Gradient estimates for heat kernels and harmonic functions

Thierry Coulhon, Renjin Jiang*, Pekka Koskela, Adam Sikora

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    17 Citations (Scopus)


    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.

    Original languageEnglish
    Article number108398
    Pages (from-to)1-67
    Number of pages67
    JournalJournal of Functional Analysis
    Issue number8
    Publication statusPublished - 1 May 2020


    • Harmonic functions
    • Heat kernels
    • Li-Yau estimates
    • Riesz transform


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