Logarithmic Sobolev inequalities on Lie groups

Marianna Chatzakou, Aidyn Kassymov, Michael Ruzhansky

Published 2021-06-29Version 1

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, {\bf very interestingly}, with the Gaussian measure on the first stratum of the group. 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 give weighted versions of log-Sobolev and Nash inequalities for general Lie groups.