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

TL;DR

This paper studies the mapping properties of nonlocal Schr"odinger operators with certain potentials, establishing weak Harnack inequalities, decay estimates for fundamental solutions, and $L^p$ bounds for the inverse operator.

Contribution

It introduces new $L^p$ and $L^p-L^q$ mapping results for the inverse of nonlocal Schr"odinger operators with specific potentials, along with inequalities and decay estimates.

Findings

Established weak Harnack inequality for solutions.

Derived improved decay estimates for fundamental solutions.

Proved $L^p$ and $L^p-L^q$ mapping properties of the inverse operator.

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 $L_K+V$.