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.
Categories: math.FA math.CA