Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators

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].