Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradient

TL;DR

This paper establishes precise two-sided bounds and regularity properties for the heat kernel of the spectral fractional Laplacian with gradient perturbations in bounded domains, advancing understanding of such operators.

Contribution

It provides sharp heat kernel estimates and gradient regularity results for spectral fractional Laplacians with time-dependent gradient perturbations in bounded domains.

Findings

Sharp two-sided heat kernel estimates are proved.

Gradient estimates and Hölder continuity of the heat kernel's gradient are established.

Results apply to spectral fractional Laplacians with time-dependent gradient perturbations.

Abstract

By using Duhamel's formula, we prove sharp two-sided estimates for the heat kernel of spectral fractional Laplacian with time-dependent gradient perturbation in bounded $C^{1,1}$ domains. Moreover, we also obtain gradient estimate as well as H\"older continuity of the gradient of the heat kernel.