Urbina-Romero Gaussian Harmonic Analysis

1. Auflage 2019
ISBN: 978-3-030-05597-4
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 477 Seiten

Reihe: Springer Monographs in Mathematics

Authored by a ranking authority in Gaussian harmonic analysis, this book embodies a state-of-the-art entrée at the intersection of two important fields of research: harmonic analysis and  probability. The book is intended for a very diverse audience, from graduate students all the way to researchers working in a broad spectrum of areas in analysis. Written with the graduate student in mind, it is assumed that the reader has familiarity with the basics of real analysis as well as with classical harmonic analysis, including Calderón-Zygmund theory; also some knowledge of basic orthogonal polynomials theory would be convenient. The monograph  develops the main topics of classical harmonic analysis (semigroups, covering lemmas, maximal functions, Littlewood-Paley functions, spectral multipliers, fractional integrals and fractional derivatives, singular integrals) with respect to the Gaussian measure. The text provide an updated exposition, as self-contained as possible, of all the topics in Gaussian harmonic analysis that up to now are mostly scattered in research papers and sections of books; also an exhaustive bibliography for further reading.  Each chapter ends with a section of notes and further results where connections between Gaussian harmonic analysis and other connected fields, points of view and alternative techniques are given. Mathematicians and researchers in several areas will find the breadth and depth of the treatment of the subject highly useful.

Autoren/Hrsg.

