Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

TL;DR

This paper establishes precise two-sided heat kernel estimates and a boundary Harnack principle for a fractional Laplacian with gradient perturbation in bounded domains, extending understanding of such operators with Kato class drifts.

Contribution

It provides the first sharp two-sided heat kernel estimates and boundary Harnack principle for the fractional Laplacian with gradient perturbation in bounded $C^{1,1}$ domains.

Findings

Sharp two-sided heat kernel estimates derived.

Boundary Harnack principle with explicit decay rate established.

Results applicable to operators with Kato class drifts.

Abstract

Suppose that $d\geq2$ and $\alpha\in(1,2)$. Let D be a bounded $C^{1,1}$ open set in $\mathbb{R}^d$ and b an $\mathbb{R}^d$-valued function on $\mathbb{R}^d$ whose components are in a certain Kato class of the rotationally symmetric \alpha-stable process. In this paper, we derive sharp two-sided heat kernel estimates for $\mathcal{L}^b=\Delta^{\alpha/2}+b\cdot\nabla$ in D with zero exterior condition. We also obtain the boundary Harnack principle for $\mathcal{L}^b$ in D with explicit decay rate.