Heat kernel estimates for regional fractional Laplacians with multi-singular critical potentials in $C^{1,{\hat{I}}^2}$ open sets

Abstract

Let $D$ be an open set of ${\mathbb{R}}^d$, $\hat{I}\pm \in (0,2)$ and let ${\mathcal{L}}_{\hat{I}}{\pm }^D$ be the generator of the censored $\hat{I}\pm$-stable process in $D$. In this paper, we establish sharp two-sided heat kernel estimates for ${\mathcal{L}}_{\hat{I}}{\pm }^D-\hat{I}º$, with $\hat{I}º$ being a non-negative critical potential and $D$ being a $C^{1,{\hat{I}}^2}$ open set, ${\hat{I}}^2\in ((\hat{I}\pm -1{)}_{+},1]$. The potential $\hat{I}º$ can exhibit multi-singularities and our regularity assumption on $D$ is weaker than the regularity assumed in earlier literature on heat kernel estimates of fractional Laplacians.