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

2019/03/24 by Alonso-Ruiz, Patricia, Baudoin, Fabrice, Chen, Li +3
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.1903.10078

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-Émery 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 L¹ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in L¹. The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

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