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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 27 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1445951157

Digital Object Identifier
doi:10.2969/jmsj/06741485

Mathematical Reviews number (MathSciNet)
MR3417504

Zentralblatt MATH identifier
1331.35152

Subjects
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}

Keywords
generalized capacity heat kernel Poincaré inequality Harnack inequality cutoff Sobolev inequality

Citation

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. https://projecteuclid.org/euclid.jmsj/1445951157


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