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Local Nash Inequality and Inhomogeneity of Heat Kernels
Author(s) -
Kigami Jun
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504014807
Subject(s) - mathematics , heat kernel , nash equilibrium , riemannian manifold , mathematical analysis , ricci curvature , extension (predicate logic) , curvature , pure mathematics , mathematical optimization , geometry , computer science , programming language
The local Nash inequality is introduced as a natural extension of the classical Nash inequality yielding a space‐homogeneous upper heat kernel estimate. The local Nash inequality contains local information of the heat kernel and is a necessary condition for the space‐inhomogeneous heat kernel estimate involving the volume of balls like the one obtained by Li and Yau for a complete Riemannian manifold with non‐negative Ricci curvature. Under the volume doubling property, the local Nash inequality combined with the exit time estimate is shown to be equivalent to a sub‐Gaussian off‐diagonal upper estimate of the heat kernel allowing space‐inhomogeneity. 2000 Mathematics Subject Classification 60J35, 47D07 (primary), 28A80, 58J35 (secondary).