A characterization of Hardy spaces associated with certain Schr\"odinger operators

Let $\{K_t{\}}_{t>0}$ be the semigroup of linear operators generated by a Schr"odinger operator $-L=\mathrm{\Delta }-V(x)$ on ${\mathbb{R}}^d$, $d\ge 3$, where $V(x)\ge 0$ satisfies ${\mathrm{\Delta }}^{-1}V\in L^{\mathrm{\infty }}$. We say that an $L^1$-function $f$ belongs to the Hardy space $H_L^1$ if the maximal function ${\mathcal{M}}_{L}f(x)={\sup }_{t>0}\mathrm{\mid }{K}_{t}f(x)\mathrm{\mid }$ belongs to $L^1({\mathbb{R}}^d)$. We prove that the operator $(-\mathrm{\Delta }{)}^{1\text{slash}2}{L}^{-1\text{slash}2}$ is an isomorphism of the space $H_L^1$ with the classical Hardy space $H^1(\mathbb R^d)$whoseinverseis$L^{1\slash 2} (-\Delta)^{-1\slash 2}$. As a corollary we obtain that the space $H_L^1$ is characterized by the Riesz transforms $R_j=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}{L}^{-1\text{slash}2}$.