Heat kernel estimates for regional fractional Laplacians with multi-singular critical potentials in C^1,β open sets

Let D be an open set of R^d, α∈(0,2) and let L_α^D be the generator of the censored α-stable process in D. In this paper, we establish sharp two-sided heat kernel estimates for L_α^D−κ, with κ being a non-negative critical potential and D being a C^1,β open set, β∈((α−1)₊,1]. The potential κ 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.