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