DOCSLIB.ORG

Lecture Notes on Harmonic Analysis Updated: April 28, 2020 © Chengchun HAO Institute of Mathematics AMSS CAS - Iii - 4.4

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes on Harmonic Analysis Updated: April 28, 2020 © Chengchun HAO Institute of Mathematics AMSS CAS - Iii - 4.4 GRADUATE LECTURES IN ANALYSIS LLECTUREECTURE NNOTESOTES ON ON HHARMONICARMONIC AANALYSISNALYSIS CHENGCHUN HAO INSTITUTE OF MATHEMATICS LECTURE NOTES ON HARMONIC ANALYSIS Chengchun Hao Institute of Mathematics, AMSS Chinese Academy of Sciences Email: [email protected] Updated: April 28, 2020 Preface Harmonic analysis, as a subfield of analysis, is particularly interested in the study of quantitative properties on functions, and how these quantitative proper- ties change when apply various operators. In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience. Most of the material in these notes are excerpted from the book of Stein [Ste70], the book of Stein and Weiss [SW71], the books of Grafakos [Gra14a, Gra14b] and the book of Wang-Huo-Hao-Guo [WHHG11], etc. with some necessary modification. Please email me ([email protected]) with corrections or suggested improvements of any kinds. Chengchun Hao Beijing April 28, 2020 References [Gra14a] Loukas Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014. [Gra14b] Loukas Grafakos. Modern Fourier analysis, volume 250 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014. [Ste70] Elias M. Stein. Singular integrals and differentiability properties of func- tions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. [SW71] Elias M. Stein and Guido Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. [WHHG11] Baoxiang Wang, Zhaohui Huo, Chengchun Hao, and Zihua Guo. Har- monic analysis method for nonlinear evolution equations, volume I. World Scientific Publishing Co. Pte. Ltd., 2011. Contents Preface i 1. The Weak Lp Spaces and Interpolation 1 1.1. The distribution function and weak Lp .................. 1 1.2. Complex method: Riesz-Thorin and Stein interpolation theorems .. 9 1.2.1. Riesz-Thorin interpolation theorem ............... 9 1.2.2. Stein interpolation theorem .................... 14 1.3. The decreasing rearrangement and Lorentz spaces ........... 19 1.4. Real method: Marcinkiewicz’ interpolation theorem .......... 25 References ..................................... 29 2. Fourier Transform and Tempered Distributions 31 2.1. Fourier transform of L1 functions .................... 31 2.1.1. The definition and properties ................... 31 2.1.2. Riemann-Lebesgue lemma .................... 34 2.1.3. Approximate identities ....................... 35 2.1.4. Fourier inversion .......................... 42 2.2. Fourier transform on Lp for 1 < p 6 2 .................. 45 2.3. The class of Schwartz functions ...................... 48 2.4. The class of tempered distributions .................... 54 2.5. Characterization of operators commuting with translations ...... 66 2.6. Fourier multipliers on Lp .......................... 72 References ..................................... 77 3. The Maximal Function and Calderón-Zygmund Decomposition 79 3.1. Two covering lemmas ........................... 79 3.2. Hardy-Littlewood maximal function ................... 81 3.2.1. Hardy-Littlewood maximal operator ............... 82 3.2.2. Control of other maximal operators ............... 88 3.2.3. Applications to differentiation theory .............. 89 3.2.4. An application to Sobolev’s inequality .............. 92 3.3. Calderón-Zygmund decomposition ................... 94 References ..................................... 99 4. Singular Integrals of Convolution Type 101 4.1. Poisson kernel and Hilbert transform ................... 101 4.1.1. Poisson kernel and the conjugate ................. 