Potential Analysis

, Volume 44, Issue 3, pp 601–627 | Cite as

Heat Kernel Bounds on Metric Measure Spaces and Some Applications



Let (X,d,μ) be a R C D (K,N) space with \(K\in \mathbb {R}\) and N∈[1,). We derive the upper and lower bounds of the heat kernel on (X,d,μ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L p boundedness of (local) Riesz transforms.


Metric measure space Ricci curvature Heat kernel Heat equation Riesz transform 

Mathematics Subject Classification (2010)

Primary 53C23 Secondary 35K08 35K05 42B20 47B06 


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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijingChina
  2. 2.School of MathematicsSichuan UniversityChengduChina
  3. 3.Department of MathematicsSun Yat-sen UniversityGuangzhouChina

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