Merch´an,Tom´asPh.D., May 2020 PURE MATHEMATICS SINGULAR INTEGRALS AND RECTIFIABILITY (79 pages) Dissertation Advisors: Benjamin Jaye and Fedor Nazarov The goal of this thesis is to shed further light on the connection between two basic properties of singular integral operators, the L2-boundedness and the existence in principal value. We do so by providing geometric conditions that allow us to obtain results which are valid for a wide class of kernels, and sharp for operators of great importance: the Riesz and Huovinen transforms. The content of this thesis has turned out to be essential to prove that, in the case of the Huovinen transform associated to a measure with positive and finite upper density, the existence in principal value implies the rectifiability of the underlying measure. SINGULAR INTEGRALS AND RECTIFIABILITY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Tom´asMerch´an May 2020 c Copyright All rights reserved Except for previously published materials Dissertation written by Tom´asMerch´an B.S., Universidad de Sevilla, 2015 M.S., Kent State University, 2017 Ph.D., Kent State University, 2020 Approved by Benjamin Jaye , Co-Chair, Doctoral Dissertation Committee Fedor Nazarov , Co-Chair, Doctoral Dissertation Committee Artem Zvavitch , Members, Doctoral Dissertation Committee Feodor Dragan Maxim Dzero Accepted by Andrew M. Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS TABLE OF CONTENTS . iv ACKNOWLEDGMENTS . vi 1 Introduction . 1 1.1 Basic Notation and Concepts . .3 2 Classification of Symmetric measures . 9 2.1 Introduction . .9 2.2 Examining symmetric measures outside their support . 12 2.3 The Riesz kernel . 13 2.4 The Huovinen kernel . 14 2.5 Operators of co-dimension less than one . 19 3 Small Local Action . 27 3.1 Introduction and Main Results . 27 3.2 Existence in the sense of principal value implies small local action: Proof . 31 3.3 Proof of Theorem 3.1.8 . 33 4 When does L2-boundedness imply existence in the sense of Principal Value? 37 4.1 Introduction and Main Results . 37 4.1.1 A sharp sufficient condition for the existence of principal values . 39 4.1.2 On the Huovinen kernels . 40 4.2 The general theorem . 41 4.2.1 Symmetric measures . 41 4.2.2 Scheme of a proof . 43 4.3 Main applications . 43 4.3.1 Theorem 4.1.5 . 44 iv 4.3.2 Theorem 4.1.6 . 44 4.3.3 Theorem 4.1.8 . 44 4.4 Introductory lemmas . 45 4.4.1 Thin boundary balls . 47 4.4.2 The David-Mattila toolbox . 47 4.5 Reduction to the case ν = µ in (4.1.2) . 49 4.6 The Mattila-Verdera Weak Limit . 52 4.7 Basic estimates for measures with small transportation number . 54 4.7.1 Small Boundaries . 55 4.7.2 Doubling properties . 56 4.7.3 The small density/reflection symmetry alternative . 57 4.8 Three main estimates . 58 4.8.1 Long range comparison lemma . 58 4.8.2 Intermediate contribution lemma . 60 4.8.3 Unit scale averaging lemma . 62 4.9 Gluing it all together: The Proof of Theorem 4.2.6 . 66 4.9.1 Reduction to Doubling scales . 67 4.9.2 Final Claim . 68 BIBLIOGRAPHY . 70 v ACKNOWLEDGMENTS This dissertation is a consequence of the effort and support from many people. Therefore it is only fair to take a moment to thank them. Firstly, I would like to thank my advisor, Benjamin Jaye, and my co-advisor, Fedor Nazarov. Ben is and has been an amazing advisor, who has guided me through mathematics and life in general during these past years, always there for me, whatever question (math or non-math related) that I might have. Thank you for your patience and for giving me the opportunity of working with you in this great area of mathematics. Every PhD student learns the craft of Mathematics from their advisors; that influence carries with them through their careers and surely I count myself lucky for having such a great advisor. I would also like to thank Fedja for his guidance, his patience, and always being there for me. It is a privilege for me to learn so much from you and enjoy many fruitful conversations. What I take from our conversations is much more than the insight about the topic of the day, but perspective about how a mathematician should think. Kent State has truly been a home for these five years, and I would like to thank Professor Zvavitch and Ryabogin for counsel and advice throughout the years. To Richard and Eleanor Aron, for opening their home to me. I have enjoyed tremendously the concerts, pancakes, Thanksgiving dinners, and every occasion in which we gathered. To all my friends, both in the States and in Spain, for all the support and laughs throughout the years. In this