .s; '/ WD

' .x/ kxks dxAf ,

(7.1)

  ˚  s ' xf ; x1 xf  C jx1 j1 dxA ,

(7.2)

Af nfx2Af IkxkD0g

  ' 2 D Af , s 2 C with Re.s/ > 0, and Z & A .s; / WD Anfx2AIkxf kCjx1 j1 D0g

' 2 D .A/, s 2 C with Re.s/ > 0. Notice that kxkRe.s/ , with Re.s/ > 0, is a continuous function on Af , because Af ; is a complete metric space with R .x; y/ D kx  yk, x, y 2 Af , and Af j' .x/j dxAf < 1 due to the fact that j' .x/j is a linear combination of characteristic functions of balls and the dxAf -measure of a   ball is finite. Therefore & Af .s; '/ converges for Re.s/ > 0 and any ' 2 D Af . A similar reasoning works for & A .s; /. These distributions are ‘additive’ adelic Riesz © Springer International Publishing AG 2016 W.A. ZúQniga-Galindo, Pseudodifferential Equations Over Non-Archimedean Spaces, Lecture Notes in Mathematics 2174, DOI 10.1007/978-3-319-46738-2_7

167

168

7 Final Remarks and Some Open Problems

  Q kernels. Take ' 0 .x/ D p 0, j0 p