Characterization of infinitesimal boundedness of Schrödinger operator

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