penultimate paragraph I would like to thank who I consider to be my family. A mis padres, Tom´asy Ana Mar´ıa,a mi hermana, Ana, y a toda mi familia, por estar siempre ah´ı,en lo bueno y en lo malo, apoy´andome.Lo mucho o poco que consiga siempre ser´aun ´exitocompartido con vosotros. Empec´eesta aventura hace cinco a~nos,y no siempre ha sido f´acil,tanto para m´ı como para vosotros, estar tan lejos de casa. Translation: To my parents, Tom´asy Ana Mar´ıa,to my sister, Ana, and to my whole family, for always being there for me, in the good and in the bad, supporting me. The little or much that I achieve it will always be a share success. I started this adventure five years ago and it has not always been easy, for all of us, for me to be so far from home. To my American family, Bruce, Ann, Amelia and Adam, for treating me as family from vi the very first day, for the laughs, movies, games, and everything that will come in the future. To Guillermo, my undergraduate advisor, who made it possible that I could come to Kent, and we all know how important that has been. You have helped and supported me in many different ways since I was an student in Universidad de Sevilla. I always enjoy tremendously the conversations, visits and meals. Lastly, but not least, in fact, the most, I would like to thank Louisa, the person without whom all this would not make any sense. Kent has given me a lot, but nothing in comparison to having found you. vii CHAPTER 1 Introduction In this dissertation, we are interested in the theory of singular integral operators. These are integral operators whose properties are based on subtle cancellations within the kernel. This theory started with the Hilbert transform, which was introduced by Hilbert in the early 1900's during his work in complex analysis; more specifically, for the so called Riemann-Hilbert problem. The Hilbert transform is formally defined, in the real line, by Z f(y) H(f)(x) = dy; x − y for a function f : R ! R. We observe that the above integral does not converge absolutely as a Lebesgue integral. This obstacle is a recurrent theme in the theory, and it takes some work to define these operators precisely, see subsection Introduction to Calder´on-Zygmund Theory below. The theory of singular integral operators has turned out to be a very active area of research over the past few decades, where harmonic analysis together with geometric measure theory, potential theory, operator theory, the theory of analytic functions, and partial differential equations interlace to provide beautiful and far-reaching mathematics. In this dissertation, we shall focus primarily on the connections between harmonic analysis and geometric measure theory. In particular, we will be studying the relationship between ana- lytic properties of a Calder´on-Zygmund operator (CZO) and geometric properties of an associated measure. We will work in spaces of non-homogeneous type (in which no doubling conditions1 on the underlying measure are assumed). The study of such spaces arose out of attempts to understand the removable sets for bounded analytic functions and was a primary motivating factor for the development of non-homogeneous harmonic analysis and the quantitative geometric Littlewood- Paley theory by Jones [Jo]; David and Semmes [DS]; Nazarov, Treil, and Volberg [NTV2]; and 1See [To5]. 1 Tolsa [To5], among many others. In spite of all this work, many basic questions in the field remain unsolved. Although many of our results cover a wide range of singular integral operators, two instances need to be highlighted at this point: x • The importance of the s-Riesz transform, which has kernel K(x) = , along with its jxjs+1 1 close relative the Cauchy transform, corresponding to the kernel K(z) = in C, cannot be z emphasized enough. They turn out to be essential tools and objects of study in harmonic analysis, PDE's, geometric measure theory, and potential theory, among many other areas of study. • Another operator which enjoys a prominent place in this dissertation is the Huovinen trans- zk form, whose kernel is K(z) = for k odd, which appears in [H]. The Huovinen transform jzjk+1 plays an important role in the literature, as the simplest kernel for which the curvature tech- nique, introduced into the area by Melnikov [Me], fails to be applicable. Before this dissertation, the only general criteria for the existence of singular integral operators in the sense of principal value (acting on non-doubling spaces) was a famous theorem by Mattila and Verdera (see Theorem 4.1.5, also [MV]).