Loukas Grafakos Classical Fourier Analysis Second Edition 4y Springer
1 IP Spaces and Interpolation 1 1.1 V and Weak IP 1 1.1.1 The Distribution Function 2 1.1.2 Convergence in Measure 5 1.1.3 A First Glimpse at Interpolation 8 Exercises 10 1.2 Convolution and Approximate Identities 16 1.2.1 Examples of Topological Groups 16 1.2.2 Convolution 18 1.2.3 Basic Convolution Inequalities 19 1.2.4 Approximate Identities 24 Exercises 28 1.3 Interpolation 30 1.3.1 Real Method: The Marcinkiewicz Interpolation Theorem... 31 1.3.2 Complex Method: The Riesz-Thorin Interpolation Theorem.. 34 1.3.3 Interpolation of Analytic Families of Operators 37 1.3.4 Proofs of Lemmas 1.3.5 and 1.3.8 39 Exercises 42 1.4 Lorentz Spaces 44 1.4.1 Decreasing Rearrangements 44 1.4.2 Lorentz Spaces 48 1.4.3 Duals of Lorentz Spaces 51 1.4.4 The Off-Diagonal Marcinkiewicz Interpolation Theorem... 55 Exercises 63 2 Maximal Functions, Fourier Transform, and Distributions 77 2.1 Maximal Functions 78 2.1.1 The Hardy-Littlewood Maximal Operator 78 2.1.2 Control of Other Maximal Operators 82 2.1.3 Applications to Differentiation Theory 85 Exercises 89
2.2 The Schwartz Class and the Fourier Transform 94 2.2.1 The Class of Schwartz Functions 95 2.2.2 The Fourier Transform of a Schwartz Function 98 2.2.3 The Inverse Fourier Transform and Fourier Inversion 102 2.2.4 The Fourier Transform on L l +L 2 103 Exercises 106 2.3 The Class of Tempered Distributions 109 2.3.1 Spaces of Test Functions 109 2.3.2 Spaces of Functionals on Test Functions 110 2.3.3 The Space of Tempered Distributions 112 2.3.4 The Space of Tempered Distributions Modulo Polynomials.. 121 Exercises 122 2.4 More About Distributions and the Fourier Transform 124 2.4.1 Distributions Supported at a Point 124 2.4.2 The Laplacian 125 2.4.3 Homogeneous Distributions 127 Exercises 133 2.5 Convolution Operators on LP Spaces and Multipliers 135 2.5.1 Operators That Commute with Translations 135 2.5.2 The Transpose and the Adjoint of a Linear Operator 138 2.5.3 The Spaces ^^(R") 139 2.5.4 Characterizations of ^T 1 ' 1 (R") and ^2' 2 (R") 141 2.5.5 The Space of Fourier Multipliers J% p (R n ) 143 Exercises 146 2.6 Oscillatory Integrals 148 2.6.1 Phases with No Critical Points 149 2.6.2 Sublevel Set Estimates and the Van der Corput Lemma 151 Exercises 156 Fourier Analysis on the Torus 161 3.1 Fourier Coefficients 161 3.1.1 The n-torus T" 162 3.1.2 Fourier Coefficients 163 3.1.3 The Dirichlet and Fejer Kernels 165 3.1.4 Reproduction of Functions from Their Fourier Coefficients..168 3.1.5 The Poisson Summation Formula 171 Exercises 173 3.2 Decay of Fourier Coefficients 176 3.2.1 Decay of Fourier Coefficients of Arbitrary Integrable Functions 176 3.2.2 Decay of Fourier Coefficients of Smooth Functions 179 3.2.3 Functions with Absolutely Summable Fourier Coefficients.. 183 Exercises 185 3.3 Pointwise Convergence of Fourier Series 186 3.3.1 Pointwise Convergence of the Fejer Means 186
3.3.2 Almost Everywhere Convergence of the Fejer Means 188 3.3.3 Pointwise Divergence of the Dirichlet Means 191 3.3.4 Pointwise Convergence of the Dirichlet Means 192 Exercises 193 3.4 Divergence of Fourier and Bochner-Riesz Summability 195 3.4.1 Motivation for Bochner-Riesz Summability 195 3.4.2 Divergence of Fourier Series of Integrable Functions 198 3.4.3 Divergence of Bochner-Riesz Means of Integrable Functions 203 Exercises 209 3.5 The Conjugate Function and Convergence in Norm 211 3.5.1 Equivalent Formulations of Convergence in Norm 211 3.5.2 The LP Boundedness of the Conjugate Function 215 Exercises 218 3.6 Multipliers, Transference, and Almost Everywhere Convergence... 