Classical Fourier Analysis

Transcription

1 Loukas Grafakos Classical Fourier Analysis Third Edition ~Springer

2 1 V' Spaces and Interpolation V' and Weak V' l The Distribution Function Convergence in Measure A First Glimpse at Interpolation Exercises Convolution and Approximate Identities Examples of Topological Groups Convolution Basic Convolution Inequalities Approximate Identities Exercises Interpolation Real Method: The Marcinkiewicz Interpolation Theorem Complex Method: The Riesz-Thorin Interpolation Theorem Interpolation of Analytic Families of Operators Exercises Lorentz Spaces Decreasing Rearrangements Lorentz Spaces Duals of Lorentz Spaces The Off-Diagonal Marcinkiewicz Interpolation Theorem Exercises Maximal Functions, Fourier Transfonn, and Distributions Maximal Functions The Hardy-Littlewood Maximal Operator Control of Other Maximal Operators xi

3 xii Applications to Differentiation Theory Exercises Tue Schwartz Class and the Fourier Transform l Tue Class of Schwartz Functions The Fourier Transform of a Schwartz Function The Inverse Fourier Transform and Fourier Inversion Tue Fourier Transform on L 1 + L Exercises The Class of Tempered Distributions l Spaces of Test Functions Spaces of Functionals on Test Functions The Space of Tempered Distributions Exercises More About Distributions and the Fourier Transform l Distributions Supported at a Point Tue Laplacian Homogeneous Distributions Exercises Convolution Operators on IP Spaces and Multipliers l Operators That Commute with Translations The Transpose and the Adjoint of a Linear Operator The Spaces Alp,q(Rn) Characterizations of Al 1 1 (Rn) and Af 2 2 (Rn) Tue Space of Fourier Multipliers Alp(Rn) Exercises Oscillatory Integrals l Phases with No Critical Points Sublevel Set Estimates and the Van der Corput Lemma Exercises Fourier Series Fourier Coefficients The n-torus Tn Fourier Coefficients Tue Dirichlet and Fejer Kernels Exercises Reproduction of Functions from Their Fourier Coefficients Partial sums and Fourier inversion Fourier series of square summable functions Tue Poisson Summation Formula Exercises Decay of Fourier Coefficients Decay of Fourier Coefficients of Arbitrary lntegrable Functions Decay of Fourier Coefficients of Smooth Functions

4 xiii Functions with Absolutely Summable Fourier Coefficients Exercises Pointwise Convergence of Fourier Series Pointwise Convergence of the Fejer Means Almost Everywhere Convergence of the Fejer Means Pointwise Divergence of the Dirichlet Means Pointwise Convergence of the Dirichlet Means Exercises A Tauberian theorem and Functions of Bounded Variation A Tauberian theorem The sine integral function Further properties of functions of bounded variation Gibbs phenomenon Exercises Lacunary Series and Sidon Sets Definition and Basic Properties of Lacunary Series Equivalence of LP Norms of Lacunary Series Sidon sets Exercises Topics on Fourier Series Convergence in Norm, Conjugate Function, and Bochner-Riesz Means Equi valent Formulations of Convergence in Norm The LP Boundedness of the Conjugate Function Bochner-Riesz Summability Exercises A. E. Divergence of Fourier Series and Bochner-Riesz means Divergence of Fourier Series of lntegrable Functions Divergence of Bochner-Riesz Means of Integrable Functions Exercises Multipliers, Transference, and Almost Everywhere Convergence Multipliers on the Torus Transference of Multipliers Applications of Transference Transference of Maximal Multipliers Applications to Almost Everywhere Convergence Almost Everywhere Convergence of Square Dirichlet Means Exercises

5 xiv 4.4 Applications to Geometry and Partial Differential Equations Tue Isoperimetric Inequality The Heat Equation with Periodic Boundary Condition Exercises Applications to Number theory and Ergodic theory Evaluation of the Riemann Zeta Function at even Natural numbers Equidistributed sequences Tue Number of Lattice Points inside a Ball Exercises Singular Integrals of Convolution Type Tue Hilbert Transform and the Riesz Transforms l Definition and Basic Properties of the Hilbert Transform Connections with Analytic Functions V' Boundedness of the Hilbert Transform The Riesz Transforms Exercises Homogeneous Singular Integrals and the Method of Rotations l 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 Exercises The Calder6n-Zygmund Decomposition and Singular Integrals l The Calder6n-Zygmund Decomposition General Singular Integrals U Boundedness Implies Weak TYpe ( 1, 1) Boundedness Discussion on Maximal Singular Integrals Boundedness for Maximal Singular Integrals Implies Weak Type ( 1, 1) Boundedness Exercises Sufficient Conditions for V' Boundedness l Sufficient Conditions for V' Boundedness of Singular Integrals An Example Necessity of the Cancellation Condition Sufficient Conditions for V' Boundedness of Maximal Singular Integrals Exercises Vector-Valued Inequalities l t' 2 -Valued Extensions of Linear Operators Applications and er-valued Extensions of Linear Operators

