Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities
Abstract
We introduce heat semigroupbased 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.04267
 Bibcode:
 2018arXiv181104267A
 Keywords:

 Mathematics  Functional Analysis;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Metric Geometry;
 Mathematics  Probability
 EPrint:
 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