Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $$B^{1,1/2}(X)$$ that was introduced in our previous paper. Assuming furthermore a strong Bakry-Émery curvature type condition, we prove that for $$p>1$$, the Sobolev class $$W^{1,p}(X)$$ can be identified with $$B^{p,1/2}(X)$$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.

Arxiv preprint version

To appear in Calc Var PDE