Moment estimates implied by modified log-Sobolev inequalities^∗

¹ Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland.
R.Adamczak@mimuw.edu.pl; W.Bednorz@mimuw.edu.pl
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 .
P.Wolff@mimuw.edu.pl

Received: 8 May 2016
Accepted: 7 December 2016

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.