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Stability of heat kernel estimates for symmetric non-local Dirichlet forms
About this Title
Zhen-Qing Chen, Takashi Kumagai and Jian Wang
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 271, Number 1330
ISBNs: 978-1-4704-4863-9 (print); 978-1-4704-6638-1 (online)
DOI: https://doi.org/10.1090/memo/1330
Published electronically: August 16, 2021
Keywords: Symmetric jump process,
metric measure space,
heat kernel estimate,
stability,
Dirichlet form,
cut-off Sobolev inequality,
capacity,
Faber-Krahn inequality,
Lévy system,
jumping kernel,
exit time
Table of Contents
Chapters
- 1. Introduction and Main Results
- 2. Preliminaries
- 3. Implications of heat kernel estimates
- 4. Implications of $\mathrm {CSJ}(\phi )$ and $\mathrm {J}_{\phi , \geq }$
- 5. Consequences of condition $\mathrm {J}_{\phi }$ and mean exit time condition $\mathrm {E}_{\phi }$
- 6. Applications and Examples
- 7. Appendix
- Acknowledgment
Abstract
In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for $\alpha$-stable-like processes even with $\alpha \ge 2$ when the underlying spaces have walk dimensions larger than $2$, which has been one of the major open problems in this area.- Sebastian Andres and Martin T. Barlow, Energy inequalities for cutoff functions and some applications, J. Reine Angew. Math. 699 (2015), 183–215. MR 3305925, DOI 10.1515/crelle-2013-0009
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