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Fourier analysis. Transl. from the Spanish and revised by David Cruz-Uribe. (English) Zbl 0969.42001

Graduate Studies in Mathematics. 29. Providence, RI: American Mathematical Society (AMS). xviii, 222 p. (2001).
Fourier analysis is a large branch of mathematics whose point of departure is the study of Fourier series and integrals. The purpose of this book is to develop the real variable methods introduced into Fourier analysis by A. P. Calderón and A. Zygmund in the 1950’s. The author begins in Chapter 1 with a review of Fourier series and integrals and then in Chapters 2 and 3, he introduces two operators which are basic to the field: the Hardy-Littlewood maximal function and the Hilbert transform. In subsequent Chapters 4 and 5, the author discusses singular integrals along with their modern generalizations. In Chapter 6, the author studies the relationship between \(H^1\), BMO and singular integrals, and then he introduces Littlewood-Paley theory in Chapter 8. Finally, in Chapter 9, the author presents an important result of the 80’s, the so-called \(T1\) theorem. At the end of each chapter, there is a section in which the author tries to give some idea of further results which are not discussed in the text and gives references for the interested readers.
The material in this book is almost self-contained. It is written in a rigorous, pedagogic and heuristic way. To understand the main content of this book, readers only need some basic knowledge of real analysis, distributions and tempered distributions. The topics in this book cover the essentials of the real variable methods of harmonic analysis and are of great interest in applications, for example, in partial differential equations. Thus, this is a book of interest to anyone having to work with harmonic analysis. In particular, it is an excellent textbook for graduate students of analysis.
Reviewer: Yang Dachun (Jena)

MSC:

42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
42B15 Multipliers for harmonic analysis in several variables
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B38 Linear operators on function spaces (general)
42B30 \(H^p\)-spaces
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