Singular integrals and Hardy type spaces for the inverse Gauss measure

TL;DR

This paper studies the boundedness properties of certain singular integral operators related to a weighted Laplacian on a measure inverse to a Gaussian, establishing results for Hardy spaces and weak type estimates.

Contribution

It introduces new Hardy-type spaces adapted to the inverse Gaussian measure and analyzes the boundedness of associated singular integral operators.

Findings

Boundedness of imaginary powers and Riesz transforms on Hardy spaces

Unboundedness results for certain operators

Weak type (1,1) properties of these operators

Abstract

Let $\gamma_{-1}$ be the absolutely continuous measure on $\mathbb{R}^n$ whose density is the reciprocal of a Gaussian and consider the natural weighted Laplacian $\mathcal{A}$ on $L^2(\gamma_{-1})$. In this paper, we prove boundedness and unboundedness results for the purely imaginary powers and the first order Riesz transforms associated with the translated operators $\mathcal{A}+\lambda I$, $\lambda\geq0$, from certain new Hardy-type spaces adapted to $\gamma_{-1}$ to $L^1(\gamma_{-1})$. We also investigate the weak type $(1,1)$ of these operators.