Title: Hilbert transforms -- something old, something new.

Abstract:

This talk will be a story of the Lp boundedness of Hilbert transforms.
These transforms  are linear maps on  2-integrable  functions, which keep
the analytic part of the functions and switch the sign of their anti-analytic part.
In the language of Fourier analysis for real functions,  the Hilbert
transform is a Fourier multiplier with symbol "-i sign (x)".

I plan to start with the story of  trigonometric functions and to end
with some new discovery of Hilbert transforms on free groups.