Hardy spaces H1 for Schrödinger operators with compactly supported potentials

Let L=-Δ+V be a Schrödinger operator on ℝd, d≥3, where V is a non-negative compactly supported potential that belongs to Lp for some p>d/2. Let {Kt}t>0 denote the semigroup of linear operators generated by -L. For a function f we define its H1L-norm by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\| f\|_{H^1_L}=\| \sup_{t>0} |K_t f(x)|\|_{L^1(dx)}$\end{document}. It is proved that for a properly defined weight w a function f belongs to H1L if and only if wf∈H1(ℝd), where H1(ℝd) is the classical real Hardy space.