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Estimates of heat kernels for non-local regular Dirichlet forms


Authors: Alexander Grigor’yan, Jiaxin Hu and Ka-Sing Lau
Journal: Trans. Amer. Math. Soc. 366 (2014), 6397-6441
MSC (2010): Primary 47D07; Secondary 28A80, 60J35
Published electronically: July 24, 2014
MathSciNet review: 3267014
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Abstract: In this paper we present new heat kernel upper bounds for a certain class of non-local regular Dirichlet forms on metric measure spaces, including fractal spaces. We use a new purely analytic method where one of the main tools is the parabolic maximum principle. We deduce an off-diagonal upper bound of the heat kernel from the on-diagonal one under the volume regularity hypothesis, restriction of the jump kernel and the survival hypothesis. As an application, we obtain two-sided estimates of heat kernels for non-local regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than $ 2$.


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Additional Information

Alexander Grigor’yan
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
Email: grigor@math.uni-bielefeld.de

Jiaxin Hu
Affiliation: Department of Mathematical Sciences, and Mathematical Sciences Center, Tsinghua University, Beijing 100084, People’s Republic of China
Email: hujiaxin@mail.tsinghua.edu.cn

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Email: kslau@math.cuhk.edu.hk

DOI: https://doi.org/10.1090/S0002-9947-2014-06034-0
Keywords: Heat kernel, non-local Dirichlet form, effective resistance
Received by editor(s): September 23, 2011
Received by editor(s) in revised form: November 13, 2012
Published electronically: July 24, 2014
Additional Notes: The first author was supported by SFB 701 of the German Research Council (DFG) and the Grants from the Department of Mathematics and IMS of CUHK
The second author was supported by NSFC (Grant No. 11071138, 11271122), SFB 701 and the HKRGC Grant of CUHK. The second author is the corresponding author
The third author was supported by the HKRGC Grant, the Focus Investment Scheme of CUHK, and also by the NSFC (no. 11171100, 11371382)
Article copyright: © Copyright 2014 American Mathematical Society