Heat kernel on smooth metric measure spaces and applications

We derive a Harnack inequality for positive solutions of the f -heat equation and Gaussian upper and lower bound estimates for the f -heat kernel on complete smooth metric measure spaces with Bakry–Émery Ricci curvature bounded below. Both upper and lower bound estimates are sharp when the Bakry–Éme...

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Bibliographic Details
Published in:Mathematische annalen 2016-06, Vol.365 (1-2), p.309-344
Main Authors: Wu, Jia-Yong, Wu, Peng
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:We derive a Harnack inequality for positive solutions of the f -heat equation and Gaussian upper and lower bound estimates for the f -heat kernel on complete smooth metric measure spaces with Bakry–Émery Ricci curvature bounded below. Both upper and lower bound estimates are sharp when the Bakry–Émery Ricci curvature is nonnegative. The main argument is the De Giorgi–Nash–Moser theory. As applications, we prove an L f 1 -Liouville theorem for f -subharmonic functions and an L f 1 -uniqueness theorem for f -heat equations when f has at most linear growth. We also obtain eigenvalues estimates and f -Green’s function estimates for the f -Laplace operator.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-015-1289-6