Dual review of *Classical and Modern Fourier Analysis* by Loukas Grafakos and Real-Variable Methods in Harmonic Analysis by Alberto Torchinsky.

When I was young, the primary source book for Fourier analysis was Antoni Zygmund's two-volume classic *Trigonometric Series*, 2nd edition, published in 1959. A lot has happened in this subject since then, and many fine books have been published in the interim. In this review, I will focus on two recently published books.

The books by Grafakos and Torchinsky were published eighteen years apart, so it seems unfair to compare them. However, I want to give good guidance to potential students and other readers. Both books are encyclopedic sources of a large amount of important modern Fourier analysis, but my blunt advice to a budding analyst would be to study Grafakos's book and keep a copy of Torchinsky's book handy. Both books have lots of good exercises, many with hints. Many of Torchinsky's hints are quite detailed.

Torchinsky claims that his book is user-friendly, but I did not find it as friendly as Grafakos's book. Most of the book is presented in the classical theorem-proof style without much explanation of strategies and tactics. The chapters end with historical notes that are good accounts of the very old history, though on page 23 he refers to work Fourier (1768-1830) did in 1867. Later chapters deal with more recent history, especially results in the 1960's to 1980's. Grafakos's book also has historical notes, but the focus is on more recent work. Grafakos's book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. The treatment is thoroughly modern with free use of operators and functional analysis. Moreover, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises. In fact, he indicates that they took nearly triple the time and effort of the rest of the book. The book includes useful appendices, which make the book self-contained. The only exception is the Banach-Alaoglu Theorem, which is used on page 139 but is missing from Appendix G.

Grafakos's book includes a slick treatment of some classical "real variables." For example, maximal functions are used to study differentiation theory (pages 85-88), and lacunary sequences are used to provide continuous nowhere-differentiable functions (page 238).

After suitable preliminaries, both books cover many of the most important topics in the subject, for example, Hilbert and Riesz transforms, Calderón-Zygmund analysis, Hardy-Littlewood maximal functions, the class BMO of functions of bounded mean oscillation, the Marcinkiewicz interpolation theorem, Littlewood-Paley theory, and Hardy spaces of several variables. Sometimes it is better to start with Torchinsky to see the essential ideas before studying the corresponding topic in Grafakos. For example, compare the T(1) theorem in Torchinsky (page 413) with that in Grafakos (page 591). There are many other topics in Grafakos. The major topics covered in Torchinsky, that are not in Grafakos, are harmonic and subharmonic functions (chapter 7) and boundary-value problems on C^{1}-domains (chapter 17).

Grafakos's index is quite good, but I had a couple of difficulties. Neither quasi-Banach space nor quasi-norm appear in the index. [A quasi-norm satisfies ||x+y|| ≤ c ( ||x||+||y|| ) for some positive constant c.] The word "control" is used a lot and is used in the following mathematical sense: B controls A means that either A ≤ cB (pointwise control) or ||A|| ≤ c||B|| (norm control). In both cases c is a positive constant independent of the variable quantities A and B. There is also a handy Index of Notation.

I conclude by mentioning two fine relatively-recent books: *A Course in Abstract Harmonic Analysis* (1995) by Gerald B. Folland (from CRC Press) and A Panorama of Harmonic Analysis (1999) by Steven G. Krantz (in the MAA's *Carus Mathematical Monographs* series). Both books provide modern treatments; neither book has any exercises. Krantz's book covers many topics covered by the books under review. Folland's book focuses on locally compact groups and representation theory.

Kenneth A. Ross ([email protected]) taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. His most recent work has been on Markov chains and random walks on finite groups and other algebraic systems. He is the author of the book *Elementary analysis: the theory of calculus* (1980, now in 14th printing), co-author of *Discrete Mathematics* (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).