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 $-L=\Delta - V(x)$ on $\mathbb R^d$, $d\geq 3$, where $V(x)\geq 0$ satisfies $\Delta^{-1} V\in L^\infty$. We say that an $L^1$-function $f$ belongs to the Hardy space $H^1_L$ if the maximal function $\mathcal M_L f(x) = \sup_{t>0} |K_tf(x)|$ belongs to $L^1(\mathbb R^d)$. We prove that the operator $(-\Delta)^{1\slash 2} L^{-1\slash 2}$ is an isomorphism of the space $H^1_L$ with the classical Hardy space $H^1(\mathbb R^d)$ whose inverse is $L^{1\slash 2} (-\Delta)^{-1\slash 2}$. As a corollary we obtain that the space $H^1_L$ is characterized by the Riesz transforms $R_j=\frac{\partial}{\partial x_j}L^{-1\slash 2}$.