Basic Ideas and Tools

Abstract

In this chapter we will introduce the basic definitions, theorems, and some of the analytic techniques and tools that will be used throughout the book. Here we will consider harmonic functions and their basic properties, the Poisson kernel, the Dirichlet problem, HP spaces, and singular integrals. We will then state and prove some of the classical results relating the nontangential maximal function and Lusin area function. Our goal is not to give a comprehensive introduction to these topics, but rather to introduce, as quickly and efficiently as possible, the requisite background, both mathematical and historical, for what follows in the subsequent chapters. We provide proofs for most of the results in this chapter, especially those concerning harmonic functions on half spaces, since these occupy center stage throughout this monograph. However, it will be impossible, in the space of an introductory chapter, to present complete proofs of all the material mentioned above. Although we will be thorough with the development of harmonic functions, we will merely give references for some of the real analysis tools we will use, in particular, those results readily attainable in the literature. (We do assume that the reader is familiar with the rudiments of analysis.) Readers knowledgeable on these topics may skip this chapter, although it should serve such readers as a convenient reference. Those wishing a complete and comprehensive introduction to these topics are advised to consult some of the numerous texts already in existence: [ABR], [Du], [Fo], [Ga], [Ho], [Jou], [Koo], [St4], [St6], [SW2], [To] and [Zy2]. As we discussed in the Preface, and as the title clearly indicates probabilistic ideas and techniques play an essential role in what we do in this monograph. Much of the material of this chapter can also be presented from this point of view. In order to maintain this chapter as short and as elementary as possible, we decided to present the analytic point of view in this introduction and refer the interested reader to [Dur] or [Bas2] for the probailistic approach.