  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.