6 xv General Banach-Valued Extensions Exercises Vector-Valued Singular Integrals , l Banach-Valued Singular Integral Operators Applications Vector-Valued Estimates for Maximal Functions Exercises Littlewood-Paley Theory and Multipliers Littlewood-Paley Theory The Littlewood-Paley Theorem Vector-Valued Analogues V' Estimates for Square Functions Associated with Dyadic Sums Lack of Orthogonality on [)' Exercises \vo Multiplier Theorems Tue Marcinkiewicz Multiplier Theorem on R Tue Marcinkiewicz Multiplier Theorem on R" The Mihlin-Hörmander Multiplier Theorem on R" Exercises Applications of Littlewood-Paley Theory l Estimates for Maximal Operators Estimates for Singular Integrals with Rough Kernels An Almost Orthogonality Principle on V' Exercises The Haar System, Conditional Expectation, and Martingales I Conditional Expectation and Dyadic Martingale Differences Relation Between Dyadic Martingale Differences and Haar Functions The Dyadic Martingale Square Function Almost Orthogonality Between the Littlewood-Paley Operators and the Dyadic Martingale Difference Operators Exercises Tue Spherical Maximal Function Introduction of the Spherical Maximal Function The First Key Lemma Tue Second Key Lemma Completion of the Proof Exercises Wavelets and Sampling l Some Preliminary Facts Construction of a Nonsmooth Wavelet

7 xvi Construction of a Smooth Wavelet Sampling Exercises Weighted Inequalities The Ap Condition Motivation for the Ap Condition Properties of Ap Weights Exercises Reverse Hölder lnequality for Ap Weights and Consequences The Reverse Hölder Property of Ap Weights Consequences of the Reverse Hölder Property Exercises The Aoo Condition The Class of Aoo Weights Characterizations of Aoo Weights Exercises Weighted Norm lnequalities for Singular Integrals Singular Integrals of Non Convolution type A Good Lambda Estimate for Singular Integrals Consequences of the Good Lambda Estimate Necessity of the Ap Condition Exercises Further Properties of Ap Weights Factorization of Weights Extrapolation from Weighted Estimates on a Single Vo Weighted Inequalities Versus Vector-Valued lnequalities Exercises A Gamma and Beta Functions 563 A.l A Useful Formula A.2 Definitions of I'(z) and B(z, w) A.3 Volume of the Unit Ball and Surface of the Unit Sphere A.4 Computation of Integrals Using Gamma Functions A.5 Meromorphic Extensions of B(z, w) and I'(z) A.6 Asymptotics of I' (x) as x -+ oo 567 A.7 Euler's Limit Formula for the Gamma Function A.8 Reftection and Duplication Formulas for the Gamma Function B Bessel Functions 573 B.l Definition B.2 Some Basic Properties B.3 An Interesting ldentity B.4 The Fourier Transform of Surface Measure on sn-t B.5 The Fourier Transform of a Radial Function on Rn

8 xvii B.6 Bessel Functions of Small Arguments B.7 Bessel Functions of Large Arguments B.8 Asymptotics of Bessel Functions B.9 Bessel Functions of general complex indices C Rademacher Functions 585 C.l Definition of the Rademacher Functions C.2 Khintchine's Inequalities C.3 Derivation of Khintchine's Inequalities C.4 Khintchine's Inequalities for Weak Type Spaces C.5 Extension to Several Variables D Spherical Coordinates 591 D.1 Spherical Coordinate Formula D.2 A Useful Change of Variables Formula D.3 Computation of an Integral over the Sphere D.4 The Computation of Another Integral over the Sphere D.5 Integration over a General Surface D.6 The Stereographic Projection E Some Trigonometrie ldentities and Inequalities 597 F Summation by Parts 599 G Basic Functional Analysis 601 H The Minimax Lemma Taylor's and Mean Value Theorem in Several Variables Mutlivariable Taylor's Theorem The Mean value Theorem J The Whitney Decomposition of Open Sets in Rn 609 J.1 Decomposition of Open Sets J.2 Partition of Unity adapted to Whitney cubes Glossary 613 References 617 Index 633