Improved Sobolev Inequalities and Muckenhoupt weights on stratified Lie groups

2010/07/23 by Chamorro, Diego
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1007.4086

Abstract

We study in this article the Improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces W(s;p) and Besov spaces B(-b, infty, infty). When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in Rⁿ and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p = 1, these two techniques are not available anymore and following M. Ledoux in Rⁿ, we will treat here the critical case p = 1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of Improved Sobolev inequalities using weak-type Sobolev spaces.

Related

• Functional Inequalities in Stratified Lie groups with Sobolev, Besov, Lorentz and Morrey spaces
• Mean Value Inequalities and Characterizations of Sobolev spaces on graded groups
• Logarithmic Sobolev-type inequalities on Lie groups
• Weighted L^p-Hardy and L^p-Rellich inequalities with boundary terms on stratified Lie groups
• A counterexample for Improved Sobolev Inequalities over the 2-adic group