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

我们在严格局部Dirichlet空间的通用框架中用加倍度量介绍了有界变异（BV）函数的类。在2-Poincaré不等式和弱Bakry-Émery曲率类型条件下，该BV类通过基于热半群的Besov类\（\ mathbf {B} ^ {1,1 / 2}（X）\）来识别。在我们以前的论文中介绍。进一步假设一个准Bakry-Émery曲率类型条件，我们用\（\ mathbf {B} ^ {p，1/2}（X）\来确定Sobolev类\（W ^ {1，p}（X）\））表示\（p> 1 \）。给出了根据等压线和Sobolev不等式并具有明显指数的那些鉴定的结果。