Preface Fourier Analysis is a large branch of mathematics whose point of departure is the study of Fourier series and integrals. However, it encompasses a variety of perspectives and techniques, and so many different introductions with that title are possible. The goal of this book is to study the real variable methods introduced into Fourier analysis by A. P. Calderon and A. Zygmund in the 1950's. We begin in Chapter 1 with a review of Fourier series and integrals, and then in Chapters 2 and 3 we introduce two operators which are basic to the field: the Hardy-Littlewood maximal function and the Hilbert trans- form. Even though they appeared before the techniques of Calderon and Zygmund, we treat these operators from their point of view. The goal of these techniques is to enable the study of analogs of the Hilbert transform in higher dimensions these are of great interest in applications. Such oper- ators are known as singular integrals and are discussed in Chapters 4 and 5 along with their modern generalizations. We next consider two of the many contributions to the field which appeared in the 1970's. In Chapter 6 we study the relationship between H1, BMO and singular integrals, and in Chapter 7 we present the elementary theory of weighted norm inequalities. In Chapter 8 we discuss Littlewood-Paley theory its origins date back to the 1930's, but it has had extensive later development which includes a number of applications. Those presented in this chapter are useful in the study of Fourier multipliers, which also uses the theory of weighted inequalities. We end the book with an important result of the 80's, the so-called Tl theorem, which has been of crucial importance to the field. At the end of each chapter there is a section in which we try to give some idea of further results which are not discussed in the text, and give xm
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