TY - JOUR
T1 - Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms
AU - Chen, Zhen Qing
AU - Kumagai, Takashi
AU - Wang, Jian
N1 - Funding Information:
We are grateful to the referee for helpful comments. The research of Zhen-Qing Chen is partially supported by Simons Foundation Grant 520542 , a Victor Klee Faculty Fellowship at UW , and NNSFC grant 11731009 . The research of Takashi Kumagai is supported by JSPS KAKENHI Grant Number JP17H01093 and by the Alexander von Humboldt Foundation . The research of Jian Wang is supported by the National Natural Science Foundation of China (No. 11831014 ), the Program for Probability and Statistics: Theory and Application (No. IRTL1704), and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ).
Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/11/18
Y1 - 2020/11/18
N2 - In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M,d,μ): E(f,g)=E(c)(f,g)+∫M×M(f(x)−f(y))(g(x)−g(y))J(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) is a symmetric Radon measure on M×M. Under general volume doubling condition on (M,d,μ) and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp. two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2.
AB - In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M,d,μ): E(f,g)=E(c)(f,g)+∫M×M(f(x)−f(y))(g(x)−g(y))J(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) is a symmetric Radon measure on M×M. Under general volume doubling condition on (M,d,μ) and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp. two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2.
KW - Cut-off Sobolev inequality
KW - Heat kernel estimate
KW - Metric measure space
KW - Parabolic Harnack inequality
KW - Stability
KW - Symmetric Dirichlet form
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U2 - 10.1016/j.aim.2020.107269
DO - 10.1016/j.aim.2020.107269
M3 - Article
AN - SCOPUS:85087659503
SN - 0001-8708
VL - 374
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 107269
ER -