220 3.6.1 Multipliers on the Torus 221 3.6.2 Transference of Multipliers 223 3.6.3 Applications of Transference 228 3.6.4 Transference of Maximal Multipliers 228 3.6.5 Transference and Almost Everywhere Convergence 232 Exercises 235 3.7 Lacunary Series 237 3.7.1 Definition and Basic Properties of Lacunary Series 238 3.7.2 Equivalence of LP Norms of Lacunary Series 240 Exercises 245 Singular Integrals of Convolution Type 249 4.1 The Hilbert Transform and the Riesz Transforms 249 4.1.1 Definition and Basic Properties of the Hilbert Transform... 250 4.1.2 Connections with Analytic Functions 253 4.1.3 LP Boundedness of the Hilbert Transform 255 4.1.4 The Riesz Transforms 259 Exercises 263 4.2 Homogeneous Singular Integrals and the Method of Rotations 267 4.2.1 Homogeneous Singular and Maximal Singular Integrals... 267 4.2.2 L 2 Boundedness of Homogeneous Singular Integrals 269 4.2.3 The Method of Rotations 272 4.2.4 Singular Integrals with Even Kernels 274 4.2.5 Maximal Singular Integrals with Even Kernels 278 Exercises 284 4.3 The Calderdn-Zygmund Decomposition and Singular Integrals... 286 4.3.1 The Calderon-Zygmund Decomposition 286 4.3.2 General Singular Integrals 289 4.3.3 U Boundedness Implies Weak Type (1,1) Boundedness...290 4.3.4 Discussion on Maximal Singular Integrals 293
4.3.5 Boundedness for Maximal Singular Integrals Implies Weak Type (1,1) Boundedness 297 Exercises 302 4.4 Sufficient Conditions for LP Boundedness 305 4.4.1 Sufficient Conditions for LP Boundedness of Singular Integrals 305 4.4.2 An Example 308 4.4.3 Necessity of the Cancellation Condition 309 4.4.4 Sufficient Conditions for LP Boundedness of Maximal Singular Integrals 310 Exercises 314 4.5 Vector-Valued Inequalities 315 4.5.1 i! 2 -Valued Extensions of Linear Operators 316 4.5.2 Applications and r -Valued Extensions of Linear Operators..319 4.5.3 General Banach-Valued Extensions 321 Exercises 327 4.6 Vector-Valued Singular Integrals 329 4.6.1 Banach-Valued Singular Integral Operators 329 4.6.2 Applications 332 4.6.3 Vector-Valued Estimates for Maximal Functions 334 Exercises 337 Littlewood-Paley Theory and Multipliers 341 5.1 Littlewood-Paley Theory 341 5.1.1 The Littlewood-Paley Theorem 342 5.1.2 Vector-Valued Analogues 347 5.1.3 LP Estimates for Square Functions Associated with Dyadic Sums 348 5.1.4 Lack of Orthogonality on LP 353 Exercises 355 5.2 Two Multiplier Theorems 359 5.2.1 The Marcinkiewicz Multiplier Theorem on R 360 5.2.2 The Marcinkiewicz Multiplier Theorem on R" 363 5.2.3 The Hormander-Mihlin Multiplier Theorem on R" 366 Exercises 371 5.3 Applications of Littlewood-Paley Theory 373 5.3.1 Estimates for Maximal Operators 373 5.3.2 Estimates for Singular Integrals with Rough Kernels 375 5.3.3 An Almost Orthogonality Principle on LP 379 Exercises 381 5.4 The Haar System, Conditional Expectation, and Martingales 383 5.4.1 Conditional Expectation and Dyadic Martingale Differences.. 384 5.4.2 Relation Between Dyadic Martingale Differences and Haar Functions 385 5.4.3 The Dyadic Martingale Square Function 388
xv 5.4.4 Almost Orthogonality Between the Littlewood-Paley,. Operators and the Dyadic Martingale Difference Operators... 391 Exercises 394 5.5 The Spherical Maximal Function 395 5.5.1 Introduction of the Spherical Maximal Function 395 5.5.2 The First Key Lemma 397 5.5.3 The Second Key Lemma 399 5.5.4 Completion of the Proof 400 Exercises 400 5.6 Wavelets 402 5.6.1 Some Preliminary Facts 403 5.6.2 Construction of a Nonsmooth Wavelet 404 5.6.3 Construction of a Smooth Wavelet 406 5.6.4 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"" 1 428 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
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 457 1.1 The Classical Schur Lemma 457 1.2 Schur's Lemma for Positive Operators 457 1.3 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