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

ISSN0022-1236

eISSN1096-0783

Publication year2020

Volume278

Issue number8

Article number108398

PublisherElsevier

Publication countryUnited States

Publication languageEnglish

DOIhttps://doi.org/10.1016/j.jfa.2019.108398

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


Abstract

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