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Obtaining upper bounds of heat kernels from lower bounds
Author(s) -
Grigor′yan Alexander,
Hu Jiaxin,
Lau KaSing
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20215
Subject(s) - heat kernel , mathematics , upper and lower bounds , diagonal , dirichlet distribution , kernel (algebra) , dirichlet form , ball (mathematics) , measure (data warehouse) , metric (unit) , combinatorics , mathematical analysis , geometry , operations management , database , computer science , economics , boundary value problem
We show that a near‐diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on‐diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off‐diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near‐diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. © 2007 Wiley Periodicals, Inc.