Gradient estimates for heat kernels and harmonic functions
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
Published inJournal of Functional Analysis
© 2019 Elsevier Inc.
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  and Auscher-Coulhon . 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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Related funder(s)Academy of Finland
Funding program(s)Centre of Excellence, AoF
Additional information about fundingT. Coulhon and A. Sikora were partially supported by Australian Research Council Discovery grant DP130101302. This research was undertaken while T. Coulhon was employed by the Australian National University. R. Jiang was partially supported by NNSF of China (11922114 & 11671039), P. Koskela was partially supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (project No. 307333). ...
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