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

Michael V. Ruzhansky, Nurgissa Yessirkegenov

2021

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DOI 10.1142/s0219199721500619

Data confidence 70 medium

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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.