A simplified characterization of stable-like heat kernel estimates

TL;DR

This paper provides a simplified characterization of stable-like heat kernel estimates on metric measure spaces, linking capacity bounds to heat kernel behavior through a novel non-local Whitney technique.

Contribution

It introduces a non-local Whitney blending method and characterizes stable-like heat kernel estimates via capacity bounds and Sobolev inequalities.

Findings

Stable-like heat kernel estimates are characterized by capacity upper bounds.

A new non-local Whitney technique is developed for analysis.

The paper confirms a conjecture by Grigor'yan, Hu, and Hu.

Abstract

We study heat kernel estimates for symmetric pure jump processes on general metric measure spaces. Building on recent progress in the local setting due to S.~Eriksson-Bique, we develop a non-local version of the Whitney blending technique and use it to relate stable-like heat kernel estimates to capacity upper bounds. Under two-sided stable-like bounds on the jump kernel, we show that a capacity upper bound across annuli implies a cutoff Sobolev inequality, and we obtain a characterization of stable-like heat kernel estimates in terms of these conditions. As a consequence, we give an affirmative answer to a conjecture of A. Grigor'yan, E. Hu, and J. Hu.