Urbina-Romero, Wilfredo

Inhaltsverzeichnis

1;Foreword;7
2;Preface;10
3;Contents;16
4;1 Preliminary Results: The Gaussian Measure and HermitePolynomials;19
4.1;1.1 The Gaussian Measure;19
4.2;1.2 Estimates for the Gaussian Measure of Balls in Rd and the Doubling Condition;21
4.3;1.3 Hermite Polynomials;30
4.3.1;Hermite Polynomials in One Variable;30
4.3.2;Hermite Polynomials in d Variables;40
4.4;1.4 Notes and Further Results;43
5;2 The Ornstein–Uhlenbeck Operator and the Ornstein–Uhlenbeck Semigroup;49
5.1;2.1 The Ornstein–Uhlenbeck Operator;49
5.2;2.2 Definition and Basic Properties of the Ornstein–Uhlenbeck Semigroup;56
5.3;2.3 The Hypercontractivity Property for the Ornstein–Uhlenbeck Semigroup and the Logarithmic Sobolev Inequality;75
5.4;2.4 Applications of the Hypercontractivity Property;82
5.5;2.5 Notes and Further Results;84
6;3 The Poisson–Hermite Semigroup;95
6.1;3.1 Definition and Basic Properties;95
6.2;3.2 Characterization of ?2?t2 + L-Harmonic Functions;105
6.3;3.3 Generalized Poisson–Hermite Semigroups;110
6.4;3.4 Conjugate Poisson–Hermite Semigroup;112
6.5;3.5 Notes and Further Results;116
7;4 Covering Lemmas, Gaussian Maximal Functions, and Calderón–Zygmund Operators;117
7.1;4.1 Covering Lemmas with Respect to the Gaussian Measure;117
7.2;4.2 Hardy–Littlewood Maximal Function with Respect to the Gaussian Measure and Its Variants;130
7.3;4.3 The Maximal Functions of the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups;144
7.3.1;The Continuity Properties of the Ornstein–Uhlenbeck Maximal Function;144
7.3.2;The Continuity Properties of the Poisson–Hermite Maximal Function;167
7.4;4.4 The Local and Global Regions;168
7.5;4.5 Calderón–Zygmund Operators and the Gaussian Measure;169
7.6;4.6 The Non-tangential Maximal Functions for the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups;183
7.6.1;The Non-tangential Ornstein–Uhlenbeck Maximal Function;183
7.6.2;The Non-tangential Poisson–Hermite Maximal Function;188
7.7;4.7 Radial and Non-tangential Convergence of the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups;190
7.8;4.8 Notes and Further Results;197
8;5 Littlewood–Paley–Stein Theory with Respect to theGaussian Measure;210
8.1;5.1 The Gaussian Littlewood–Paley g Function and Its Variants;210
8.2;5.2 The Higher Order Gaussian Littlewood–Paley g Functions;229
8.3;5.3 The Gaussian Lusin Area Function;238
8.4;5.4 Notes and Further Results;242
9;6 Spectral Multiplier Operators with Respect to theGaussian Measure;247
9.1;6.1 Gaussian Spectral Multiplier Operators;247
9.2;6.2 Meyer's Multipliers;248
9.3;6.3 Gaussian Laplace Transform Type Multipliers;249
9.4;6.4 Functional Calculus for the Ornstein–Uhlenbeck Operator;252
9.5;6.5 Notes and Further Results;257
10;7 Function Spaces with Respect to the Gaussian Measure;261
10.1;7.1 Gaussian Lebesgue Spaces Lp(?d);261
10.2;7.2 Gaussian Sobolev Spaces L?p(?d);262
10.3;7.3 Gaussian Tent Spaces T1,q(?d);263
10.4;7.4 Gaussian Hardy Spaces H1(?d);272
10.5;7.5 Gaussian BMO(?d) Spaces;282
10.6;7.6 Gaussian Lipschitz Spaces Lip?(?);284
10.7;7.7 Gaussian Besov–Lipschitz Spaces Bp,q?(?d);291
10.8;7.8 Gaussian Triebel–Lizorkin Spaces Fp,q?(?d);301
10.9;7.9 Notes and Further Results;317
11;8 Gaussian Fractional Integrals and Fractional Derivatives,and Their Boundedness on Gaussian Function Spaces;319
11.1;8.1 Riesz and Bessel Potentials with Respect to the GaussianMeasure;319
11.1.1;Gaussian Riesz Potentials;319
11.1.2;Gaussian Bessel Potentials;325
11.2;8.2 Fractional Derivatives with Respect to the Gaussian Measure;326
11.2.1;Gaussian Riesz Fractional Derivate;326
11.2.2;Gaussian Bessel Fractional Derivates;329
11.3;8.3 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Lipschitz Spaces;330
11.4;8.4 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Besov–Lipschitz Spaces;334
11.5;8.5 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Triebel–Lizorkin Spaces;353
11.6;8.6 Notes and Further Results;368
12;9 Singular Integrals with Respect to the Gaussian Measure;374
12.1;9.1 Definition and Boundedness Properties of the Gaussian Riesz Transforms;375
12.2;9.2 Definition and Boundedness Properties of the Higher-Order Gaussian Riesz Transforms;381
12.3;9.3 Alternative Gaussian Riesz Transforms;393
12.4;9.4 Definition and Boundedness Properties of General Gaussian Singular Integrals;401
12.5;9.5 Notes and Further Results;410
13;Correction to: Gaussian Harmonic Analysis;422
14;Appendix;433
14.1;10.1 The Gamma Function and Related Functions;433
14.2;10.2 Classical Orthogonal Polynomials;434
14.2.1;Hermite Polynomials;435
14.2.2;Laguerre Polynomials;437
14.2.3;Generalized Hermite Polynomials;440
14.2.4;Jacobi Polynomials;442
14.3;10.3 Doubling Measures;445
14.4;10.4 Density Theorems for Positive Radon Measures;446
14.5;10.5 Classical Semigroups in Analysis: The Heat and the Poisson Semigroups;452
14.5.1;The Heat Semigroup;452
14.5.2;The Poisson Semigroup;459
14.6;10.6 Interpolation Theory;462
14.7;10.7 Hardy's Inequalities;464
14.8;10.8 Natanson's Lemma and Generalizations;465
14.9;10.9 Forward Differences;468
15;References;472
16;Glossary of Symbols;488
17;Index;499