Mikel Florez will defend his thesis on Monday, July 7th

The defence will take place at Sala Adela Room at the Faculty of Science and Technology of Leioa EHU Campus at 11:00 h

Mikel Florez has been a PhD student at the Basque Center for Applied Mathematics (BCAM) since April 2021. His research lies in the area of Harmonic Analysis, with a particular focus on the theory of singular integrals, directional singular integrals, square functions, and time-frequency analysis. His work has been carried out under the supervision of Prof. Luz Roncal and Prof. Ioannis Parissis.

He obtained his Bachelor’s degree in Mathematics from the University of the Basque Country (UPV/EHU) in 2019 and his Master’s degree from the Complutense University of Madrid (UCM) in 2020.

His thesis, titled "Topics in harmonic analysis related to Rubio de Francia square functions and directional singular integrals," will be defended on July 7th, 2025 at 11:00 in the Adela Moyua Room, Faculty of Science and Technology, Leioa.

On behalf of all members of BCAM, we would like to wish him all the best for the future, both professionally and personally.

Abstract

Firstly, we study maximal directional singular integral operators in R^n defined by a Hörmander-Mihlin multiplier on an (n-1)-dimensional subspace which act trivially in the perpendicular direction. The choice of subspace depends measurably on the first n-1 variables of R^n. Assuming the subspace to be non-degenerate in the sense that it is contained in a subspace of R^n away of a cone around e_n and the function f to be frequency supported in a cone away from R^(n-1), we prove L^p bounds for these operators when p > 3/2. If we assume, additionally, that the Fourier transform of f is supported in a single frequency band, we are able to extend the boundedness range to p > 1. The non-degeneracy assumption cannot, in general, be removed, even in the band-limited case.

Secondly, we study one-dimensional square functions in the spirit of Rubio de Francia. Let P_ω f be the Fourier restriction of f ∈ L^2(R) to an interval ω ⊂ R. If Ω is an arbitrary collection of pairwise disjoint intervals, the square function of {P_ω f: ω ∈ Ω} is termed the Rubio de Francia square function T^Ω. In this thesis we prove a pointwise bound for T^Ω by a sparse operator involving local L^2-averages. A pointwise bound for the smooth version of T^Ω by a sparse square function is also proved. These pointwise localization principles lead to quantitative L^p(w), p > 2, and weighted weak-type (p,p), p ≥ 2, norm inequalities for T^Ω. In particular, the obtained weak L^p(w) norm bounds are new for p ≥ 2 and sharp for p > 2. The proofs rely on sparse bounds for abstract balayages of Carleson sequences, oscillation inequalities, local orthogonality and time-frequency analysis discretization techniques. The thesis also contains two results related to the outstanding conjecture that T^Ω is bounded on L^2(w) if and only if w ∈ A_1. The conjecture is verified for radially decreasing, even A_1 weights, and in full generality for the Walsh group analogue of T^Ω.