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

TL;DR

This paper extends Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces to variable exponents, establishing inclusion relations and interpolation results under certain regularity conditions.

Contribution

Introduction of variable Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces with regularity conditions, expanding the functional analysis framework in Gaussian settings.

Findings

Defined variable Gaussian Besov-Lipschitz spaces

Established inclusion relations among these spaces

Proved interpolation results for the new spaces

Abstract

In a previous paper two of the authors introduced and study Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha}(\gamma_{d})$ and Gaussian Triebel-Lizorkin spaces $F_{p,q}^{\alpha}(\gamma_{d})$. Now, in this paper we introduce the variable Gaussian Besov-Lipschitz spaces $B_{p(\cdot),q(\cdot)}^{\alpha}(\gamma_{d})$ and the variable Gaussian Triebel-Lizorkin spaces $F_{p(\cdot),q(\cdot)}^{\alpha}(\gamma_{d}),$ that is to say, Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces with variable exponents, under certain additional regularity conditions on the exponents $p(\cdot)$ and $q(\cdot)$ introduced by Dalmasso and Scotto. Trivially, they include the Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha}(\gamma_{d})$ and Gaussian Triebel-Lizorkin spaces $F_{p,q}^{\alpha}(\gamma_{d})$. We consider some inclusion relations of those spaces and finally we also prove some interpolation results for them.