Classical Fourier Analysis

Transcription

1 Loukas Grafakos Classical Fourier Analysis Second Edition 4y Springer

2 1 IP Spaces and Interpolation V and Weak IP The Distribution Function Convergence in Measure A First Glimpse at Interpolation 8 Exercises Convolution and Approximate Identities Examples of Topological Groups Convolution Basic Convolution Inequalities Approximate Identities 24 Exercises Interpolation Real Method: The Marcinkiewicz Interpolation Theorem Complex Method: The Riesz-Thorin Interpolation Theorem Interpolation of Analytic Families of Operators Proofs of Lemmas and Exercises Lorentz Spaces Decreasing Rearrangements Lorentz Spaces Duals of Lorentz Spaces The Off-Diagonal Marcinkiewicz Interpolation Theorem Exercises 63 2 Maximal Functions, Fourier Transform, and Distributions Maximal Functions The Hardy-Littlewood Maximal Operator Control of Other Maximal Operators Applications to Differentiation Theory 85 Exercises 89

3 2.2 The Schwartz Class and the Fourier Transform The Class of Schwartz Functions The Fourier Transform of a Schwartz Function The Inverse Fourier Transform and Fourier Inversion The Fourier Transform on L l +L Exercises The Class of Tempered Distributions Spaces of Test Functions Spaces of Functionals on Test Functions The Space of Tempered Distributions The Space of Tempered Distributions Modulo Polynomials Exercises More About Distributions and the Fourier Transform Distributions Supported at a Point The Laplacian Homogeneous Distributions 127 Exercises Convolution Operators on LP Spaces and Multipliers Operators That Commute with Translations The Transpose and the Adjoint of a Linear Operator The Spaces ^^(R") Characterizations of ^T 1 ' 1 (R") and ^2' 2 (R") The Space of Fourier Multipliers J% p (R n ) 143 Exercises Oscillatory Integrals Phases with No Critical Points Sublevel Set Estimates and the Van der Corput Lemma 151 Exercises 156 Fourier Analysis on the Torus Fourier Coefficients The n-torus T" Fourier Coefficients The Dirichlet and Fejer Kernels Reproduction of Functions from Their Fourier Coefficients The Poisson Summation Formula 171 Exercises Decay of Fourier Coefficients Decay of Fourier Coefficients of Arbitrary Integrable Functions Decay of Fourier Coefficients of Smooth Functions Functions with Absolutely Summable Fourier Coefficients Exercises Pointwise Convergence of Fourier Series Pointwise Convergence of the Fejer Means 186

4 3.3.2 Almost Everywhere Convergence of the Fejer Means Pointwise Divergence of the Dirichlet Means Pointwise Convergence of the Dirichlet Means 192 Exercises Divergence of Fourier and Bochner-Riesz Summability Motivation for Bochner-Riesz Summability Divergence of Fourier Series of Integrable Functions Divergence of Bochner-Riesz Means of Integrable Functions 203 Exercises The Conjugate Function and Convergence in Norm Equivalent Formulations of Convergence in Norm The LP Boundedness of the Conjugate Function 215 Exercises Multipliers, Transference, and Almost Everywhere Convergence Multipliers on the Torus Transference of Multipliers Applications of Transference Transference of Maximal Multipliers Transference and Almost Everywhere Convergence 232 Exercises Lacunary Series Definition and Basic Properties of Lacunary Series Equivalence of LP Norms of Lacunary Series 240 Exercises 245 Singular Integrals of Convolution Type The Hilbert Transform and the Riesz Transforms Definition and Basic Properties of the Hilbert Transform Connections with Analytic Functions LP Boundedness of the Hilbert Transform The Riesz Transforms 259 Exercises Homogeneous Singular Integrals and the Method of Rotations Homogeneous Singular and Maximal Singular Integrals L 2 Boundedness of Homogeneous Singular Integrals The Method of Rotations Singular Integrals with Even Kernels Maximal Singular Integrals with Even Kernels 278 Exercises The Calderdn-Zygmund Decomposition and Singular Integrals The Calderon-Zygmund Decomposition General Singular Integrals U Boundedness Implies Weak Type (1,1) Boundedness Discussion on Maximal Singular Integrals 293

