Let L1 be a nonnegative self-adjoint operator in L2(ℝn) satisfying the Davies-Gaffney estimates and L2 a second order divergence form elliptic operator with complex bounded measurable coefficients. A typical example of L1 is the Schrödinger operator −Δ+V, where Δ is the Laplace operator on ℝn and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$0\le V\in L^{1}_{\mathop{\mathrm{loc}}} ({\mathbb{R}}^{n})$\end{document}. Let \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$H^{p}_{L_{i}}(\mathbb{R}^{n})$\end{document} be the Hardy space associated to Li for i∈{1, 2}. In this paper, the authors prove that the Riesz transform \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$D (L_{i}^{-1/2})$\end{document} is bounded from \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$H^{p}_{L_{i}}(\mathbb{R}^{n})$\end{document} to the classical weak Hardy space WHp(ℝn) in the critical case that p=n/(n+1). Recall that it is known that \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$D(L_{i}^{-1/2})$\end{document} is bounded from \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$H^{p}_{L_{i}}(\mathbb{R}^{n})$\end{document} to the classical Hardy space Hp(ℝn) when p∈(n/(n+1), 1].
