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

Abstract

Let $D$ be an open set of $\mathbb{R}^d$, $\alpha\in (0, 2)$ and let $\mathcal{L}_{\alpha}^D$ be the generator of the censored $\alpha$-stable process in $D$. In this paper, we establish sharp two-sided heat kernel estimates for $\mathcal{L}_{\alpha}^D-\kappa$, with $\kappa$ being a non-negative critical potential and $D$ being a $C^{1, \beta}$ open set, $\beta \in ((\alpha-1)_+,1]$. The potential $\kappa$ 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.