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.

中文翻译：

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狄利克雷空间 I 上通过热半群的 Besov 类：Sobolev 型不等式

我们在 Dirichlet 空间的一般框架中引入了基于热半群的 Besov 类。研究了这些类的一般性质，并获得了该空间尺度中热半群的定量正则化估计。作为本文的一个亮点，我们获得了 Sobolev 不等式的一个深远的 $L^p$-模拟物 $p\ge 1$，N. Varopoulos 在超收缩性假设下证明了 $p=2$热半群。$p=1$ 的情况特别有趣，因为它会产生等周型不等式。