Part I. Fourier series and Hilbert Transform -- Hardy-Littlewood maximal function -- Fourier Series -- Hilbert Transform -- Part II. The Carleson-Hunt Theorem -- The Basic Step -- Maximal inequalities -- Growth of Partial Sums -- Carleson Analysis of the Function -- Allowed pairs -- Pair Interchange Theorems -- All together -- Part III. Consequences -- Some spaces of functions -- The Maximal Operator of Fourier series.

This book contains a detailed exposition of Carleson-Hunt theorem following the proof of Carleson: to this day this is the only one giving better bounds. It points out the motivation of every step in the proof. Thus the Carleson-Hunt theorem becomes accessible to any analyst.The book also contains the first detailed exposition of the fine results of Hunt, Sjölin, Soria, etc on the convergence of Fourier Series. Its final chapters present original material. With both Fefferman's proof and the recent one of Lacey and Thiele in print, it becomes more important than ever to understand and compare these two related proofs with that of Carleson and Hunt. These alternative proofs do not yield all the results of the Carleson-Hunt proof. The intention of this monograph is to make Carleson's proof accessible to a wider audience, and to explain its consequences for the pointwise convergence of Fourier series for functions in spaces near $äcal Lü^1$, filling a well-known gap in the literature.