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

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,,$\text{ in }{\text{BR}}^n$\end{equation*} where $V\in {\text{rh}}^q$ for $q>\text{f}n2s$ and $0<s<1$. We denote the solution of the above equation by ${\text{cS}}_{V}f:=u$, which is called {\it the inverse of the nonlocal Schr"odinger operator $L_K+V$ with potential $V$}; that is, ${\text{cS}}_V=(L_K+V{)}^{-1}$. Then we obtain a weak Harnack inequality of weak subsolutions of the nonlocal equation \begin{equation}\begin{cases}L_K u+V u=0,,&$\text{ in }\text{Om}\text{,}$ \quad u=g,,&$\text{ in }{\text{BR}}^n\text{s}\text{Om}\text{,}$ \end{cases}\end{equation} where $g\in H^s(\BR^n)$and$\Om$isaboundedopendomainin$\BR^n$ with Lipschitz boundary, and also get an improved decay of a fundamental solution ${\text{fe}}_V$ for $L_K+V$. Moreover, we obtain $L^p$ and $L^p-L^q$ mapping properties of the inverse ${\text{cS}}_V$ of the nonlocal Schr"odinger operator $L_K+V$.