On the Riesz Transforms for the inverse Gauss measure

TL;DR

This paper investigates the boundedness properties of Riesz transforms linked to a specific inverse Gaussian measure, establishing weak type (1,1) results for first and second order transforms and showing higher order transforms lack this property.

Contribution

It provides new results on the weak type (1,1) boundedness of Riesz transforms for the inverse Gaussian measure, especially distinguishing between low and high order transforms.

Findings

First and second order Riesz transforms are of weak type (1,1)

Higher order Riesz transforms are not of weak type (1,1)

Results extend understanding of harmonic analysis under inverse Gaussian measures

Abstract

Let $\gamma_{-1}$ be the absolutely continuous measure on $\mathbb{R}^n$ whose density is the reciprocal of a Gaussian function. Let further $\mathscr{A}$ be the natural self-adjoint Laplacian on $L^2(\gamma_{-1})$. In this paper, we prove that the Riesz transforms associated with $\mathscr{A}$ of order one or two are of weak type $(1,1)$, but that those of higher order are not.