Sharp bounds for the Green’s function and the heat kernel

Li, Peter; Tam, Luen Fai; Wang, Jiaping

doi:10.4310/mrl.1997.v4.n4.a13

Cited by 38 publications

(29 citation statements)

References 11 publications

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“…The rest of the proof to (i) is the same as in the proof of Theorem 1.4 of [8]. The proof of the second part follows from estimates given in Theorem 2.1 of Li-Tam-Wang [5]. We leave the details to the interested readers.…”

Section: In Particularñ(h T) Is Bounded From Below If and Only If Mmentioning

confidence: 89%

Addenda to “The entropy formula for linear heat equation”

2004

J Geom Anal

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ABSTRACT. We add two sections to [8] The relation with Li-Yau's gradient estimatesIn this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li-Yau's gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash's 'entropy quantity' − M H log H dv in the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9].Let u(x, t) be a positive solution toSimple calculation shows thatMath Subject Classifications. Primary: 58G11.

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Section: In Particularñ(h T) Is Bounded From Below If and Only If Mmentioning

confidence: 89%

Addenda to “The entropy formula for linear heat equation”

2004

J Geom Anal

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“…The following Lemma was essentially proved in [CM97] firstly, which used Gromov-Hausdorff convergence. Our statement followed from the intrinsic argument in [LTW97].…”

Section: The Estimates Of Green Function and Its Relativesmentioning

confidence: 99%

Integral of Scalar Curvature on Non-parabolic Manifolds

2019

J Geom Anal

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“…(3.10) Remark 3.13. We should mention that the idea of proof of the theorem is from [24]. Note that, due to our calculation, on the one hand, the constant C N in (3.8) above depends only on N, while the constant in the expression of the heat kernel upper estimate in [24, Theorem 2.1] depends not only on N but also on κ ∞ , and on the other hand, from Lemma 3.6, we have…”

Section: )mentioning

confidence: 99%

Sharp heat kernel bounds and entropy in metric measure spaces

2017

Sci. China Math.

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We establish sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying RCD(0, N) ( equivalently, RCD * (0, N)) condition with N ∈ N \ {1} and having maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the (minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in Riemannian manifolds and also in metric measure spaces obtained recently by the author with R. Jiang and H. Zhang in [21].

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Sharp bounds for the Green’s function and the heat kernel

Cited by 38 publications

References 11 publications

Addenda to “The entropy formula for linear heat equation”

Addenda to “The entropy formula for linear heat equation”

Integral of Scalar Curvature on Non-parabolic Manifolds

Sharp heat kernel bounds and entropy in metric measure spaces

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