Riesz transforms associated to Schrödinger operators with negative potentials

Abstract

The goal of this paper is to study the Riesz transforms $\nabla A^{-1/2}$ where $A$ is the Schrödinger operator $-\Delta-V$, $V\ge 0$, under different conditions on the potential $V$. We prove that if $V$ is strongly subcritical, $\nabla A^{-1/2}$ is bounded on $L^p(\mathbb{R}^N)$, $N\ge3$, for all $p\in(p_0';2]$ where $p_0'$ is the dual exponent of $p_0$ where $2<\frac{2N}{N-2}<p_0<\infty$; and we give a counterexample to the boundedness on $L^p(\mathbb{R}^N)$ for $p\in(1;p'_0)\cup(p_{0*};\infty)$ where $p_{0*}:=\frac{p_0N}{N+p_0}$ is the reverse Sobolev exponent of $p_0$. If the potential is strongly subcritical in the Kato subclass $K_N^{\infty}$, then $\nabla A^{-1/2}$ is bounded on $L^p(\mathbb{R}^N)$ for all $p\in(1;2]$, moreover if it is in $L^{N/2}_w(\mathbb{R}^N)$ then $\nabla A^{-1/2}$ is bounded on $L^p(\mathbb{R}^N)$ for all $p\in(1;N)$. We prove also boundedness of $V^{1/2}A^{-1/2}$ with the same conditions on the same spaces. Finally we study these operators on manifolds. We prove that our results hold on a class of Riemannian manifolds.