  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.
