Javier Canto received a Bachelor’s degree in Mathematics from the University of Basque Country in 2016 and in 2017 he obtained a Master’s degree in Mathematics and Applications by the Universidad Autónoma de Madrid.
In 2018 he joined Basque Center for Applied Mathematics – BCAM as a PhD student in the Harmonic Analysis Research line.
His doctoral thesis, Geometric Harmonic Analysis: Cp weights, John–Nirenbger estimates and Hajlasz capacity density condition, has been supervised by Carlos Pérez (BCAM-UPV/EHU) and Kangwei Li, former Juan de la Cierva Postdoc at BCAM.
The defense will take place in the seminar room of the Department of Mathematics at UPV/EHU, and will be streaming through the platform Webex. It will take place on Friday, December 17 at 12:00, and users will be able to follow it live using the following link: https://ehu.webex.com/ehu/j.php?MTID=m5af171f02e473cc3ee25abf086563d4a
On behalf of all BCAM members, we would like to wish Javier the best of luck in his upcoming thesis defense.
PhD thesis Title:
Geometric Harmonic Analysis: Cp weights, John–Nirenbger estimates and Hajlasz capacity density conditions.
In this thesis, several concepts concerning Harmonic Analysis are developed. First, we develop a systematic treatment of the Cp class of weights, delivering a Cp constant that quantifies the weights in this class, along with a quantitative reverse Hölder inequality for these weights. We also give several quantitative weighted norm inequalities for different operators, such as Calderón-Zygmund operators or the Hardy-Littelwood maximal operator. We also study some extensions of the John-Nirenberg theorem, that give some weighted inequalities, and some maximal estimates. We also show how the BMO condition can be relaxed to more weak norms concerning convex functions. Finally, we introduce Hajlasz capacity density conditions in abstract metric spaces, which are geometric conditions concerning non-local Hajlasz gradients. We show that these density conditions are self-improving.