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 \(\mathscr{A}\) 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.