A characterization of Hardy spaces associated with certain Schrödinger operators

Abstract

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