Javier Martinez Perales received a Bachelor’s degree in Mathematics from the University of Málaga in 2015. In 2016 he obtained a Master’s degree in Mathematics and Applications from the Universities of Málaga, Cádiz, Granada, Jaén and Almeria.
He joined the Basque Center for Applied Mathematics – BCAM in 2016 as a PhD student in the framework of the “La Caixa” Severo Ochoa Fellowship within the Harmonic Analysis research line. During this period, he has spent three months at Universidad Nacional de la Plata, Universidad de Buenos Aires and Universidad Nacional del Sur, in Argentina.
His PhD thesis, Generalized Poincaré-Sobolev inequalities, has been supervised by Carlos Pérez Moreno (BCAM-UPV/EHU) and Luz Roncal Gómez (BCAM).
Due to the COVID-19 pandemic, the defense will be held online, through the platform BBCollaborate. It will take place on Tuesdays, December 15th at 16:00, and users will be able to follow it live using the following link: https://eu.bbcollab.com/guest/5c9b386726ee48bd9ce9067638de6cb0
On behalf of all BCAM members, we would like to wish Javier the best of luck in his upcoming thesis defense.
PhD thesis Title:
Generalized Poincaré-Sobolev inequalities
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.