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

@article{AlonsoRuiz2020BesovCV, title={Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates}, author={Patricia Alonso-Ruiz and Fabrice Baudoin and Li Chen and Luke G. Rogers and Nageswari Shanmugalingam and Alexander Teplyaev}, journal={Calculus of Variations and Partial Differential Equations}, year={2020}, volume={59}, pages={1-32} }

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 $$\mathbf {B}^{1,1/2}(X)$$ B 1 , 1 / 2 ( X ) that was introduced in our previous paper. Assuming furthermore a quasi Bakry–Émery curvature type condition, we identify the Sobolev class $$W^{1,p}(X)$$ W 1 , p…

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