5 4.3.5 Boundedness for Maximal Singular Integrals Implies Weak Type (1,1) Boundedness 297 Exercises Sufficient Conditions for LP Boundedness Sufficient Conditions for LP Boundedness of Singular Integrals An Example Necessity of the Cancellation Condition Sufficient Conditions for LP Boundedness of Maximal Singular Integrals 310 Exercises Vector-Valued Inequalities i! 2 -Valued Extensions of Linear Operators Applications and r -Valued Extensions of Linear Operators General Banach-Valued Extensions 321 Exercises Vector-Valued Singular Integrals Banach-Valued Singular Integral Operators Applications Vector-Valued Estimates for Maximal Functions 334 Exercises 337 Littlewood-Paley Theory and Multipliers Littlewood-Paley Theory The Littlewood-Paley Theorem Vector-Valued Analogues LP Estimates for Square Functions Associated with Dyadic Sums Lack of Orthogonality on LP 353 Exercises Two Multiplier Theorems The Marcinkiewicz Multiplier Theorem on R The Marcinkiewicz Multiplier Theorem on R" The Hormander-Mihlin Multiplier Theorem on R" 366 Exercises Applications of Littlewood-Paley Theory Estimates for Maximal Operators Estimates for Singular Integrals with Rough Kernels An Almost Orthogonality Principle on LP 379 Exercises The Haar System, Conditional Expectation, and Martingales Conditional Expectation and Dyadic Martingale Differences Relation Between Dyadic Martingale Differences and Haar Functions The Dyadic Martingale Square Function 388

6 xv Almost Orthogonality Between the Littlewood-Paley,. Operators and the Dyadic Martingale Difference Operators Exercises The Spherical Maximal Function Introduction of the Spherical Maximal Function The First Key Lemma The Second Key Lemma Completion of the Proof 400 Exercises Wavelets Some Preliminary Facts Construction of a Nonsmooth Wavelet Construction of a Smooth Wavelet A Sampling Theorem 410 Exercises 411 A Gamma and Beta Functions 417 A.I A Useful Formula 417 A.2 Definitions of T(z) and B(z, w) 417 A.3 Volume of the Unit Ball and Surface of the Unit Sphere 418 A.4 Computation of Integrals Using Gamma Functions 419 A.5 Meromorphic Extensions of B(z, w) and -T(z) 420 A.6 Asymptotics of F(x) as x > 420 A.7 Euler's Limit Formula for the Gamma Function 421 A.8 Reflection and Duplication Formulas for the Gamma Function 424 B Bessel Functions 425 B.I. Definition 425 B.2 Some Basic Properties 425 B.3 An Interesting Identity 427 B.4. The Fourier Transform of Surface Measure on S"" B.5 The Fourier Transform of a Radial Function on R" 428 B.6 Bessel Functions of Small Arguments 429 B.7 Bessel Functions of Large Arguments 430 B.8 Asymptotics of Bessel Functions 431 C Rademacher Functions 435 C.I Definition of the Rademacher Functions 435 C.2 Khintchine's Inequalities 435 C.3 Derivation of Khintchine's Inequalities 436 C.4 Khintchine's Inequalities for Weak Type Spaces 438 C.5 Extension to Several Variables 439

7 xvi Contents D Spherical Coordinates 441 D.I Spherical Coordinate Formula 441 D.2 A Useful Change of Variables Formula 441 D.3 Computation of an Integral over the Sphere 442 D.4 The Computation of Another Integral over the Sphere 443 D.5 Integration over a General Surface 444 D.6 The Stereographic Projection 444 E Some Trigonometric Identities and Inequalities 447 F Summation by Parts 449 G Basic Functional Analysis 451 H The Minimax Lemma 453 I The Schur Lemma The Classical Schur Lemma Schur's Lemma for Positive Operators An Example 460 J The Whitney Decomposition of Open Sets in R" 463 K Smoothness and Vanishing Moments 465 K.1 The Case of No Cancellation 465 K.2 The Case of Cancellation 466 K.3 The Case of Three Factors 467 Glossary 469 References 473 Index... * 485