New Gaussian Riesz transforms on variable Lebesgue spaces

TL;DR

This paper establishes conditions for the boundedness of Gaussian maximal functions and higher order Riesz transforms on variable Lebesgue spaces, extending previous work on Gaussian harmonic analysis.

Contribution

It introduces new boundedness criteria for Gaussian Riesz transforms on variable Lebesgue spaces, generalizing earlier results to higher order transforms.

Findings

Boundedness of Gaussian maximal function on variable Lebesgue spaces established

Higher order Riesz transforms are bounded under new conditions

Extends Gaussian harmonic analysis to variable exponent settings

Abstract

We give sufficient conditions on the exponent $p: \mathbb R^d\rightarrow [1,\infty)$ for the boundedness of the non-centered Gaussian maximal function on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb R^d, \gamma_d)$, as well as of the new higher order Riesz transforms associated with the Ornstein-Uhlenbeck semigroup, which are the natural extensions of the supplementary first order Gaussian Riesz transforms defined by A. Nowak and K. Stempak in \cite{nowakstempak}.