Heat Kernels and Green Functions on Metric Measure Spaces

2014 ◽  
Vol 66 (3) ◽  
pp. 641-699 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Jiaxin Hu
Keyword(s):  
Exit Time ◽  
Metric Measure Spaces ◽  
Doubling Property ◽  
Technical Tool ◽  
Capacity Estimate ◽  
Gaussian Estimate ◽  
Mean Exit Time ◽  
Measure Spaces ◽  
Volume Doubling ◽  
Main Technical Tool

AbstractWe prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball.

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 Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

2021 ◽  
Author(s):  
Hiroaki Aikawa ◽  
Michiel van den Berg ◽  
Jun Masamune
Keyword(s):  
Heat Semigroup ◽  
Doubling Property ◽  
Dirichlet Laplacian ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Connected Sets ◽  
Gaussian Estimate ◽  
Negative Constant ◽  
Volume Doubling ◽  
Dirichlet Heat Kernel

AbstractLet M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $$L^2(D)$$ L 2 ( D ) , and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.

scholarly journals An alternative capacity in metric measure spaces

2021 ◽  
Vol 18 (2) ◽  
pp. 196-208
Author(s):  
Olli Martio
Keyword(s):  
Measure Space ◽  
Lipschitz Functions ◽  
Metric Measure Space ◽  
Good Measure ◽  
Metric Measure Spaces ◽  
Doubling Property ◽  
Condenser Capacity ◽  
Regularity Properties ◽  
Measure Spaces ◽  
Curve Family

A new condenser capacity $\CMp(E,G)$ is introduced as an alternative to the classical Dirichlet capacity in a metric measure space $X$. For $p>1$, it coincides with the $M_p$-modulus of the curve family $\Gamma(E,G)$ joining $\partial G$ to an arbitrary set $E \subset G$ and, for $p = 1$, it lies between $AM_1(\Gamma(E,G))$ and $M_1(\Gamma(E,G))$. Moreover, the $\CMp(E,G)$-capacity has good measure theoretic regularity properties with respect to the set $E$. The $\CMp(E,G)$-capacity uses Lipschitz functions and their upper gradients. The doubling property of the measure $\mu$ and Poincar\'e inequalities in $X$ are not needed.

scholarly journals Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

2017 ◽  
Vol 272 (8) ◽  
pp. 3311-3346 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Eryan Hu ◽  
Jiaxin Hu
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Non Local

scholarly journals 1PT004 Mean Exit Time Analysis about Water Molecules around Membrane Surfaces(The 50th Annual Meeting of the Biophysical Society of Japan)

2012 ◽  
Vol 52 (supplement) ◽  
pp. S84
Author(s):  
Eiji Yamamoto ◽  
Takuma Akimoto ◽  
Yoshinori Hirano ◽  
Masato Yasui ◽  
Kenji Yasuoka
Keyword(s):  
Annual Meeting ◽  
Exit Time ◽  
Water Molecules ◽  
Time Analysis ◽  
Membrane Surfaces ◽  
Mean Exit Time ◽  
Biophysical Society

scholarly journals Square-integrable representations of non-unimodular groups

1976 ◽  
Vol 15 (1) ◽  
pp. 1-12 ◽  
Author(s):  
A.L. Carey
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Hilbert Algebra ◽  
Technical Tool ◽  
Integrable Representation ◽  
Hilbert Algebras ◽  
Orthogonality Relations ◽  
Square Integrable Representation ◽  
Main Technical Tool

In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.

scholarly journals A note on the extension of BV functions in metric measure spaces

2008 ◽  
Vol 340 (1) ◽  
pp. 197-208 ◽  
Author(s):  
Annalisa Baldi ◽  
Francescopaolo Montefalcone
Keyword(s):  
Metric Measure Spaces ◽  
Bv Functions ◽  
Measure Spaces
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