101 4.1.2. Hilbert transform .......................... 105 4.1.3. Lp boundedness of Hilbert transform .............. 109 4.1.4. The maximal Hilbert transform and its Lp boundedness ... 111 4.2. Calderón-Zygmund singular integrals .................. 114 4.3. L2 boundedness of homogeneous singular integrals .......... 120 Lecture Notes on Harmonic Analysis Updated: April 28, 2020 © Chengchun HAO Institute of Mathematics AMSS CAS - iii - 4.4. Riesz transforms and spherical harmonics ................ 125 4.4.1. Riesz transforms .......................... 125 4.4.2. Spherical harmonics and higher Riesz transforms ....... 130 4.4.3. Equivalence between two classes of transforms ........ 139 4.5. The method of rotations and singualr integral with odd kernels ... 141 4.6. Singular integral operators with Dini-type condition .......... 144 4.6.1. Lp boundedness of homogeneous singular integrals ...... 144 4.6.2. The maximal singular integral operator ............. 146 4.7. Vector-valued analogues .......................... 150 References ..................................... 153 5. Littlewood-Paley Theory and Multiplier Theorems 155 5.1. Three approach functions and Lp boundedness ............. 155 5.2. Mikhlin and Hörmander multiplier theorem .............. 166 5.3. The partial sums operators ......................... 170 5.4. The dyadic decomposition ......................... 173 5.5. Marcinkiewicz multiplier theorem .................... 180 References ..................................... 183 6. Smoothness and Function Spaces 185 6.1. Riesz potentials and fractional integrals ................. 185 6.2. Bessel potentials ............................... 189 6.3. Sobolev spaces ................................ 192 6.4. The smooth dyadic decomposition .................... 198 6.5. Besov spaces and Triebel-Lizorkin spaces ................ 205 6.6. Embedding and interpolation of spaces ................. 210 6.7. Differential-difference norm on Besov spaces .............. 214 6.8. The realization of homogeneous Besov spaces for PDEs ........ 218 6.9. Hölder spaces ................................ 228 References ..................................... 236 7. BMO and Carleson Measures 237 7.1. The sharp maximal functions and BMO spaces ............. 237 7.2. John-Nirenberg theorem .......................... 243 7.3. Non-tangential maximal functions and Carleson measures ...... 248 7.4. BMO functions and Carleson measures ................. 253 References ..................................... 255 Index 257 I The Weak Lp Spaces and Interpolation 1.1. The distribution function and weak Lp ............... 1 1.2. Complex method: Riesz-Thorin and Stein interpolation theorems 9 1.2.1. Riesz-Thorin interpolation theorem ............. 9 1.2.2. Stein interpolation theorem .................. 14 1.3. The decreasing rearrangement and Lorentz spaces ........ 19 1.4. Real method: Marcinkiewicz’ interpolation theorem ....... 25 References ................................... 29 § 1.1 The distribution function and weak Lp In this chapter, we will consider general measurable functions in measurable spaces instead of Rn only. A σ-algebra on a set X is a collection of subsets of X that includes the empty subset, is closed under complement, countable unions and countable intersections. A measure space is a set X equipped with a σ-algebra of subsets of it and a function µ σ [0; 1] µ(?) = 0 from the -algebra to that0 satisfies1 and [1 X1 @ A µ Bj = µ(Bj) j=1 j=1 for any sequence fBjg of pairwise disjoint elements of the σ-algebra. The function µ is called a positive measure on X and elements of the σ-algebra of X are called measurable sets. Measure spaces will be assumed to be complete, i.e., subsets of the σ-algebra of measure zero also