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

With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry–Émery curvature type condition...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 60; no. 5
Main Authors Alonso-Ruiz, Patricia, Baudoin, Fabrice, Chen, Li, Rogers, Luke, Shanmugalingam, Nageswari, Teplyaev, Alexander
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2021
Springer Nature B.V
Subjects
Analysis
Calculus of Variations and Optimal Control; Optimization
Control
Dirichlet problem
Estimates
Fractals
Kernels
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Semigroups
Systems Theory
Theoretical
26A45
31E05
31C25
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Summary:With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the 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 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.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-021-02041-2