$L^p$ mapping properties for nonlocal Schr\"odinger operators with certain potential

Abstract

In this paper, we consider nonlocal Schr\"odinger equations with certain potentials $V$ given by an integro-differential operator $L_K$ as follows; \begin{equation*}L_K u+V u=f\,\,\text{ in $\BR^n$ }\end{equation*} where $V\in\rh^q$ for $q>\f{n}{2s}$ and $0<s<1$. We denote the solution of the above equation by $\cS_V f:=u$, which is called {\it the inverse of the nonlocal Schr\"odinger operator $L_K+V$ with potential $V$}; that is, $\cS_V=(L_K+V)^{-1}$. Then we obtain a weak Harnack inequality of weak subsolutions of the nonlocal equation \begin{equation}\begin{cases}L_K u+V u=0\,\,&\text{ in $\Om$,} \quad u=g\,\,&\text{ in $\BR^n\s\Om$,} \end{cases}\end{equation} where $g\in H^s(\BR^n)$ and $\Om$ is a bounded open domain in $\BR^n$ with Lipschitz boundary, and also get an improved decay of a fundamental solution $\fe_V$ for $L_K+V$. Moreover, we obtain $L^p$ and $L^p-L^q$ mapping properties of the inverse $\cS_V$ of the nonlocal Schr\"odinger operator $L_K+V$.