Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates
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 Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.
Keywords
Cite
@article{arxiv.1903.10078,
title = {Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates},
author = {Patricia Alonso-Ruiz and Fabrice Baudoin and Li Chen and Luke Rogers and Nageswari Shanmugalingam and Alexander Teplyaev},
journal= {arXiv preprint arXiv:1903.10078},
year = {2022}
}
Comments
The notes arXiv:1806.03428 will be divided in a series of papers. This is the third paper. v2: Final version v3: The proof of Theorem 3.9 contained an error which is corrected in this version