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

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  • Huaiqian Li

    (Sichuan University
    Macquarie University)

Abstract

The dimension-free Harnack inequality for the heat semigroup is established on the $$\mathrm{{RCD}}(K,\infty )$$ RCD ( K , ∞ ) 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.

Suggested Citation

  • Huaiqian Li, 2016. "Dimension-Free Harnack Inequalities on $$\hbox {RCD}(K, \infty )$$ RCD ( K , ∞ ) Spaces," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1280-1297, December.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:4:d:10.1007_s10959-015-0621-0
    DOI: 10.1007/s10959-015-0621-0
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