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On the Lipschitz continuity of the heat kernel

Patrizio Bifulco, Delio Mugnolo

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 Hölder 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 Sierpiński gasket, and to structurally damped wave equations. An extension to non-autonomous problems is also discussed.

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 Hölder 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 Sierpiński gasket, and to structurally damped wave equations. An extension to non-autonomous problems is also discussed.
Paper Structure (10 sections, 12 theorems, 77 equations)

This paper contains 10 sections, 12 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. The setup
  3. Main Results
  4. Heat flows in the non-smooth setting
  5. Non-autonomous problems
  6. Some applications
  7. Laplacians on bounded domains
  8. Schrödinger operators on metric graphs
  9. Laplacians on fractals
  10. Structurally damped wave equations

Key Result

Lemma 2.1

For any $r \in [1,\infty)$ and any $\sigma$-finite measure space $(X,\mu)$, there is an isometric isomorphism between $L^\infty(X;L^{r'}(X))$ and $\mathcal{L}(L^r(X);L^\infty(X))$.

Theorems & Definitions (31)

  • Lemma 2.1
  • Definition 1
  • Lemma 2.2
  • proof
  • Theorem 3.1: Hölder continuity in one coordinate
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • ...and 21 more