  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.
