Quantitative Boundedness of Littlewood--Paley Functions on Weighted Lebesgue Spaces in the Schrödinger Setting

Abstract

Let $L:=-\Delta +V$ be the Schrödinger operator on ${\mathbb{R}}^n$ with $n\ge 3$, where $V$ is a non-negative potential which belongs to certain reverse Hölder class $R{H}_q({\mathbb{R}}^n)$ 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 $L$.

26 pages; Submitted