Optimal gradient estimates of heat kernels of stable-like operators

TL;DR

This paper establishes the optimal gradient estimates for heat kernels associated with stable-like operators, using a counterexample to demonstrate the sharpness of these estimates.

Contribution

It provides the first known optimal gradient estimate for heat kernels of stable-like operators, confirming the bounds are sharp.

Findings

Optimal gradient estimates are proven for heat kernels of stable-like operators.

A counterexample demonstrates the sharpness of the estimates.

The results improve understanding of heat kernel behavior for non-local operators.

Abstract

In this note we show the optimal gradient estimate for heat kernels of stable-like operators by providing a counterexample.