Characterization of infinitesimal boundedness of Schrödinger operator

Abstract

In this paper, we characterize the weighted infinitesimal boundedness: for $0<\alpha <n$ and $1<p<\infty$, $$\Vert V\varphi {\Vert }_{L^p(w)}^p\le \epsilon \Vert (-\Delta {)}^{\frac{\alpha }{2}}\varphi {\Vert }_{L^p(w)}^p+C(\epsilon )\Vert \varphi {\Vert }_{L^p(w)}^p.$$ In particular, we extend the classical result due to Maz'ya and Verbitsky by using Carleson condition, localization estimates and capacity theory.

18 pages