belong to the σ-algebra. A measure space X isS called σ-finite if there is a sequence of measurable subsets 1 1 Xn of it such that X = n=1 Xn and µ(Xn) < . A real-valued function f on a measure space is called measurable if the set fx 2 X : f(x) > λg is measurable for all real numbers λ. A complex-valued function is measurable if and only if its real and imaginary parts are measurable. We adopt the usual convention that two functions are considered equal if they agree except on a set of µ-measure zero. For p 2 [1; 1), we denote by Lp(X; dµ) (or simply Lp(dµ), Lp(X) or even Lp) the Lebesgue-space of (all equivalence classes of) scalar-valued µ-measurable functions f on X, such that Z 1/p p kfkp = jf(x)j dµ X 1 is finite. For p = 1, L consists of all µ-measurable and bounded functions. Then we write kfk1 = ess sup jf(x)j = inffB > 0 : µ(fx : jf(x)j > Bg) = 0g: X Lecture Notes on Harmonic Analysis Updated: April 28, 2020 © Chengchun HAO Institute of Mathematics AMSS CAS - 1 - - 2 - Chengchun HAO It is well-known that Lp(X; µ) is a Banach space for any p 2 [1; 1] (i.e., the Riesz-Fisher theorem). For any p 2 (1; 1), we define the Hölder conjugate number 0 p 0 1 10 0 0 2 1 p = p−1 . Moreover, we set 1 = and = 1, so that (p ) = p for all p [1; ]. Hölder’s inequality says that for all p 2 [1; 1] and all measurable function f, g on (X; µ), we have kfgk1 6 kfkpkfkp0 : 0 0 It is also a well-known fact that the dual (Lp) of Lp is isometric to Lp for all p 2 (1; 1) and also when p = 1 if X is σ-finite. Furthermore, the Lp norm of a function can be obtained via duality when p 2 (1; 1) as follows: Z kfkp = sup fgdµ : k k g p0 =1 X The endpoint cases p = 1; 1 also work if X is σ-finite. It is often convenient to work with functions that are only locally in some Lp space. We give the definition in the following. Definition 1.1.1. 2 1 p p For p [1; ), the space Lloc(X; µ) or simply Lloc(X) is the set of all Rn Lebesgue measurable functionsZf on that satisfy jfjpdµ < 1 (1.1.1) K for any compact subset K ⊂ X. Functions that satisfy
Load more

Recommended publications
  • HARMONIC ANALYSIS AS the EXPLOITATION of SYMMETRY-A HISTORICAL SURVEY Preface 1. Introduction 2. the Characters of Finite Groups
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY-A HISTORICAL SURVEY BY GEORGE W. MACKEY CONTENTS Preface 1. Introduction 2. The Characters of Finite Groups and the Connection with Fourier Analysis 3. Probability Theory Before the Twentieth Century 4. The Method of Generating Functions in Probability Theory 5. Number Theory Before 1801 6. The Work of Gauss and Dirichlet and the Introduction of Characters and Harmonic Analysis into Number Theory 7. Mathematical Physics Before 1807 8. The Work of Fourier, Poisson, and Cauchy, and Early Applications of Harmonic Analysis to Physics 9. Harmonic Analysis, Solutions by Definite Integrals, and the Theory of Functions of a Complex Variable 10. Elliptic Functions and Early Applications of the Theory of Functions of a Complex Variable to Number Theory 11. The Emergence of the Group Concept 12. Introduction to Sections 13-16 13. Thermodynamics, Atoms, Statistical Mechanics, and the Old Quantum Theory 14. The Lebesgue Integral, Integral Equations, and the Development of Real and Abstract Analysis 15. Group Representations and Their Characters 16. Group Representations in Hilbert Space and the Discovery of Quantum Mechanics 17. The Development of the Theory of Unitary Group Representations Be­ tween 1930 and 1945 Reprinted from Rice University Studies (Volume 64, Numbers 2 and 3, Spring- Summer 1978, pages 73 to 228), with the permission of the publisher. 1980 Mathematics Subject Classification. Primary 01,10,12, 20, 22, 26, 28, 30, 35, 40, 42, 43, 45,46, 47, 60, 62, 70, 76, 78, 80, 81, 82.
