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