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 defined by

the (Walsh-)Carleson operator here is thus

and in order to prove a.e. for one can prove that

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

to hold on every measurable (of finite measure).

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