    [Show full text]
  • Harmonic Analysis from Fourier to Wavelets
    STUDENT MATHEMATICAL LIBRARY ⍀ IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 63 Harmonic Analysis From Fourier to Wavelets María Cristina Pereyra Lesley A. Ward American Mathematical Society Institute for Advanced Study Harmonic Analysis From Fourier to Wavelets STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 63 Harmonic Analysis From Fourier to Wavelets María Cristina Pereyra Lesley A. Ward American Mathematical Society, Providence, Rhode Island Institute for Advanced Study, Princeton, New Jersey Editorial Board of the Student Mathematical Library Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell Series Editor for the Park City Mathematics Institute John Polking 2010 Mathematics Subject Classification. Primary 42–01; Secondary 42–02, 42Axx, 42B25, 42C40. The anteater on the dedication page is by Miguel. The dragon at the back of the book is by Alexander. For additional information and updates on this book, visit www.ams.org/bookpages/stml-63 Library of Congress Cataloging-in-Publication Data Pereyra, Mar´ıa Cristina. Harmonic analysis : from Fourier to wavelets / Mar´ıa Cristina Pereyra, Lesley A. Ward. p. cm. — (Student mathematical library ; 63. IAS/Park City mathematical subseries) Includes bibliographical references and indexes. ISBN 978-0-8218-7566-7 (alk. paper) 1. Harmonic analysis—Textbooks. I. Ward, Lesley A., 1963– II. Title. QA403.P44 2012 515.2433—dc23 2012001283 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
    [Show full text]
  • Lecture Notes: Harmonic Analysis
    Lecture notes: harmonic analysis Russell Brown Department of mathematics University of Kentucky Lexington, KY 40506-0027 August 14, 2009 ii Contents Preface vii 1 The Fourier transform on L1 1 1.1 Definition and symmetry properties . 1 1.2 The Fourier inversion theorem . 9 2 Tempered distributions 11 2.1 Test functions . 11 2.2 Tempered distributions . 15 2.3 Operations on tempered distributions . 17 2.4 The Fourier transform . 20 2.5 More distributions . 22 3 The Fourier transform on L2. 25 3.1 Plancherel's theorem . 25 3.2 Multiplier operators . 27 3.3 Sobolev spaces . 28 4 Interpolation of operators 31 4.1 The Riesz-Thorin theorem . 31 4.2 Interpolation for analytic families of operators . 36 4.3 Real methods . 37 5 The Hardy-Littlewood maximal function 41 5.1 The Lp-inequalities . 41 5.2 Differentiation theorems . 45 iii iv CONTENTS 6 Singular integrals 49 6.1 Calder´on-Zygmund kernels . 49 6.2 Some multiplier operators . 55 7 Littlewood-Paley theory 61 7.1 A square function that characterizes Lp ................... 61 7.2 Variations . 63 8 Fractional integration 65 8.1 The Hardy-Littlewood-Sobolev theorem . 66 8.2 A Sobolev inequality . 72 9 Singular multipliers 77 9.1 Estimates for an operator with a singular symbol . 77 9.2 A trace theorem. 87 10 The Dirichlet problem for elliptic equations. 91 10.1 Domains in Rn ................................ 91 10.2 The weak Dirichlet problem . 99 11 Inverse Problems: Boundary identifiability 103 11.1 The Dirichlet to Neumann map . 103 11.2 Identifiability . 107 12 Inverse problem: Global uniqueness 117 12.1 A Schr¨odingerequation .
    [Show full text]
  • Radial Basis Functions
    Acta Numerica (2000), pp. 1–38 c Cambridge University Press, 2000 Radial basis functions M. D. Buhmann Mathematical Institute, Justus Liebig University, 35392 Giessen, Germany E-mail: [email protected] Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper. CONTENTS 1 Introduction 1 2 Convergence rates 5 3 Compact support 11 4 Iterative methods for implementation 16 5 Interpolation on spheres 25 6 Applications 28 References 34 1. Introduction There is a multitude of ways to approximate a function of many variables: multivariate polynomials, splines, tensor product methods, local methods and global methods. All of these approaches have many advantages and some disadvantages, but if the dimensionality of the problem (the number of variables) is large, which is often the case in many applications from stat- istics to neural networks, our choice of methods is greatly reduced, unless 2 M. D. Buhmann we resort solely to tensor product methods.
