Heat kernel analysis on diamond fractals

Abstract

This paper presents a detailed analysis of the heat kernel on an (N×N)-parameter family of compact metric measure spaces which do not satisfy the volume doubling property. In particular, uniform bounds of the heat kernel, its Lipschitz continuity and the continuity of the corresponding heat semigroup are studied; a specific example is presented revealing a logarithmic correction. The estimates are applied to derive functional inequalities of interest in describing the convergence to equilibrium of the diffusion process.