Gaussian Harmonic Analysis (eBook)

Foreword 7
Preface 10
Contents 16
1 Preliminary Results: The Gaussian Measure and HermitePolynomials 19
1.1 The Gaussian Measure 19
1.2 Estimates for the Gaussian Measure of Balls in Rd and the Doubling Condition 21
1.3 Hermite Polynomials 30
Hermite Polynomials in One Variable 30
Hermite Polynomials in d Variables 40
1.4 Notes and Further Results 43
2 The Ornstein–Uhlenbeck Operator and the Ornstein–Uhlenbeck Semigroup 49
2.1 The Ornstein–Uhlenbeck Operator 49
2.2 Definition and Basic Properties of the Ornstein–Uhlenbeck Semigroup 56
2.3 The Hypercontractivity Property for the Ornstein–Uhlenbeck Semigroup and the Logarithmic Sobolev Inequality 75
2.4 Applications of the Hypercontractivity Property 82
2.5 Notes and Further Results 84
3 The Poisson–Hermite Semigroup 95
3.1 Definition and Basic Properties 95
3.2 Characterization of ?2?t2 + L-Harmonic Functions 105
3.3 Generalized Poisson–Hermite Semigroups 110
3.4 Conjugate Poisson–Hermite Semigroup 112
3.5 Notes and Further Results 116
4 Covering Lemmas, Gaussian Maximal Functions, and Calderón–Zygmund Operators 117
4.1 Covering Lemmas with Respect to the Gaussian Measure 117
4.2 Hardy–Littlewood Maximal Function with Respect to the Gaussian Measure and Its Variants 130
4.3 The Maximal Functions of the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups 144
The Continuity Properties of the Ornstein–Uhlenbeck Maximal Function 144
The Continuity Properties of the Poisson–Hermite Maximal Function 167
4.4 The Local and Global Regions 168
4.5 Calderón–Zygmund Operators and the Gaussian Measure 169
4.6 The Non-tangential Maximal Functions for the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups 183
The Non-tangential Ornstein–Uhlenbeck Maximal Function 183
The Non-tangential Poisson–Hermite Maximal Function 188
4.7 Radial and Non-tangential Convergence of the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups 190
4.8 Notes and Further Results 197
5 Littlewood–Paley–Stein Theory with Respect to theGaussian Measure 210
5.1 The Gaussian Littlewood–Paley g Function and Its Variants 210
5.2 The Higher Order Gaussian Littlewood–Paley g Functions 229
5.3 The Gaussian Lusin Area Function 238
5.4 Notes and Further Results 242
6 Spectral Multiplier Operators with Respect to theGaussian Measure 247
6.1 Gaussian Spectral Multiplier Operators 247
6.2 Meyer's Multipliers 248
6.3 Gaussian Laplace Transform Type Multipliers 249
6.4 Functional Calculus for the Ornstein–Uhlenbeck Operator 252
6.5 Notes and Further Results 257
7 Function Spaces with Respect to the Gaussian Measure 261
7.1 Gaussian Lebesgue Spaces Lp(?d) 261
7.2 Gaussian Sobolev Spaces L?p(?d) 262
7.3 Gaussian Tent Spaces T1,q(?d) 263
7.4 Gaussian Hardy Spaces H1(?d) 272
7.5 Gaussian BMO(?d) Spaces 282
7.6 Gaussian Lipschitz Spaces Lip?(?) 284
7.7 Gaussian Besov–Lipschitz Spaces Bp,q?(?d) 291
7.8 Gaussian Triebel–Lizorkin Spaces Fp,q?(?d) 301
7.9 Notes and Further Results 317
8 Gaussian Fractional Integrals and Fractional Derivatives,and Their Boundedness on Gaussian Function Spaces 319
8.1 Riesz and Bessel Potentials with Respect to the GaussianMeasure 319
Gaussian Riesz Potentials 319
Gaussian Bessel Potentials 325
8.2 Fractional Derivatives with Respect to the Gaussian Measure 326
Gaussian Riesz Fractional Derivate 326
Gaussian Bessel Fractional Derivates 329
8.3 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Lipschitz Spaces 330
8.4 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Besov–Lipschitz Spaces 334
8.5 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Triebel–Lizorkin Spaces 353
8.6 Notes and Further Results 368
9 Singular Integrals with Respect to the Gaussian Measure 374
9.1 Definition and Boundedness Properties of the Gaussian Riesz Transforms 375
9.2 Definition and Boundedness Properties of the Higher-Order Gaussian Riesz Transforms 381
9.3 Alternative Gaussian Riesz Transforms 393
9.4 Definition and Boundedness Properties of General Gaussian Singular Integrals 401
9.5 Notes and Further Results 410
Correction to: Gaussian Harmonic Analysis 422
Appendix 433
10.1 The Gamma Function and Related Functions 433
10.2 Classical Orthogonal Polynomials 434
Hermite Polynomials 435
Laguerre Polynomials 437
Generalized Hermite Polynomials 440
Jacobi Polynomials 442
10.3 Doubling Measures 445
10.4 Density Theorems for Positive Radon Measures 446
10.5 Classical Semigroups in Analysis: The Heat and the Poisson Semigroups 452
The Heat Semigroup 452
The Poisson Semigroup 459
10.6 Interpolation Theory 462
10.7 Hardy's Inequalities 464
10.8 Natanson's Lemma and Generalizations 465
10.9 Forward Differences 468
References 472
Glossary of Symbols 488
Index 499