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Pointwise convergence of Fourier series. (English) Zbl 1003.42001
Lecture Notes in Mathematics. 1785. Berlin: Springer. xviii, 175 p. EUR 27.95/net; sFr. 46.50; £19.50; \$ 36.80 (2002).
The book under review grew out of the author’s attempt in August 1998 to compare the proofs by L. Carleson and Ch. Fefferman on the pointwise convergence of Fourier series with the proof by M. Lacey and C. Thiele on the boundedness of the bilinear Hilbert transform. The author admits that “my summer vacation would not suffice to understand Carleson’s proof…I’ve been working on this project for almost two years and lectured on it at the University of Seville from February to June 2000”.
N. Luzin conjectured in 1913 that every function in $$L^2[-\pi,\pi]$$ has an a.e. convergent Fourier series. In 1923, A. N. Kolmogorov gave an example of a function in $$L^1[-\pi,\pi]$$ whose Fourier series diverges a.e. Later on, Kolmogorov improved this example to diverge everywhere.
In 1966 L. Carleson justified Luzin’s conjecture. Next year, R. A. Hunt proved the a.e. convergence of the Fourier series of every $$f\in L^p[-\pi,\pi]$$ for $$1< p\leq\infty$$. Actually, Hunt proved that every function in $$L(\log L)^2$$ has an a.e. convergent Fourier series. P. Sjölin, in 1969, sharpened this result as follows: every function in $$L\log L\log\log L$$ has an a.e. convergent Fourier series. The latest result in this direction is due to N. Yu. Antonov (in 1966), who proved the same convergence result for functions in $$L\log L\log\log\log L$$.
In the other direction, the results of Kolmogorov were sharpened by V. I. Prohorenko (in 1968), Yung-Ming Chen (in 1969), and K. Tandori (in 1969), each of them constructured functions in $$L(\log\log L)^{1-\varepsilon}$$ with a.e. divergent Fourier series, $$0< \varepsilon< 1$$. The latest result in this direction is due to S. V. Konyagin (in 1999) who obtained the same divergence result for the spaces $$L\varphi(L)$$, whenever $$\varphi$$ satisfies the growth condition $$\varphi(t)= o\{(\log t/\log\log t)^{1/2}\}$$ as $$t\to\infty$$.
The book consists of three parts. The first part gives a concise review of the Hardy-Littlewood maximal function, Fourier series, and Hilbert transform. These are needed in the proof of the Carleson-Hunt theorem, whose exposition is contained in the second part. This part comprises more than half of the book. The third part is dedicated to deriving some consequences of the proof of the Carleson-Hunt theorem.
The book is meant for graduate students who want to understand one of the greatest achievements of the twentieth century. Since its publication in 1966, Carleson’s theorem has acquired a reputation of being an isolated result, very technical, and not profitable to study. Therefore, it is of great significance that the author of the present book explains the motivation of each step in the proof of this fundamental theorem, and his presentation is always clear and well-readable. The better understanding is also enhanced by figures and a flow diagram of the proof. The text is supplemented by a List of Notations, References (containing 50 items), Comments, and a Subject Index.
Reviewer’s remark: The following two papers should be added to the References: V. I. Prohorenko, “Divergent Fourer series” (Russian) [Math. USSR, Sb. 4, 167-180 (1968); translation from Mat. Sb., n. Ser. 75(117), 185-198 (1968; Zbl 0179.09801)]; K. Tandori, “Ein Divergenzsatz für Fourierreihen” [Acta Sci. Math. 30, 43-48 (1969; Zbl 0175.35301)]. The correct spelling of the family name of the author of reference  is “Máté”.

##### MSC:
 42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces 42A20 Convergence and absolute convergence of Fourier and trigonometric series
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