Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs

2019 ◽  
Vol 2019 (757) ◽  
pp. 89-130 ◽  
Author(s):  
Paul Horn ◽  
Yong Lin ◽  
Shuang Liu ◽  
Shing-Tung Yau
Keyword(s):  
Heat Kernel ◽  
Positive Curvature ◽  
Type Theorem ◽  
Heat Semigroup ◽  
Logarithmic Sobolev Inequalities ◽  
Harnack Inequalities ◽  
Negatively Curved ◽  
Gaussian Estimate ◽  
Heat Kernel Estimate ◽  
Volume Doubling

AbstractStudying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality {\mathrm{CDE}^{\prime}(n,0)}, which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying {\mathrm{CDE}^{\prime}(n,0)}) also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. As a consequence, we obtain that the dimension of the space of harmonic functions on graphs with polynomial growth is finite. In the Riemannian setting, this was originally a conjecture of Yau, which was proved in that context by Colding and Minicozzi. Under the assumption that a graph has positive curvature, we derive a Bonnet–Myers-type theorem. That is, we show the diameter of positively curved graphs is finite and bounded above in terms of the positive curvature. This is accomplished by proving some logarithmic Sobolev inequalities.

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 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 Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

2020 ◽  
Vol 59 (3) ◽  
Author(s):  
Patricia Alonso-Ruiz ◽  
Fabrice Baudoin ◽  
Li Chen ◽  
Luke Rogers ◽  
Nageswari Shanmugalingam ◽  
...  
Keyword(s):  
Heat Kernel ◽  
Heat Semigroup ◽  
Dirichlet Spaces ◽  
Kernel Estimates ◽  
Besov Class ◽  
Heat Kernel Estimates ◽  
Bv Functions

scholarly journals A note on bounded variation and heat semigroup on Riemannian manifolds

2007 ◽  
Vol 76 (1) ◽  
pp. 155-160 ◽  
Author(s):  
A. Carbonaro ◽  
G. Mauceri
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Bounded Variation ◽  
Riemannian Manifolds ◽  
Heat Semigroup ◽  
Functions Of Bounded Variation ◽  
Fixed Radius ◽  
Lower Bound Estimate

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

scholarly journals Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

2021 ◽  
Vol 60 (5) ◽  
Author(s):  
Patricia Alonso-Ruiz ◽  
Fabrice Baudoin ◽  
Li Chen ◽  
Luke Rogers ◽  
Nageswari Shanmugalingam ◽  
...  
Keyword(s):  
Heat Kernel ◽  
Heat Semigroup ◽  
Dirichlet Spaces ◽  
Kernel Estimates ◽  
Besov Class ◽  
Heat Kernel Estimates ◽  
Bv Functions

scholarly journals Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
M. E. Amendola ◽  
L. Rossi ◽  
A. Vitolo
Keyword(s):  
Maximum Principle ◽  
Type Theorem ◽  
Unbounded Domains ◽  
Local Maximum ◽  
Maximum Principles ◽  
Gradient Term ◽  
Harnack Inequalities ◽  
General Unbounded Domains ◽  
Second Order Elliptic Equations ◽  
Weak Harnack Inequality

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

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.

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