Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

TL;DR

This paper extends the theory of BV functions to fractal spaces using heat kernel estimates, establishing key properties and demonstrating density in specific fractal examples.

Contribution

It introduces a BV class in Dirichlet spaces with sub-Gaussian heat kernel estimates, linking it to Besov spaces via heat semigroup methods, and explores its properties.

Findings

Proves co-area formulas and Sobolev embeddings for BV functions.

Establishes BV class density in L^1 for certain fractals.

Analyzes BV functions on unbounded fractals like Vicsek and Sierpinski sets.

Abstract

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-\'Emery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $L^1$. The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.