    [Show full text]
  • Harmonic Analysis
    אוניברסיטת תל אביב University Aviv Tel הפקולטה להנדסה Engineering of Faculty בית הספר להנדסת חשמל Engineering Electrical of School Harmonic Analysis Fall Semester LECTURER Nir Sharon Office: 03-640-7729 Location: Room 14M, Schreiber Bldg E-mail: [email protected] COURSE DESCRIPTION The aim of this course is to introduce the fundamentals of Harmonic analysis. In particular, we focus on three main subjects: the theory of Fourier series, approximation in Hilbert spaces by a general orthogonal system, and the basics of the theory of Fourier transform. Harmonic analysis is the study of objects (functions, measures, etc.), defined on different mathematical spaces. Specifically, we study the question of finding the "elementary components" of functions, and how to analyse a given function based on its elementary components. The trigonometric system of cosine and sine functions plays a major role in our presentation of the theory. The course is intended for undergraduate students of engineering, mathematics and physics, although we deal almost exclusively with aspects of Fourier analysis that are usefull in physics and engeneering rather than those of pure mathematics. We presume knowledge in: linear algebra, calculus, elementary theory of ordinary differential equations, and some acquaintance with the system of complex numbers. COURSE TOPICS Week 1: Fourier Series of piecewise continuous functions on a symmetrical segment. Complex and real representations of the Fourier Series. Week 2: Bessel Inequality, the Riemann-Lebesgue Lemma, partial
    [Show full text]
  • Harmonic Analysis in Euclidean Spaces
    http://dx.doi.org/10.1090/pspum/035.1 HARMONIC ANALYSIS IN EUCLIDEAN SPACES Part 1 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XXXV, Part 1 HARMONIC ANALYSIS IN EUCLIDEAN SPACES AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1979 PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY HELD AT WILLIAMS COLLEGE WILLIAMSTOWN, MASSACHUSETTS JULY 10-28, 1978 EDITED BY GUIDO WEISS STEPHEN WAINGER Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 77-23480 Library of Congress Cataloging in Publication Data Symposium in Pure Mathematics, Williams College, 1978. Harmonic analysis in Euclidean spaces. (Proceedings of symposia in pure mathematics; v. 35) Includes bibliographies. 1. Harmonic analysis—Congresses. 2. Spaces, Generalized—Congresses. I. Wainger, Stephen, 1936- II. Weiss, Guido L., 1928- HI. American Mathematical Society. IV. Title. V. Series. QA403.S9 1978 515'.2433 79-12726 ISBN 0-8218-1436-2 (v.l) AMS (MOS) subject classifications (1970). Primary 22Exx, 26A51, 28A70, 30A78, 30A86, 31Bxx, 31C05, 31C99, 32A30, 32C20, 35-XX, 42-XX, 43-XX, 44-XX, 47A35, 47B35 Copyright © 1979 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. CONTENTS OF VOLUME PART 1 Dedication to Nestor Riviere vi Contents of Part 1 xix Photographs xxiii Preface xxv Chapter 1. Real harmonic analysis 3 Chapter 2. Hardy spaces and BMO 189 Chapter 3. Harmonic functions, potential theory and theory of functions of one complex variable 313 PART 2 Contents of Part 2 v Chapter 4.
    [Show full text]
  • Harmonic Analysis Is a Broad field Involving a Great Deal of Subjects Concerning the Art of Decomposing Functions Into Constituent Parts
    Lecture Notes in Harmonic Analysis Lectures by Dr. Charles Moore Throughout this document, signifies end proof, and N signifies end of example. Table of Contents Table of Contents i Lecture 1 Introduction to Fourier Analysis 1 1.1 Fourier Analysis . 1 1.2 In more general settings. 3 Lecture 2 More Fourier Analysis 4 2.1 Elementary Facts from Fourier Analysis . 4 Lecture 3 Convolving Functions 7 3.1 Properties of Convolution . 7 Lecture 4 An Application 9 4.1 Photographing a Star . 9 4.2 Results in L2 ............................. 10 Lecture 5 Hilbert Spaces 12 5.1 Fourier Series on L2 ......................... 12 Lecture 6 More on Hilbert Spaces 15 6.1 Haar Functions . 15 6.2 Fourier Transform on L2 ....................... 16 Lecture 7 Inverse Fourier Transform 17 7.1 Undoing Fourier Transforms . 17 Lecture 8 Fejer Kernels 19 8.1 Fejer Kernels and Approximate Identities . 19 Lecture 9 Convergence of Cesar´oMeans 22 9.1 Convergence of Fourier Sums . 22 Notes by Jakob Streipel. Last updated December 1, 2017. i TABLE OF CONTENTS ii Lecture 10 Toward Convergance of Partial Sums 24 10.1 Dirichlet Kernels . 24 10.2 Convergence for Continuous Functions . 25 Lecture 11 Convergence in Lp 26 11.1 Convergence in Lp .......................... 26 11.2 Almost Everywhere Convergence . 28 Lecture 12 Maximal Functions 29 12.1 Hardy-Littlewood Maximal Functions . 29 Lecture 13 More on Maximal Functions 33 13.1 Proof of Hardy-Littlewood's Theorem . 33 Lecture 14 Marcinkiewicz Interpolation 36 14.1 Proof of Marcinkiewicz Interpolation Theorem . 36 Lecture 15 Lebesgue Differentiation Theorem 38 15.1 A Note About Maximal Functions .
