Littlewood-Paley-Stein functions for Schr\"odinger operators

TL;DR

This paper investigates the boundedness of Littlewood-Paley-Stein functions associated with Schr"odinger operators on L^p spaces, establishing boundedness for p in (1,2] and characterizing conditions for p > 2.

Contribution

It provides new results on the boundedness of these functions for Schr"odinger operators, including a sharp characterization for p > 2 related to the potential V.

Findings

Boundedness on L^p for p in (1,2] is established.

For p > 2, boundedness holds only if the potential V is zero.

The results delineate the influence of the potential on function boundedness.

Abstract

We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein functions for Schr\"odinger 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 $V= 0$.