On the Bakry–Émery Condition, the Gradient Estimates and the Local-to-Global Property of \(\mathsf{RCD}^*(K,N)\) Metric Measure Spaces

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• 作者：Luigi Ambrosio ; Andrea Mondino ; Giuseppe Savaré
• 关键词：Bakry–Émery curvature bounds ; Dirichlet forms ; CD (K ; N) spaces ; Optimal transport ; 35K05 ; 35K90 ; 31C25 ; 58J35 ; 60J65
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：26
• 期：1
• 页码：24-56
• 全文大小：692 KB
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• 作者单位：Luigi Ambrosio (1)
  Andrea Mondino (2)
  Giuseppe Savaré (3)
  
  1. Scuola Normale Superiore, Pisa, Italy
  2. ETH, Zurich, Zurich, Switzerland
  3. University of Pavia, Pavia, Italy
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

We prove higher summability and regularity of \(\Gamma \big (f\big )\) for functions \(f\) in spaces satisfying the Bakry–Émery condition \(\mathsf{BE}(K,\infty )\). As a byproduct, we obtain various equivalent weak formulations of \(\mathsf{BE}(K,N)\) and we prove the Local-to-Global property of the \(\mathsf{RCD}^*(K,N)\) condition in locally compact metric measure spaces \((X,\mathsf{d},\mathfrak m)\), without assuming a priori the non-branching condition on the metric space. Keywords Bakry–Émery curvature bounds Dirichlet forms CD (K, N) spaces Optimal transport