Journal of the Mathematical Society of Japan

Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces

Alexander GRIGOR'YAN, Jiaxin HU, and Ka-Sing LAU

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We give necessary and sufficient conditions for sub-Gaussian estimates of the heat kernel of a strongly local regular Dirichlet form on a metric measure space. The conditions for two-sided estimates are given in terms of the generalized capacity inequality and the Poincaré inequality. The main difficulty lies in obtaining the elliptic Harnack inequality under these assumptions. The conditions for upper bound alone are given in terms of the generalized capacity inequality and the Faber–Krahn inequality.

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J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1485-1549.

First available in Project Euclid: 27 October 2015

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Primary: 35K08: Heat kernel
Secondary: 28A80: Fractals [See also 37Fxx] 31B05: Harmonic, subharmonic, superharmonic functions 35J08: Green's functions 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

generalized capacity heat kernel Poincaré inequality Harnack inequality cutoff Sobolev inequality


GRIGOR'YAN, Alexander; HU, Jiaxin; LAU, Ka-Sing. Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces. J. Math. Soc. Japan 67 (2015), no. 4, 1485--1549. doi:10.2969/jmsj/06741485.

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