Estimates for some rough operators with modulation symmetries

Abstract

This thesis contains three articles on inequalities for operators in Euclidean harmonic analysis.<br /> Chapter 1 consists of the article `A degree one Carleson operator along the paraboloid'. It is concerned with a problem historically motivated by the proof of Carleson's theorem, stating that the Fourier series of a square-integrable function f converges pointwise almost everywhere to f. Carleson's theorem is essentially equivalent to an estimate for the so-called maximally modulated Hilbert transform. We study very rough generalizations, namely maximally modulated singular integrals along certain submanifolds of Euclidean space. Our main result is a new Lp estimate for such operators. The proof combines an overarching strategy due to Fefferman with several in this context new ingredients, most notably so-called sparse bounds due to Oberlin and a new square function estimate.<br /> Chapter 2 contains the article `On trilinear singular Brascamp-Lieb integrals', which is joint work with Polona Durcik and Fred Yu-Hsiang Lin. It deals with a classification problem for singular Brascamp-Lieb forms and several related problems. Classical examples of such forms are paraproducts, and more singular representatives arise in connection with elliptic partial differential equations on Lipschitz domains. In this more singular context, the theory draws heavily from methods introduced first in the context of the maximal modulation operators relevant to the first article. However, these methods do not always apply, different methods are needed depending on the form in question. We solve the implied classification problem for trilinear singular Brascamp-Lieb forms, working out the relevant features of the form for different methods to apply. Then we use this new insight to prove new estimates and some abstract transference principles between different forms.<br /> Chapter 3 consists of the article `Sharp Fourier extension for functions with localized support on the circle'. This article is about the Tomas-Stein restriction inequality for the circle, one of the starting points of the area of Fourier restriction theory. Among its many applications are, much in the spirit of the motivation of our first article, some optimal convergence results for Bochner-Riesz sums of Fourier series in two dimensions. We are interested in the folklore conjecture that the optimal constant in the Tomas-Stein inequality is attained by constant functions. Our main result is that the conjectured sharp inequality certainly holds for functions supported in a small arc on the circle.<br /> The three articles are preceded by an introduction in which we give the historical motivation for the problems considered in this thesis, elaborate on their connection, and give a more detailed overview of our results.