The Boundedness of General Alternative Gaussian Singular Integrals on variable Lebesgue spaces with Gaussian measure

TL;DR

This paper investigates the boundedness of a new class of Gaussian singular integrals, called general alternative Gaussian singular integrals, on variable Lebesgue spaces with Gaussian measure, extending previous results on fixed $L^p$ spaces.

Contribution

It establishes boundedness results for these integrals on Gaussian variable Lebesgue spaces under regularity conditions on the variable exponent function.

Findings

Boundedness of the integrals on variable Lebesgue spaces established.

Extension of previous fixed $L^p$ results to variable exponent spaces.

Conditions on $p( abla)$ for boundedness are specified.

Abstract

In a previous paper, we introduced a new class of Gaussian singular integrals, that we called the general alternative Gaussian singular integrals and study the boundedness of them on $L^p(\gamma_d)$, $ 1 < p < \infty.$ In this paper, we study the boundedness of those operators on Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot)$ following a paper by E. Dalmasso and R. Scotto.