    [Show full text]
  • On Fourier Transforms of Radial Functions and Distributions
    ON FOURIER TRANSFORMS OF RADIAL FUNCTIONS AND DISTRIBUTIONS LOUKAS GRAFAKOS AND GERALD TESCHL Abstract. We find a formula that relates the Fourier transform of a radial function on Rn with the Fourier transform of the same function defined on Rn+2. This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function t 7! f(jtj) and the two-dimensional function (x1; x2) 7! f(j(x1; x2)j). We prove analogous results for radial tempered distributions. 1. Introduction The Fourier transform of a function Φ in L1(Rn) is defined by the con- vergent integral Z −2πix·ξ Fn(Φ)(ξ) = Φ(x)e dx : Rn If the function Φ is radial, i.e., Φ(x) = '(jxj) for some function ' on the line, then its Fourier transform is also radial and we use the notation Fn(Φ)(ξ) = Fn(')(r) ; where r = jξj. In this article, we will show that there is a relationship be- tween Fn(')(r) and Fn+2(')(r) as functions of the positive real variable r. We have the following result. Theorem 1.1. Let n ≥ 1. Suppose that f is a function on the real line such that the functions f(j · j) are in L1(Rn+2) and also in L1(Rn). Then we have 1 1 d (1) F (f)(r) = − F (f)(r) r > 0 : n+2 2π r dr n Moreover, the following formula is valid for all even Schwartz functions ' on the real line: 1 1 d (2) F (')(r) = F s−n+1 ('(s)sn) (r); r > 0 : n+2 2π r2 n ds 1991 Mathematics Subject Classification.
    [Show full text]
  • Group Representations and Harmonic Analysis from Euler to Langlands, Part I Anthony W
    Group Representations and Harmonic Analysis from Euler to Langlands, Part I Anthony W. Knapp roup representations and harmonic for s>1. In fact, if each factor (1 p s ) 1 on − − − analysis play a critical role in subjects the right side of (1) is expanded in geometric se- as diverse as number theory, proba- ries as 1+p s +p 2s + , then the product of − − ··· bility, and mathematical physics. A the factors for p N is the sum of those terms ≤ representation-theoretic theorem of 1/ns for which n is divisible only by primes GLanglands is a vital ingredient in the work of N; hence a passage to the limit yields (1). Wiles on Fermat's Last Theorem, and represen- ≤ Euler well knew that the sum for (s) exceeds tation theory provided the framework for the the integral prediction that quarks exist. What are group representations, why are they so pervasive in ∞ dx 1 s = mathematics, and where is their theory headed? 1 x s 1. Z − Euler and His Product Expansions This expression is unbounded as s decreases to Like much of modern mathematics, the field of 1, but the product (1) cannot be unbounded un- group representations and harmonic analysis less there are infinitely many factors. Hence (1) has some of its roots in the work of Euler. In 1737 yielded for Euler a new proof of Euclid's theo- Euler made what Weil [4] calls a “momentous dis- rem that there are infinitely many primes. In covery”, namely, to start with the function that fact, (1) implies, as Euler observed, the better the- we now know as the Riemann function 1 orem that 1/p diverges.
