On the Lipschitz continuity of the heat kernel

TL;DR

This paper establishes conditions under which heat kernels on various complex spaces exhibit Lipschitz or H"older continuity, using semigroup smoothing properties and Morrey-type inequalities.

Contribution

It provides new sufficient conditions for kernel regularity applicable to diverse settings like fractals, quantum graphs, and damped wave equations.

Findings

Lipschitz continuity of kernels on fractals and quantum graphs

H"older regularity results for pseudodifferential operators

Extension of results to non-autonomous problems

Abstract

We study integral kernels of strongly continuous semigroups on Lebesgue spaces over metric measure spaces. Based on semigroup smoothing properties and abstract Morrey-type inequalities, we give sufficient conditions for H\"older or Lipschitz continuity of the kernels. We apply our results to (pseudo)differential operators on domains and quantum graphs, to Laplacians on a class of fractals including the Sierpi\'nski gasket, and to structurally damped wave equations. An extension to non-autonomous problems is also discussed.