Moment estimates implied by modified log-Sobolev inequalities

2015/09/24 by Radosław Adamczak, Adamczak, Radosław, Witold Bednorz +3 · 1 citation
Mathematics · #26D10 #60E15 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.FA #math.PR #msc:26D10 #msc:60E15

paper · pdf · doi:10.48550/arxiv.1509.07565

arxiv created 2015/09/24 · arxiv updated 2015/09/28

Abstract

We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on ℝⁿ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of L^p-Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-type estimates for deviations of smooth (not necessarily Lipschitz) functions and measures of enlargements of sets corresponding to non-Euclidean norms. We also prove a two-level concentration result for functions of bounded Hessian and measures satisfying the classical logarithmic Sobolev inequality.

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