Volume 21, 2017
|Page(s)||467 - 494|
|Published online||08 January 2018|
Moment estimates implied by modified log-Sobolev inequalities∗
1 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland.
2 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland; and Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland .
Received: 8 May 2016
Accepted: 7 December 2016
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 Lp-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.
Mathematics Subject Classification: 60E15 / 26D10
Key words: Concentration of measure / modified logarithmic Sobolev inequalities
© EDP Sciences, SMAI, 2017
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