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I tried to read a few years ago the book "Pointwise Convergence of Fourier Series" (Springer, Juan Arias De Reyna) which is a detailed proof of the Carleson theorem, but I was lost after a few dozen pages, because I didn't see the big picture.

What are the main parts and main ideas in the proof of the Carleson theorem?

i.e. pointwise almost everywhere convergence of Fourier series of $L^2$ functions.

I remember one key element was the study of the so-called "maximal Carleson operator", but I don't remember why this operator was key. Does one of you know the key ideas of the proof?

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    $\begingroup$ You might find the following surveys helpful: arxiv.org/abs/1210.0886 and arxiv.org/abs/math/0307008 $\endgroup$
    – Mark Lewko
    Commented Feb 6, 2023 at 1:44
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    $\begingroup$ A very well-known analyst (who might or might not prefer to be anonymous here) once told me that he struggled for a long time with trying to discern the key idea(s) in Carleson's proof, and that he eventually asked Carleson, who appeared to be entirely baffled by the notion that a proof should have an idea behind it. $\endgroup$
    – Steven Landsburg
    Commented Feb 6, 2023 at 7:01
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    $\begingroup$ I've read somewhere that Carleson started out with a strategy for constructing a counterexample, and that a key insight was the revelation that he could prove his approach couldn't work. (If there is anyone out there who knows more precisely what the strategy was, I'd love to know.) At the end of Fefferman's proof he makes a few brief remarks about how he discovered his proof. $\endgroup$
    – Mark Lewko
    Commented Feb 6, 2023 at 8:52
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    $\begingroup$ There's an amusing phenomenon / evolution that tends to happen with the solution to difficult analysis problems. It takes the form of the following: (1) A mathematician spends a significant amount of time studying examples and special cases, building intuition, etc. (2) The mathematician eventually acquires the necessary understanding to solve the problem. (3) The mathematician. then works to reduce their understanding to a succinct and efficient sequence of lemmas, that can be logically verified by another mathematician... $\endgroup$
    – Mark Lewko
    Commented Feb 6, 2023 at 8:53
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    $\begingroup$ (4) Other mathematicians (or graduate students) who want to understand the solution then painfully struggle through the lemmas essentially trying to recreate the examples and intuition that motivated the original author so they can reconstruct the original author's understanding what is going on (5) If this latter person is particularly energized they might go on to write up a motivated exposition of the solution, which then overshadows the original article as the go-to place for understanding the solution. $\endgroup$
    – Mark Lewko
    Commented Feb 6, 2023 at 8:54

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