Pointwise convergence, maximal functions and regularity issues in harmonic analysis

Abstract

This cumulative thesis is dedicated to the study of different maximal operators related to pointwise convergence in Fourier Analysis and is divided in three main parts.<br /> The first part is dedicated to regularity results for maximal functions. The Hardy–Littlewood maximal function is an essential tool in establishing pointwise convergence in harmonic analysis, and recently more importance has been given to its regularity properties. We make progress in the question of estimating the variation of the maximal function in one dimension, and explore different perspectives of the regularizing properties of fractional maximal functions.<br /> The second part is aimed at maximal versions of classical Fourier restriction theorems. Although the restriction operator has been considered for the past 40 years, it was not until very recently that it was asked whether it can be defined pointwise almost everywhere. We answer this question affirmatively in the two-dimensional case, make progress on the Tomas-Stein exponent case and discuss stronger assertions about Lebesgue points of the Fourier transform.<br /> The third part of this thesis deals with the interplay between Carleson operators and the Hilbert transform along the parabola. An interesting recent conjecture states that the maximally modulated Hilbert transform along the parabola must be bounded in L^2(R^2 ). We make partial progress in this question, considering a class of functions essentially constant in directions orthogonal to any fixed line in R^2.<br /> The thesis consists of seven chapters, where Chapters 1 to 6 contain each a scientific article.<br /> In Chapter 0 we develop the historical framework and discuss the motivation for our results, connecting them to the main subject of pointwise convergence and giving a summary of the techniques used.<br /> In Chapter 1 we prove a sharp variation bound for a class of maximal functions interpolating the centered and uncentered maximal functions in one dimension. We also prove a sharp variation bound for Lipschitz truncated uncentered maximal functions. We provide counterexamples showing that our techniques are also sharp. <br /> In Chapter 2 we connect the framework of derivative estimates for fractional maximal functions to Fourier analysis tools. In particular, we prove sharp regularity bounds for certain classes of smooth fractional maximal functions, as well as regularizing bounds for the fractional spherical maximal function. <br /> In Chapter 3 we investigate the regularizing properties of the local fractional maximal function on domains, extending the previous known results to the sharp range in case the domain is smooth enough.<br /> In Chapter 4 we bridge the gap in the recently started line of research of maximal restriction estimates. In particular, we prove that H^1 −almost every point in the unit circle is a Lebesgue point of the Fourier transform of an L^p function, 1 ≤ p < 4/3 . <br /> In Chapter 5 we extend the results in the previous chapter to L^r −norm and spherical Lebesgue points of Fourier transforms of L^p functions. We also devise counterexamples to show sharpness of some of our results and impose restrictions to when the strong maximal function can satisfy full-range maximal restriction estimates.<br /> In Chapter 6 we consider a family of one-dimensional maximally modulated operators arising from the parabolic Carleson operator. We prove uniform bounds in the slope of the line, settling the degenerate case of the conjecture where the Fourier support of the function under consideration collapses into a line.