A rigidity result for metric measure spaces with Euclidean heat kernel

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 α \alpha -stable-like processes even with α ≥ 2 \alpha \ge 2 when the underlying spaces have walk dimensions larger than 2 2 , which has been one of the major open problems in this area.

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Locality of the Heat Kernel on Metric Measure Spaces

Complex Analysis and Operator Theory ◽

10.1007/s11785-017-0749-2 ◽

2017 ◽

Vol 12 (3) ◽

pp. 729-766

Author(s):

Olaf Post ◽

Ralf Rückriemen

Keyword(s):

Heat Kernel ◽

Metric Measure Spaces ◽

Measure Spaces

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Sharp heat kernel bounds and entropy in metric measure spaces

Science China Mathematics ◽

10.1007/s11425-016-0314-9 ◽

2017 ◽

Vol 61 (3) ◽

pp. 487-510 ◽

Cited By ~ 2

Author(s):

Huaiqian Li

Keyword(s):

Heat Kernel ◽

Metric Measure Spaces ◽

Measure Spaces ◽

Kernel Bounds

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Heat kernel estimates for jump processes of mixed types on metric measure spaces

Probability Theory and Related Fields ◽

10.1007/s00440-007-0070-5 ◽

2007 ◽

Vol 140 (1-2) ◽

pp. 277-317 ◽

Cited By ~ 83

Author(s):

Zhen-Qing Chen ◽

Takashi Kumagai

Keyword(s):

Heat Kernel ◽

Jump Processes ◽

Metric Measure Spaces ◽

Kernel Estimates ◽

Heat Kernel Estimates ◽

Measure Spaces ◽

Mixed Types

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The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces

Forum Mathematicum ◽

10.1515/forum-2017-0072 ◽

2018 ◽

Vol 30 (5) ◽

pp. 1129-1155 ◽

Cited By ~ 1

Author(s):

Jiaxin Hu ◽

Xuliang Li

Keyword(s):

Heat Kernel ◽

Upper Bound ◽

Dirichlet Form ◽

Measure Space ◽

Dirichlet Forms ◽

Upper Bounds ◽

Metric Measure Spaces ◽

Unified Approach ◽

Measure Spaces ◽

Non Local

AbstractWe apply the Davies method to prove that for any regular Dirichlet form on a metric measure space, an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound, a cutoff inequality on any two concentric balls, and the jump kernel upper bound, for any walk dimension. If in addition the jump kernel vanishes, that is, if the Dirichlet form is strongly local, we obtain a sub-Gaussian upper bound. This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.

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HARDY SPACES ON METRIC MEASURE SPACES WITH GENERALIZED SUB-GAUSSIAN HEAT KERNEL ESTIMATES

Journal of the Australian Mathematical Society ◽

10.1017/s144678871700012x ◽

2017 ◽

Vol 104 (2) ◽

pp. 162-194

Author(s):

LI CHEN

Keyword(s):

Hardy Space ◽

Heat Kernel ◽

Hardy Spaces ◽

Riesz Transform ◽

Square Function ◽

Space Theory ◽

Metric Measure Spaces ◽

Kernel Estimates ◽

Previous Theory ◽

Measure Spaces

Hardy space theory has been studied on manifolds or metric measure spaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However, there are natural examples where one finds a mix of both behaviours (locally Gaussian and at infinity sub-Gaussian), in which case the previous theory does not apply. Still we define molecular and square function Hardy spaces using appropriate scaling, and we show that they agree with Lebesgue spaces in some range. Besides, counterexamples are given in this setting that the $H^{p}$ space corresponding to Gaussian estimates may not coincide with $L^{p}$. As a motivation for this theory, we show that the Riesz transform maps our Hardy space $H^{1}$ into $L^{1}$.

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