Accessible proof of Carleson's $L^2$ theorem
Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$).
Some time ago I tried to read Carleson's paper, but I would say it is fairly hard to assimilate.
Is there an easier proof? Can someone point out of the core or give an outline?
What did Hunt do? Can someone give an outline of that proof?
Solution 1:
A few years after Carleson's proof Fefferman came up with a shorter proof of the $L^2$ and $L^p$ results. Later, in the context of their work in multilinear harmonic analysis Lacey and Thiele came up with a quite short and easy to understand proof for the $L^2$ theorem which is to some extent a descendant of Fefferman's proof. It's only 10 pages long and can be found in
Lacey, Michael; Thiele, Christoph (2000), "A proof of boundedness of the Carleson operator", Mathematical Research Letters 7 (4): 361–370.
(By the way, this paper has an amusing Mathscinet review which begins with "This is one of the greatest papers written in Fourier analysis.")
They also wrote an expository article describing this and a number of related results:
Lacey, Michael T. (2004), "Carleson's theorem: proof, complements, variations", Publicacions Matemàtiques 48 (2): 251–307
They also put an expanded version of this last paper on the arxiv http://arxiv.org/abs/math/0307008v4
Solution 2:
I suggest this book:
"Pointwise Convergence of Fourier Series", by Juan Arias de Reyna.
DOI: 10.1007/b83346
I think the book is very well written and its exposition is very nice. Unfortunately it does not cover Hunt's proof.