The book under review is devoted to Gaussian harmonic analysis. Classical harmonic analysis is formulated using as its reference measure the Lebesgue measure in \({\mathbb R}^d\); Gaussian harmonic analysis is formulated using the Gaussian probability measure \(\gamma_d(dx)=\frac{1}{\pi^d} e^{-|x|^2}dx\) in \({\mathbb R}^d\). There are many books dealing with classical harmonic analysis (it is enough to mention the classical books of E. M. Stein), but rather few books on Gaussian harmonic analysis have been published (including two books by the author written in Spanish).
The main goal in Gaussian harmonic analysis is to get analogs to the classical notions with respect to the Gaussian measure. But there are big differences between the Gaussian and the Lebesgue measures: the lack of translation invariance and the doubling property of the underlying measure make many arguments of the classical analysis totally useless in the Gaussian case. Therefore, the required proof methods are completely different.
The structure of the book is as follows. It consists of nine chapters and one appendix.
Chapter 1 focuses on the study of properties of the Gaussian measure in \({\mathbb R}^d, d \geq 1\), and Hermite polynomials, which are orthogonal polynomials with respect to the Gaussian measure.
In Chapter 2, the author defines and studies the Ornstein-Uhlenbeck operator \(L\) and the Ornstein-Uhlenbeck semigroup. They are analogous, in the Gaussian harmonic analysis, to the Laplacian and the heat semigroup in the classical case. The hypercontractivity of the Ornstein-Uhlenbeck semigroup is proved and some of its applications are presented.
In Chapter 3, the author studies the Poisson-Hermite semigroup (this is analogous to the classical Poisson semigroup), its basic properties, the characterization of \(\frac{\partial^2}{\partial t^2}+L\)-harmonic functions, and the conjugate Poisson-Hermite semigroup.
Chapter 4 contains the study of covering lemmas, the Hardy-Littlewood maximal function with respect to the Gaussian measure, and its variants. Additionally, the maximal functions of the Ornstein-Uhlenbeck and Poisson-Hermite semigroups, and their nontangential versions are introduced and their properties are investigated. As a consequence, results on the non-tangential convergence for the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups are obtained. The final part of this chapter contains a discussion of the Calderón-Zygmund operators and their behavior with respect to the Gaussian measure.
Chapter 5 is devoted to the Gaussian Littlewood-Paley-Stein theory.
In Chapter 6, the spectral multiplier operators for Hermite polynomial expansions are studied. The author considers Meyer’s multiplier theorem and spectral multipliers of Laplace transform type. In both cases, the boundedness in \(L^p(\gamma_d)\), for \(1 < p < \infty\) is proved. For the case of spectral multipliers of Laplace transform type, the boundedness in the case \(p = 1\) is also proved. The discussion of the fact that the Ornstein-Uhlenbeck operator has a bounded holomorphic functional calculus is included.
In Chapter 7, the analogs of classical functional spaces with respect to the Gaussian measure are under discussion (Lebesgue spaces, Sobolev spaces, sent spaces, Hardy spaces, bounded mean oscillation (BMO) spaces, Lipschitz spaces, Besov-Lipschitz spaces and Triebel-Lizorkin spaces). Most of the time, even if the spaces look similar to the classical ones, the proofs are different, mainly because the Gaussian measure is not invariant by translation.
In Chapter 8, several important operators in Gaussian harmonic analysis are introduced: Riesz and Bessel potentials with respect to the Ornstein-Uhlenbeck operator, Riesz and Bessel fractional derivatives. Their regularity on Gaussian Lipschitz spaces, on Gaussian Besov-Lipschitz spaces, and on Gaussian Triebel-Lizorkin spaces is proved. The results obtained are essentially similar to the classical results.
The aim of Chapter 9 is to develop the theory of singular integrals with respect to the Gaussian measure. The exposition begins with the study of the Gaussian Riesz transform, then the higher-order Gaussian Riesz transforms, and finally, the author considers a fairly general class of Gaussian singular integrals. The full proofs of the boundedness properties in each case are presented, even though the Gaussian Riesz transform and higher-order Gaussian Riesz transforms are particular cases of the general class of Gaussian singular integrals.
The Appendix contains the basics of the following topics: Gamma function and related functions, main properties and formulas of all the classical orthogonal polynomials, doubling measures in a general setting, classical semigroups in analysis (the heat and the Poisson semigroups), interpolation theory, Hardy’s inequalities, Natanson’s lemma and some of its generalizations, forward differences.
This well-written and organized (mainly self-contained) book is a reader-friendly manual in the field of Gaussian harmonic analysis. It can be recommended for experts and for graduate, postgraduate and doctoral students.