Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

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• 作者：Niko Marola ; Michele Miranda Jr. ; Nageswari Shanmugalingam
• 关键词：Bakry–Émery condition ; Bounded variation ; Dirichlet form ; Doubling measure ; Heat kernel ; Heat semigroup ; Isoperimetric inequality ; Metric space ; Perimeter ; Poincaré inequality ; Sets of finite perimeter ; Total variation
• 刊名：Potential Analysis
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：45
• 期：4
• 页码：609-633
• 全文大小：415 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Potential Theory
  Probability Theory and Stochastic Processes
  Geometry
  Functional Analysis
  
• 出版者：Springer Netherlands
• ISSN：1572-929X
• 卷排序：45

文摘

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N1,1-spaces) and the theory of heat semigroups (a concept related to N1,2-spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting.