Topics in harmonic analysis related to Rubio de Francia square functions and directional singular integrals

• Autores: Mikel Flórez-Amatriain
• Directores de la Tesis: Ioannis Parissis (dir. tes.) , Luz Roncal Gómez (codir. tes.) , Francisco Javier Gutierrez García (tut. tes.)
• Lectura: En la Universidad del País Vasco - Euskal Herriko Unibertsitatea ( España ) en 2025
• Idioma: inglés
• Enlaces
  • Tesis en acceso abierto en: ADDI
• Resumen

  • This thesis focuses on the Lp-boundedness of maximal directional singular integral operators and Rubio de Francia square functions. A key technique is a time-frequency analysis discretization of the operators of interest. Firstly, we study maximal directional singular integral operators in ℝn 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 ℝn. Assuming the subspace to be non-degenerate in the sense that it is contained in a subspace of ℝn away of a cone around en and the function f to be frequency supported in a cone away from ℝn-1, we prove Lp bounds for these operators when p>3/2. If we assume, additionally, that f is frequency 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. In this thesis we prove pointwise bounds for the smooth and rough Rubio de Francia square functions by sparse operators involving local L2-averages. These pointwise localization principles lead to quantitative Lp(w), p>2, and weighted weak-type (p,p), p≥2, norm inequalities for the square functions. The thesis also contains two results related to the outstanding conjecture that the Rubio de Francia square function is bounded on L2(w) if and only if w∈A1. The conjecture is verified for radially decreasing, even A1 weights, and in full generality for the Walsh group analogue of the square function.