Logarithmic Sobolev inequalities on Lie groups

Abstract

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 Grosstype 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 R n 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.

Logarithmic Hardy-Rellich inequalities on Lie groups

“…Similarly to [CKR21b] for (weighted) logarithmic Sobolev inequalities, here we prove a version of (1.13) on stratified Lie groups. In this case, if G is a stratified Lie group of homogeneous dimension Q, the inequality (1.13) takes the form…”

“…In this paper we continue the investigation of the logarithmic functional inequalities in the setting of Lie groups. In [CKR21b], we have obtained the logarithmic Sobolev, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups, and in this paper we deal with what can be called (weighted) logarithmic Hardy-Rellich inequalities, building on the terminology proposed by Del Pino, Dolbeault, Filippas and Tertikas in [DDFT10]. The scope of results and the method of proof are rather different, meriting an independent presentation.…”

“…In [CKR21b], we have proved a family of weighted Gross type logarithmic Sobolev inequalities, allowing also for different choices of weights. For example, consider the family of Euclidean norms on R n given by |x| r = (|x…”

“…with some (specified) normalisation constant γ r . Then in [CKR21b], we have obtained the following weighted Gross type logarithmic Sobolev inequalities:…”

“…Inequality (1.19) can be compared with the logarithmic Sobolev inequality on graded Lie groups established in [CKR21b], in a particular (unweighted) case, for 1 < p < ∞ and 0 < a < Q p , given by…”