scholarly journals Stability of heat kernel estimates for symmetric non-local Dirichlet forms

2021 ◽  
Vol 271 (1330) ◽  
Author(s):  
Zhen-Qing Chen ◽  
Takashi Kumagai ◽  
Jian Wang
Keyword(s):  
Heat Kernel ◽  
Jump Processes ◽  
Metric Measure Spaces ◽  
Kernel Estimates ◽  
Open Problems ◽  
Content Type ◽  
Heat Kernel Estimates ◽  
Measure Spaces ◽  
Non Local ◽  
Volume Doubling

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.

scholarly journals Heat kernel estimates for time fractional equations

2018 ◽  
Vol 30 (5) ◽  
pp. 1163-1192 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Panki Kim ◽  
Takashi Kumagai ◽  
Jian Wang
Keyword(s):  
Heat Kernel ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fundamental Solutions ◽  
Metric Measure Spaces ◽  
Kernel Estimates ◽  
Heat Kernel Estimates ◽  
Fractional Equations ◽  
Measure Spaces ◽  
General Time

AbstractIn this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.

scholarly journals Gaussian heat kernel estimates: From functions to forms

2020 ◽  
Vol 2020 (761) ◽  
pp. 25-79
Author(s):  
Thierry Coulhon ◽  
Baptiste Devyver ◽  
Adam Sikora
Keyword(s):  
Riemannian Manifold ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Compact Riemannian Manifold ◽  
Kernel Estimates ◽  
Doubling Property ◽  
Negative Part ◽  
Heat Kernel Estimates ◽  
Volume Doubling ◽  
Upper Estimate

AbstractOn a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.

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

2018 ◽  
Vol 30 (5) ◽  
pp. 1129-1155 ◽  
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.

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

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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