# Ptolemaics meetings 4 & 5 & 6 ; pt I

These last ones have been quite interesting meetings, I’m happy about how the whole thing is turning out. Sadly I’m very slow at typing and working out the ideas, so I have to include three different meetings in one. Since the notes are getting incredibly long, I’ll have to split it in at least two parts.I include the pdf version of it, in case it makes it any easier to read.

ptolemaics meeting 4 & 5 & 6 pt I

Let me get finally into the time frequency of the Walsh phase plane. I won’t include many proofs as they are already well written in Hytönen’s notes (see previous post). My main interest here is the heuristic interpretation of them (disclaimer: you might think I’m bullshitting you at a certain point, but I’m probably not). Ideally, it would be very good to be able to track back the train of thoughts that went in Fefferman’s and Thiele-Lacey’s proofs.

Sorry if the pictures are shit, I haven’t learned how to draw them properly using latex yet.

1. Brush up

Recall we have Walsh series for functions ${f \in L^2(0,1)}$ defined by

$\displaystyle W_N f(x) = \sum_{n=0}^{N}{\left\langle f,w_n\right\rangle w_n(x)},$

the (Walsh-)Carleson operator here is thus

$\displaystyle \mathcal{C}f(x) = \sup_{N\in \mathbb{N}}{|W_N f(x)|},$

and in order to prove ${W_N f(x) \rightarrow f(x)}$ a.e. for ${N\rightarrow +\infty}$ one can prove that

$\displaystyle \|\mathcal{C}f\|_{L^{2,\infty}(0,1)} \lesssim \|f\|_{L^2(0,1)}.$

There’s a general remark that should be done at this point: the last inequality is equivalent to

$\displaystyle \left|\left\langle\mathcal{C}f, \chi_E\right\rangle\right| = \left|\int_{E}{\mathcal{C}f}\,dx\right| \lesssim |E|^{1/2}\|f\|_{L^2(0,1)}$

to hold on every measurable ${E}$ (of finite measure).
As I said in the previous post, the goal for now is to understand the proof of ${L^2\rightarrow L^{2,\infty}}$ boundedness of Carleson’s operator (through time freq. analysis). As an introduction to the real thing we’ve started from the simpler case of the Walsh transform, or Walsh series, or Walsh phase plane, whatever you want to call it. It’s easier because all the nasty technicalities disappear but the ideas needed are already in there, that’s why it propaedeutic. We’re following Hytönen’s notes as suggested by Tuomas (you can find them here: http://wiki.helsinki.fi/pages/viewpage.action?pageId=79564963). An alternative is Tao’s lecture notes (lecture 5 in particular) for course Math254A W’ 01 (http://www.math.ucla.edu/~tao/254a.1.01w/) which are quite nice – as all of his stuff. The main differences are in that Hytönen proves every single statement, and he deals with the Walsh series (analogue of the Fourier series) while Tao deals with the Walsh transform (analogue of the Fourier transform). Also, Hytönen then goes on to prove the full euclidean case, while Tao doesn’t.
The Walsh operators are best described as operators on the real line with a different field structure. One works on ${\mathbb{Z}_2[[X]]}$, i.e. the Laurent series with coefficient in ${\mathbb{Z}_2}$, which can be identified with the (binary expression of) positive reals by
$\displaystyle a_N \cdots a_0 . a_{-1} a_{-2} \cdots \equiv a_N X^N + \ldots a_1 X + a_0 + \frac{a_{-1}}{X} + \frac{a_{-2}}{X^2} + \ldots .$