Grafakos L. Fundamentals of Fourier Analysis 2024

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Grafakos L. Fundamentals of Fourier Analysis 2024

Textbook in PDF format Preface Introductory Material A Review of Lebesgue Spaces The Distribution Function and Weak Lp Spaces Real Interpolation The Hardy–Littlewood Maximal Operator The Lebesgue Differentiation Theorem Convolution Smoothness and Smooth Functions with Compact Support Schwartz Functions Approximate Identities Fourier Transforms, Tempered Distributions, Approximate Identities The Fourier Transform on L1 Fourier Inversion The Fourier Transform on L2 Complex Interpolation and the Hausdorff–Young Inequality Approximate Identities and Almost Everywhere Convergence Tempered Distributions Basic Operations with Tempered Distributions Lp Fourier Multipliers Van der Corput Lemma Singular Integrals The Hilbert Transform Homogeneous Singular Integrals and Riesz Transforms Calderón–Zygmund Singular Integrals L2 Boundedness of Calderón–Zygmund Operators The Calderón–Zygmund Decomposition L2 Boundedness Implies Lp Boundedness The Hilbert Transform and the Poisson Kernel Maximal Singular Integrals Vector-Valued Singular Integrals and Littlewood–Paley Theory The Vector-Valued Calderón–Zygmund Theorem Applications of Vector-Valued Inequalities A Matrix-Valued Calderón–Zygmund Theorem and Its Applications Littlewood–Paley Theory Reverse Littlewood–Paley Inequalities Littlewood–Paley Theory of Product Type Fractional Integrability or Differentiability and Multiplier Theorems Powers of the Laplacian and Riesz Potentials Bessel Potentials The Mikhlin and Hörmander Multiplier Theorems Sobolev Spaces Interpolation of Analytic Families of Operators The Calderón–Torchinsky Multiplier Theorem The Marcinkiewicz Multiplier Theorem Bounded Mean Oscillation Basic Properties of Functions of Bounded Mean Oscillation The John–Nirenberg Theorem Dyadic Maximal Functions and Dyadic BMO The Sharp Maximal Function Interpolation Using BMO Hardy Spaces Smoothness and Cancellation Definition of Hardy Spaces and Preliminary Estimates Hp Atoms Grand Maximal Function The Whitney Decomposition of Open Sets Atomic Decomposition of H1 Singular Integrals on the Hardy Space H1 Duality Between H1 and BMO Weighted Inequalities Appearance of Weights The Ap Condition Properties of Ap Weights Strong-Type Ap Estimates The Jones Factorization of Weights Reverse Hölder Property of Ap Weights Weighted Estimates for Singular Integral Operators Historical Notes Orthogonal Matrices Subharmonic Functions Poisson Kernel on the Unit Strip Density for Subadditive Operators Transposes and Adjoints of Linear Operators Faà di Bruno Formula Besicovitch Covering Lemma Glossary References Index