On the Riesz Transforms for the inverse Gauss measure

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.