A1 Journal article (refereed)
Gradient estimates for heat kernels and harmonic functions (2020)

Coulhon, T., Jiang, R., Koskela, P., & Sikora, A. (2020). Gradient estimates for heat kernels and harmonic functions. Journal of Functional Analysis, 278(8), Article 108398. https://doi.org/10.1016/j.jfa.2019.108398

JYU authors or editors

Publication details

All authors or editorsCoulhon, Thierry; Jiang, Renjin; Koskela, Pekka; Sikora, Adam

Journal or seriesJournal of Functional Analysis



Publication year2020


Issue number8

Article number108398


Publication countryUnited States

Publication languageEnglish


Publication open accessNot open

Publication channel open access

Publication is parallel published (JYX)https://jyx.jyu.fi/handle/123456789/67655

Publication is parallel publishedhttps://arxiv.org/abs/1703.02152


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 L2-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p∈(2,∞]:
(i) (Gp): Lp-estimate for the gradient of the associated heat semigroup;
(ii) (RHp): Lp-reverse Hölder inequality for the gradients of harmonic functions;
(iii) (Rp): Lp-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 L2-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.

Keywordsdifferential geometrypartial differential equationspotential theoryharmonic analysis (mathematics)

Free keywordsharmonic functions; heat kernels; Li-Yau estimates; Riesz transform

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Ministry reportingYes

Reporting Year2020

JUFO rating2

Last updated on 2024-22-04 at 23:18