Cheeger-harmonic functions in metric measure spaces revisited

Jiang, Renjin

doi:10.1016/j.jfa.2013.11.022

Cited by 54 publications

(32 citation statements)

References 28 publications

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“…is a generalization of the notion of the Ricci curvature and is a powerful tool in geometric analysis, where Γ is the carré du champ operator and H t is the corresponding heat flow; see [9,10]. On metric measure spaces, it was shown in [30,27] that a locally doubling measure, a local weak L 2 -Poincaré inequality and a Bakry-Émery type inequality are sufficient to guarantee the Lipschitz continuity of Cheeger-harmonic functions. Since the CD(K, N), CD * (K, N) conditions include Finsler geometry, it is not known if the Bakry-Émery type conditions hold under them.…”

Section: Introductionmentioning

confidence: 99%

The Li–Yau inequality and heat kernels on metric measure spaces

Jiang

2015

Journal de Mathématiques Pures et Appliquées

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Please cite this article in press as: R. Jiang, The Li-Yau inequality and heat kernels on metric measure spaces, J. Math. Pures Appl. (2014), http://dx.Suppose that (X, d) is connected, complete and separable, and supp μ = X. We prove that the Li-Yau inequality for the heat flow holds true on (X, d, μ) when K ≥ 0. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flow are established on (X, d, μ) for general K ∈ R. Large time behaviors of heat kernels are also studied.On suppose que (X, d) est connexe, complet et séparable et supp μ = X. Nous démontrons que l'inégalité de Li-Yau pour le flot de la chaleur sur (X, d, μ) est satisfaite lorsque K ≥ 0. De plus, nousétablissons une inégalité de type Baudoin-Garofalo ainsi que des inégalités de Harnack pour ce flot dans la situation plus générale où K ∈ R. Le comportement en temps long des noyaux de la chaleur est aussiétudié.

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Section: Introductionmentioning

confidence: 99%

The Li–Yau inequality and heat kernels on metric measure spaces

Jiang

2015

Journal de Mathématiques Pures et Appliquées

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“…We shall see in Lemma 2.3 below that, under (D) and (P 2 ), (RH ∞ ) is equivalent to Yau's gradient estimate (Y ∞ ) with K = 0. See [30,72,112,114] for more about (Y ∞ ) and (RH ∞ ). Actually, a more natural formulation for the reverse L p -Hölder inequality for gradients of harmonic functions is…”

Section: Background and Main Resultsmentioning

confidence: 99%

“…For degenerate equations satisfying condition (7.2) for some A 2 -weight or qc-weight, although the heat kernel and harmonic functions are known to be Hölder continuous (cf. [17,104,105,106]), harmonic functions and the heat kernel are not Lipschitz in general; see the examples from the introductions of [72,78] for instance.…”

Section: Degenerate (Sub-)elliptic/parabolic Equationsmentioning

confidence: 99%

Gradient estimates for heat kernels and harmonic functions

Coulhon

Jiang

Koskela

et al. 2020

Journal of Functional Analysis

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Let (X, d, µ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a "carré du champ". Assume that (X, d, µ, E ) supports a scale-invariant L 2 -Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2, ∞]: (i) (G p ): L p -estimate for the gradient of the associated heat semigroup; (ii) (RH p ): L p -reverse Hölder inequality for the gradients of harmonic functions; (iii) (R p ): L p -boundedness of the Riesz transform (p < ∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p ∈ (2, ∞), (i), (ii) (iii) are equivalent, while for p = ∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L 2 -Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p = ∞, while for p ∈ (2, ∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.

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“…In the recent remarkable work of Zhang and Zhu [48], the authors proved the important interior Lipschitz regularity of harmonic mappings from certain Alexandrov spaces to NPC spaces. Parallel to the mapping case, the theory of harmonic functions on singular metric spaces also gained growing interest in the last twenty years; see for instance [24,28,30,43] and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%