分次群和基态上的临界Gagliardo-Nirenberg，Trudinger，Brezis-Gallouet-Wainger不等式
Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states
摘要：
本文研究分次李群上正Rockland算子的临界Gagliardo-Nirenberg不等式、Trudinger型不等式和Brezis-Gallouet-Wainger不等式，其中包括$\mathbb R^n$、Heisenberg和一般分层李群的情形.作为应用，利用临界Gagliardo-Nirenberg不等式，得到了非线性Schr dinger型方程最小能量解的存在性.我们还表示在临界Gagliardo-Nirenberg和Trudinger不等式的变分形式，以及在相应的非线性次椭圆方程的基态解的最佳常数。所得结果在一般的分层李群（齐次卡诺群）的情形下已经是新的.在新的技术方法中，我们还将Folland对Hölder空间的分析从分层李群推广到一般齐次李群。
英文摘要：
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 includes the cases of $\mathbb R^n$, 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.
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