Gaussian Harmonic Analysis

Foreword
Preface
Contents
1 Preliminary Results: The Gaussian Measure and HermitePolynomials
1.1 The Gaussian Measure
1.2 Estimates for the Gaussian Measure of Balls in Rd and the Doubling Condition
1.3 Hermite Polynomials
Hermite Polynomials in One Variable
Hermite Polynomials in d Variables
1.4 Notes and Further Results
2 The Ornstein–Uhlenbeck Operator and the Ornstein–Uhlenbeck Semigroup
2.1 The Ornstein–Uhlenbeck Operator
2.2 Definition and Basic Properties of the Ornstein–Uhlenbeck Semigroup
2.3 The Hypercontractivity Property for the Ornstein–Uhlenbeck Semigroup and the Logarithmic Sobolev Inequality
2.4 Applications of the Hypercontractivity Property
2.5 Notes and Further Results
3 The Poisson–Hermite Semigroup
3.1 Definition and Basic Properties
3.2 Characterization of ∂2∂t2 + L-Harmonic Functions
3.3 Generalized Poisson–Hermite Semigroups
3.4 Conjugate Poisson–Hermite Semigroup
3.5 Notes and Further Results
4 Covering Lemmas, Gaussian Maximal Functions, and Calderón–Zygmund Operators
4.1 Covering Lemmas with Respect to the Gaussian Measure
4.2 Hardy–Littlewood Maximal Function with Respect to the Gaussian Measure and Its Variants
4.3 The Maximal Functions of the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups
The Continuity Properties of the Ornstein–Uhlenbeck Maximal Function
The Continuity Properties of the Poisson–Hermite Maximal Function
4.4 The Local and Global Regions
4.5 Calderón–Zygmund Operators and the Gaussian Measure
4.6 The Non-tangential Maximal Functions for the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups
The Non-tangential Ornstein–Uhlenbeck Maximal Function
The Non-tangential Poisson–Hermite Maximal Function
4.7 Radial and Non-tangential Convergence of the Ornstein–Uhlenbeck and Poisson–Hermite Semigroups
4.8 Notes and Further Results
5 Littlewood–Paley–Stein Theory with Respect to theGaussian Measure
5.1 The Gaussian Littlewood–Paley g Function and Its Variants
5.2 The Higher Order Gaussian Littlewood–Paley g Functions
5.3 The Gaussian Lusin Area Function
5.4 Notes and Further Results
6 Spectral Multiplier Operators with Respect to theGaussian Measure
6.1 Gaussian Spectral Multiplier Operators
6.2 Meyer's Multipliers
6.3 Gaussian Laplace Transform Type Multipliers
6.4 Functional Calculus for the Ornstein–Uhlenbeck Operator
6.5 Notes and Further Results
7 Function Spaces with Respect to the Gaussian Measure
7.1 Gaussian Lebesgue Spaces Lp(γd)
7.2 Gaussian Sobolev Spaces Lβp(γd)
7.3 Gaussian Tent Spaces T1,q(γd)
7.4 Gaussian Hardy Spaces H1(γd)
7.5 Gaussian BMO(γd) Spaces
7.6 Gaussian Lipschitz Spaces Lipα(γ)
7.7 Gaussian Besov–Lipschitz Spaces Bp,qα(γd)
7.8 Gaussian Triebel–Lizorkin Spaces Fp,qα(γd)
7.9 Notes and Further Results
8 Gaussian Fractional Integrals and Fractional Derivatives,and Their Boundedness on Gaussian Function Spaces
8.1 Riesz and Bessel Potentials with Respect to the GaussianMeasure
Gaussian Riesz Potentials
Gaussian Bessel Potentials
8.2 Fractional Derivatives with Respect to the Gaussian Measure
Gaussian Riesz Fractional Derivate
Gaussian Bessel Fractional Derivates
8.3 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Lipschitz Spaces
8.4 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Besov–Lipschitz Spaces
8.5 Boundedness of Fractional Integrals and Fractional Derivatives on Gaussian Triebel–Lizorkin Spaces
8.6 Notes and Further Results
9 Singular Integrals with Respect to the Gaussian Measure
9.1 Definition and Boundedness Properties of the Gaussian Riesz Transforms
9.2 Definition and Boundedness Properties of the Higher-Order Gaussian Riesz Transforms
9.3 Alternative Gaussian Riesz Transforms
9.4 Definition and Boundedness Properties of General Gaussian Singular Integrals
9.5 Notes and Further Results
Correction to: Gaussian Harmonic Analysis
Appendix
10.1 The Gamma Function and Related Functions
10.2 Classical Orthogonal Polynomials
Hermite Polynomials
Laguerre Polynomials
Generalized Hermite Polynomials
Jacobi Polynomials
10.3 Doubling Measures
10.4 Density Theorems for Positive Radon Measures
10.5 Classical Semigroups in Analysis: The Heat and the Poisson Semigroups
The Heat Semigroup
The Poisson Semigroup
10.6 Interpolation Theory
10.7 Hardy's Inequalities
10.8 Natanson's Lemma and Generalizations
10.9 Forward Differences
References
Glossary of Symbols
Index