Some results on Gaussian Besov-Lipschitz spaces and Gaussian Triebel-Lizorkin spaces

TL;DR

This paper develops Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces using Hermite polynomial expansions, exploring their properties, operator continuity, and inclusion relations, with potential extensions to other semigroup frameworks.

Contribution

It introduces new definitions and analyzes properties of Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces, including operator continuity and space inclusions, within Gaussian harmonic analysis.

Findings

Established inclusion relations among Gaussian function spaces.

Proved continuity of key operators like Ornstein-Uhlenbeck semigroup.

Showed Gaussian Sobolev spaces are contained within these new spaces.

Abstract

In this paper we define Besov-Lipschitz and Triebel-Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces $L^p_\alpha(\gamma_d)$ are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusions semigroups.