This is the students seminar held by Dmitriy Stolyarov

The aim is the celebrated Carleson’s theorem and some things around it. The main paper we are going to read is https://arxiv.org/abs/math/0307008 . Here are the topics of the talks:

1. Kolmogorov’s counter-example. A nice exposition of this classical construction is presented in Zygmund’s book: https://en.wikipedia.org/wiki/Trigonometric_Series (browse «divergence of Fourier series in the first volume»); the book is translated into Russian.
2. Stein’s principle about almost everywhere convergence (a.e. convergence is equivalent to weak type inequalities). See the original paper https://www1.cmc.edu/pages/faculty/MOneill/Math%20138/papers138/Stein.pdf
3. Martingales: definition of a martingale, martingale transform, orthogonality of martingale differences, and Doob’s maximal inequalities. This is written in almost any book on probability theory, for example, I like the exposition in the second volume of Shiryaev’s book: https://www.springer.com/gp/book/9780387722078
4. Carleson’s theorem: preparation (discretization, invariance, and the Carleson operator). See Lacey’s paper above and also Grafakos’s book: https://www.springer.com/gp/book/9781441918567
5. Carleson’s theorem: the tree lemma (and some relations with martingales discussed in the third point)
6. Combinatorics of the Carleson theorem: size and density lemmas
7. Hunt’s theorem (Carleson theorem in L_p)