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

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 B1,1/2(X) that was introduced in our previous paper. Assuming furthermore a quasi Bakry–Émery curvature type condition, we identify the Sobolev class W1,p(X) with Bp,1/2(X) for $$p>1$$ . Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.