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Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in ℝn

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Abstract

In this paper, the modified logarithmic Sobolev inequalities and transportation cost inequalities for measures with density e  − V in ℝn are established. It is proved by using Prékopa–Leindler inequalities following the idea of Bobkov–Ledoux, but a different type of condition is used which recovers Bakry–Emery criterion. As an application, we establish the modified logarithmic Sobolev and transportation cost inequalities for probability measures \(e^{-|x|^p} {\rm{d}}x/Z_p\) with p > 1 in ℝn, and give out explicit estimates for their constants.

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Authors and Affiliations

  1. School of Mathematics, Beijing Normal University, Beijing, China

    Jinghai Shao

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  1. Jinghai Shao
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Correspondence to Jinghai Shao.

Additional information

This work is supported by NSFC (No. 10721091), 973-Project (No.2006CB805901) and DFMEC (NO. 20070027007).

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Shao, J. Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in ℝn . Potential Anal 31, 183–202 (2009). https://doi.org/10.1007/s11118-009-9131-y

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  • DOI: https://doi.org/10.1007/s11118-009-9131-y

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