Heat kernel estimates for critical fractional diffusion operator

TL;DR

This paper constructs and analyzes the heat kernel for a fractional diffusion operator with perturbations, providing sharp estimates and gradient bounds, advancing understanding of such operators in mathematical analysis.

Contribution

It introduces a novel construction of the heat kernel for a perturbed fractional Laplacian with precise estimates and gradient bounds.

Findings

Sharp two-sided heat kernel estimates

Gradient estimates for the heat kernel

Construction of heat kernel for perturbed fractional Laplacian

Abstract

In this work we construct the heat kernel of the 1/2-order Laplacian perturbed by the first-order gradient term in H\"older space and the zero-order potential term in generalized Kato's class, and obtain sharp two-sided estimates as well as the gradient estimate of the heat kernel.