    [Show full text]
  • A TECHNICAL SURVEY of HARMONIC ANALYSIS 1. General
    A TECHNICAL SURVEY OF HARMONIC ANALYSIS TERENCE TAO Abstract. A rather technical description of some of the active areas of re- search in modern real-variable harmonic analysis, and some of the techniques being developed. Note: references are completely missing; I may update this later. 1. General overview Very broadly speaking, harmonic analysis is centered around the analysis (in par- ticular, quantitative estimates) of functions (and transforms of those functions) on such domains as Euclidean spaces Rd, integer lattices Zd, periodic tori Td = Rd/Zd, or other groups G (which could be anything from Lie groups, to finite groups, to infinite dimensional groups), or manifolds M, or subdomains Ω of these spaces, or perhaps more exotic sets such as fractals. These functions typically take values in the real numbers R or complex numbers C, or in other finite-dimensional spaces (e.g. one may consider k-forms on a manifold M, although strictly speaking these are sections of a bundle rather than functions); they may even take values in infinite-dimensional Hilbert or Banach spaces. The range space is usually a Banach space1; this allows one to take finite or infinite linear combinations of these func- tions, and be able to take limits of sequences of functions. Rather than focusing on very special examples of functions (e.g. explicit algebraic functions), harmonic analysis is often more concerned with generic functions in a certain regularity class (e.g. in Ck(M), the space of k times continuously differentiable functions on a manifold M) or a certain integrability class (e.g. Lp(M), the class of pth-power integrable functions on M).
    [Show full text]
  • Advances in Harmonic Analysis and Partial Differential Equations
    748 Advances in Harmonic Analysis and Partial Differential Equations AMS Special Session Harmonic Analysis and Partial Differential Equations April 21–22, 2018 Northeastern University, Boston, MA Donatella Danielli Irina Mitrea Editors Advances in Harmonic Analysis and Partial Differential Equations AMS Special Session Harmonic Analysis and Partial Differential Equations April 21–22, 2018 Northeastern University, Boston, MA Donatella Danielli Irina Mitrea Editors 748 Advances in Harmonic Analysis and Partial Differential Equations AMS Special Session Harmonic Analysis and Partial Differential Equations April 21–22, 2018 Northeastern University, Boston, MA Donatella Danielli Irina Mitrea Editors EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 31A10, 33C10, 35G20, 35P20, 35S05, 39B72, 42B35, 46E30, 76D03, 78A05. Library of Congress Cataloging-in-Publication Data Names: AMS Special Session on Harmonic Analysis and Partial Differential Equations (2018 : Northeastern University). | Danielli, Donatella, 1966- editor. | Mitrea, Irina, editor. Title: Advances in harmonic analysis and partial differential equations : AMS special session on Harmonic Analysis and Partial Differential Equations, April 21-22, 2018, Northeastern University, Boston, MA / Donatella Danielli, Irina Mitrea, editors. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Con- temporary mathematics, 0271-4132 ; volume 748 | Includes bibliographical references.
    [Show full text]
  • Harmonic Analysis Associated with a Discrete Laplacian
    HARMONIC ANALYSIS ASSOCIATED WITH A DISCRETE LAPLACIAN OSCAR´ CIAURRI, T. ALASTAIR GILLESPIE, LUZ RONCAL, JOSE´ L. TORREA, AND JUAN LUIS VARONA Abstract. It is well-known that the fundamental solution of ut(n; t) = u(n + 1; t) − 2u(n; t) + u(n − 1; t); n 2 Z; −2t with u(n; 0) = δnm for every fixed m 2 Z, is given by u(n; t) = e In−m(2t), where Ik(t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series X −2t Wtf(n) = e In−m(2t)f(m): m2Z By using semigroup theory, this formula allows us to analyze some op- erators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted p ` (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essen- tially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz trans- forms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel func- tions. 1. Introduction The purpose of this paper is the analysis of several operators associated with the discrete Laplacian ∆df(n) = f(n + 1) − 2f(n) + f(n − 1); n 2 Z; in a similar way to the analysis of classical operators associated with the @2 Euclidean Laplacian ∆ = @x2 . Among the operators that we study are the (discrete) fractional Laplacian, maximal heat and Poisson semigroups, This paper has been published in: J.
    [Show full text]