Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

Abstract

We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching $L^p$-analogue, $p\ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest since it yields isoperimetric type inequalities.

The notes arXiv:1806.03428 will be divided in a series of papers. This is the first paper. V3: Some typos are corrected. V4. To appear in Journal of Functional Analysis