Improved Sobolev inequalities: the case p = 1 and generalizations to classical Lorentz spaces

Diego Chamorro (1) , Anca-Nicoleta Marcoci (2) , Liviu-Gabriel Marcoci (2)

1 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
2 UTCB - Technical University of Civil Engineering of Bucharest = Université Technique de Génie Civil de Bucarest

We give a general treatment of refined Sobolev inequalities in the case p = 1 and when p > 1 we study these inequalities using as base space classical Lorentz spaces associated to a weight from the Ariño-Muckenhoupt class Bp. The arguments used for the case p = 1 rely essentially on spectral theory while the ideas behind the case p > 1 are based on pointwise estimates and on the boundedness of Hardy-Littlewood maximal function. As a by-product we will also consider Morrey-Sobolev inequalities. This arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups.

Mots clés: Improved Sobolev inequalities; Sobolev spaces; Besov spaces; Classical Lorentz spaces; Stratified Lie groups

Domaines: Analyse fonctionnelle [math.FA]

Diego Chamorro, Anca-Nicoleta Marcoci, Liviu-Gabriel Marcoci. Improved Sobolev inequalities: the case p = 1 and generalizations to classical Lorentz spaces. 2015. ⟨hal-01148723v1⟩