Logarithmic Sobolev-Type Inequalities on Lie Groups

Chatzakou M. Kassymov A. Ruzhansky M.
September 2024 Springer

Journal of Geometric Analysis
2024 #34 Issue 9

In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, log-Sobolev (weighted and unweighted), log-Gagliardo–Nirenberg and log-Caffarelli–Kohn–Nirenberg inequalities on graded Lie groups. Furthermore, on stratified groups, we show that one of the obtained inequalities is equivalent to a Gross-type log-Sobolev inequality with the horizontal gradient. As a result, we obtain the Gross log-Sobolev inequality on general stratified groups but, very interestingly, with the Gaussian measure on the first stratum of the group. Moreover, our methods also yield weighted versions of the Gross log-Sobolev inequality. In particular, we also obtain new weighted Gross-type log-Sobolev inequalities on Rⁿ for arbitrary choices of homogeneous quasi-norms. As another consequence we derive the Nash inequalities on graded groups and an example application to the decay rate for the heat equations for sub-Laplacians on stratified groups. We also obtain weighted versions of log-Sobolev and Nash inequalities for general Lie groups.

28C20 , 35A23 , 35R03 , Graded groups , Gross inequality , Homogeneous groups , Lie groups , Log-Caffarelli-Kohn-Nirenberg inequality , Log-Gagliardo-Nirenberg inequality , Log-Sobolev inequality , Nash inequality , Stratified groups

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Department of Mathematics - Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
Institute of Mathematics and Mathematical Modeling, 28 Shevchenko str., Almaty, 050010, Kazakhstan
School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom

Department of Mathematics - Analysis
Institute of Mathematics and Mathematical Modeling
School of Mathematical Sciences

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