Littlewood-Paley-Stein functions for Schrödinger operators

Abstract

We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein functions for Schrödinger operators $-\Delta +V$ with nonnegative potential $V$. These functions are proved to be bounded on $L^p$ for all $p\in (1,2]$. The situation for $p>2$ is different. We prove for a class of potentials that the boundedness on $L^p$ for some $p>d$ holds if and only if $V=0$.