# 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.

1. Is there an easier proof? Can someone point out of the core or give an outline?

2. What did Hunt do? Can someone give an outline of that proof?

• +1, I seem to be not the only one that finds it hard to digest those proofs. – Jonas Teuwen Nov 27 '10 at 21:42
• @AD: Btw, I think the (partial) proof in Grafakos' Modern Fourier Analysis (chapter 11) is reasonably readable. – Jonas Teuwen Nov 27 '10 at 21:52
• @Jonas T: I have a feeling that it is not just you and me who find this difficult. – AD. Nov 28 '10 at 3:36

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

• For the lazy, here is the first paper, and here is an alternate link for the second paper from Lacey's own webpage. – J. M. is a poor mathematician Nov 28 '10 at 15:47

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.