Quantitative Boundedness of Littlewood--Paley Functions on Weighted Lebesgue Spaces in the Schr\"{o}dinger Setting

TL;DR

This paper establishes quantitative bounds for Littlewood--Paley functions related to Schr"odinger operators on weighted Lebesgue spaces, expanding understanding of harmonic analysis in the Schr"odinger setting.

Contribution

It provides new quantitative weighted boundedness results for Littlewood--Paley functions associated with Schr"odinger operators on $L^p(w)$ spaces.

Findings

Boundedness of $g_L$, $S_L$, and $g_{L,\lambda}^*$ on weighted spaces.

Results depend explicitly on weight characteristics.

Extends classical harmonic analysis to Schr"odinger operators.

Abstract

Let $L:=-\Delta+V$ be the Schr\"{o}dinger operator on $\mathbb{R}^n$ with $n\geq 3$, where $V$ is a non-negative potential which belongs to certain reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with $q\in (n/2,\,\infty)$. In this article, the authors obtain the quantitative weighted boundedness of Littlewood--Paley functions $g_L$, $S_L$ and $g_{L,\,\lambda}^\ast$, associated to $L$, on weighted Lebesgue spaces $L^p(w)$, where $w$ belongs to the class of Muckenhoupt $A_p$ weights adapted to $L$.