##
**Classical Fourier analysis.
2nd ed.**
*(English)*
Zbl 1220.42001

Graduate Texts in Mathematics 249. New York, NY: Springer (ISBN 978-0-387-09431-1/hbk). xvi, 489 p. (2008).

This is the first volume of the second edition of the author’s book [“Classical and Modern Fourier Analysis” (Upper Saddle River, NJ: Pearson/Prentice Hall) (2004; Zbl 1148.42001)], but with contents expanded. This book is intended to present the selected topics in depth and to stimulate further study in Fourier analysis. The distinctive features of this book are that proofs are provided in great detail and a large amount of exercises of varying difficulty were carefully prepared by the author; especially for the difficult ones hints or references are given.

This volume puts emphasis on the classical real variable methods in Euclidean Fourier analysis. It consists of 5 chapters, with each chapter followed by historical notes and further related results.

The first chapter focuses on the real analysis of Lebesgue and Lorentz spaces, including convolutions, approximations of the identity, and the Marcinkiewicz and Riesz-Thorin interpolation theorems.

In Chapter 2, the author discusses the Hardy-Littlewood maximal function, the basic facts about the Fourier transform on \(\mathbb R^n\), and distributions. This chapter concludes with a section on the oscillatory integrals.

Chapter 3 deals with Fourier analysis on the torus \(\mathbb T^n\). Standard material is followed by various convergence results, multipliers on the torus, and transference results linking Fourier analysis on \(\mathbb R^n\) and on \(\mathbb T^n\).

Chapter 4 presents the standard Calderón-Zygmund theory for singular integrals with convolution kernels. The last section of this chapter is concerning vector-valued extensions.

Chapter 5 treats the orthogonality properties of the Fourier transform, including the Littlewood-Paley theory, the classical multiplier theory, the spherical maximal function, and some results on wavelets.

This book is very interesting and useful. It is not only a good textbook, but also an indispensable and valuable reference for researchers who are working on analysis and partial differential equations. The readers will certainly benefit a lot from the detailed proofs and the numerous exercises.

This volume puts emphasis on the classical real variable methods in Euclidean Fourier analysis. It consists of 5 chapters, with each chapter followed by historical notes and further related results.

The first chapter focuses on the real analysis of Lebesgue and Lorentz spaces, including convolutions, approximations of the identity, and the Marcinkiewicz and Riesz-Thorin interpolation theorems.

In Chapter 2, the author discusses the Hardy-Littlewood maximal function, the basic facts about the Fourier transform on \(\mathbb R^n\), and distributions. This chapter concludes with a section on the oscillatory integrals.

Chapter 3 deals with Fourier analysis on the torus \(\mathbb T^n\). Standard material is followed by various convergence results, multipliers on the torus, and transference results linking Fourier analysis on \(\mathbb R^n\) and on \(\mathbb T^n\).

Chapter 4 presents the standard Calderón-Zygmund theory for singular integrals with convolution kernels. The last section of this chapter is concerning vector-valued extensions.

Chapter 5 treats the orthogonality properties of the Fourier transform, including the Littlewood-Paley theory, the classical multiplier theory, the spherical maximal function, and some results on wavelets.

This book is very interesting and useful. It is not only a good textbook, but also an indispensable and valuable reference for researchers who are working on analysis and partial differential equations. The readers will certainly benefit a lot from the detailed proofs and the numerous exercises.

Reviewer: Yang Dachun (Beijing)

### MSC:

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |

42Bxx | Harmonic analysis in several variables |