Singular integrals and Hardy type spaces for the inverse Gauss measure

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 $\mathcal{A}+\lambda I$, $\lambda \ge 0$, 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.

arXiv comment34 pages