On the Riesz transforms for the inverse Gauss measure

Abstract

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