Riesz Transforms of Schrödinger Operators on Manifolds

We consider Schrödinger operators A=−Δ+V on Lp(M) where M is a complete Riemannian manifold of homogeneous type and V=V+−V− is a signed potential. We study boundedness of Riesz transform type operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla A^{-\frac{1}{2}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|V|^{\frac{1}{2}}A^{-\frac{1}{2}}$\end{document} on Lp(M). When V− is strongly subcritical with constant α∈(0,1) we prove that such operators are bounded on Lp(M) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\in(p_{0}', 2]$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{0}'=1$\end{document} if N≤2, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{0}'=(\frac{2N}{(N-2)(1-\sqrt{1-\alpha })})' \in (1, 2)$\end{document} if N>2. We also study the case p>2. With additional conditions on V and M we obtain boundedness of ∇A−1/2 and |V|1/2A−1/2 on Lp(M) for p∈(1,inf (q1,N)) where q1 is such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla(-\Delta)^{-\frac{1}{2}}$\end{document} is bounded on Lr(M) for r∈[2,q1).