Characterization of infinitesimal boundedness of Schr\"{o}dinger operator

Abstract

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