Critical Gagliardo–Nirenberg, Trudinger, Brezis–Gallouet–Wainger inequalities on graded groups and ground states

Michael Ruzhansky,Nurgissa Yessirkegenov

doi:10.1142/s0219199721500619

Abstract

In this paper, we investigate critical Gagliardo–Nirenberg, Trudinger-type and Brezis–Gallouet–Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of [Formula: see text], Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo–Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo–Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland’s analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

Highlights

In this paper we investigate critical Gagliardo-Nirenberg, Trudingertype and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which includes the cases of Rn, Heisenberg, and general stratified Lie groups
In [LL13], the authors developed a rearrangement-free argument without using symmetrization to establish the Trudinger-Moser inequalities in the unbounded space Rn including Adams type inequalities on the higher order derivatives
The graded groups form the subclass of homogeneous nilpotent Lie groups admitting homogeneous hypoelliptic left-invariant differential operators ([Mil80], [tER97], see a discussion in [FR16, Section 4.1])

Summary

Preliminaries

Following Folland and Stein [FS82, Chapter 1] and the recent exposition in [FR16, Chapter 3] we recall that G is a graded (Lie) group if its Lie algebra g admits a gradation. ∀X, Y ∈ g, r > 0, [DrX, DrY ] = Dr[X, Y ], as usual [X, Y ] := XY − Y X is the Lie bracket One can extend these dilations through the exponential mapping to the group G by. The homogeneous and inhomogeneous Sobolev spaces Lpa(G) and Lpa(G) based on the positive left-invariant hypoelliptic differential operator R have been extensively analysed in [FR17] and [FR16, Section 4.4] to which we refer for the details of their properties. They generalise the Sobolev spaces based on the sub-Laplacian on stratified groups analysed by Folland in [Fol75]. We refer to the above papers for (noncritical) Sobolev inequalities in the setting of graded groups, and to [RTY20] to the determination of the best constants in non-critical Sobolev and Gagliardo-Nirenberg inequalities on graded groups

Critical Gagliardo-Nirenberg and Trudinger inequalities

Brezis-Gallouet-Wainger inequalities

Best constants and nonlinear Schrodinger type equations