On the Lipschitz continuity of the heat kernel

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.