Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation

TL;DR

This paper provides a direct proof of sharp two-sided estimates and gradient bounds for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in certain open sets, relaxing previous regularity assumptions.

Contribution

It offers a new, direct proof of heat kernel estimates for the fractional Laplacian with gradient perturbation, extending results to less regular domains.

Findings

Established sharp two-sided heat kernel estimates

Derived gradient estimates for the heat kernel

Extended results to $C^{1, heta}$ domains with weaker regularity

Abstract

We give a direct proof of the sharp two-sided estimates, recently established in [4,9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require $D$ to be $C^{1,\theta}$ for some $\theta\in (\alpha/2, 1]$.