Generalized Poincaré-Sobolev inequalities
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Poincaré-Sobolev inequalities are very powerful tools in mathematical analysis which have been extensively used for the study of differential equations and their validity is intimately related with the geometry of the underlying space. In particular, and since their applicability as part of the Moser iteration method, their weighted counterparts are of interest for applications. The goal of this dissertation is to present a self-contained study of Poincaré-Sobolev inequalities, weights and the combination of both under the framework of the abstract theory of generalized Poincaré-Sobolev inequalities. To this end, the basic aspects on the theory of Poincaré-Sobolev inequalities and the theory of Muckenhoupt weights is presented. In relation with these, the class of functions with bounded mean oscillations is studied, together with a new characterization of it through some boundedness properties of commutators of fractional integrals. A unified study of classical and fractional weighted Poincaré-Sobolev inequalities, as well as a study of Muckenhoupt weights in relation with functions with bounded mean oscillations is carried out by using new self-improving techniques.