Riesz transforms associated to Schr\"odinger operators with negative potentials

Abstract

The goal of this paper is to study the Riesz transforms $\na A^{-1/2}$ where $A$ is the Schr\"odinger operator $-\D-V, V\ge 0$, under different conditions on the potential $V$. We prove that if $V$ is strongly subcritical, $\na A^{-1/2}$ is bounded on $L^p(\R^N)$, $N\ge3$, for all $p\in(p_0';2]$ where $p_0'$ is the dual exponent of $p_0$ where $2<\frac{2N}{N-2}