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 quasi Bakry-'Emery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.
