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.

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