On the Lipschitz continuity of the heat kernel

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.