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

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

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