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Dimension-Free Harnack Inequalities on \(\hbox {RCD}(K, \infty )\) Spaces

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Abstract

The dimension-free Harnack inequality for the heat semigroup is established on the \(\mathrm{{RCD}}(K,\infty )\) space, which is a non-smooth metric measure space having the Ricci curvature bounded from below in the sense of Lott–Sturm–Villani plus the Cheeger energy being quadratic. As its applications, the heat semigroup entropy-cost inequality and contractivity properties of the semigroup are studied, and a strong-enough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.

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Acknowledgments

The author would like to thank Dr. Dejun Luo for his careful reading of the first version of this work in which the reference measure is assumed to be a probability measure. And thanks are also given to an anonymous referee for his or her careful reading and meticulous comments, which make the presentation of this paper better.

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  1. Huaiqian Li

    Present address: Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia

Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Huaiqian Li

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Correspondence to Huaiqian Li.

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Partially supported by the National Natural Science Foundation of China (NSFC) No. 11401403 and the Australian Research Council (ARC) Grant DP130101302.

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Li, H. Dimension-Free Harnack Inequalities on \(\hbox {RCD}(K, \infty )\) Spaces. J Theor Probab 29, 1280–1297 (2016). https://doi.org/10.1007/s10959-015-0621-0

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  • DOI: https://doi.org/10.1007/s10959-015-0621-0

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