MATH 711: Topics in Harmonic Analysis I (Fall 2007)

Instructor: Camil Muscalu

Meeting Time & Room

This course is the first in a sequence of advanced classes which I plan to have over the next years. It could be equally titled "Lectures on multi-linear singular integrals." Its scope is not only to present classical and well known results of the theory, but also to describe and motivate more modern and quite recent developments of it. A list of possible topics should include:

1. A "trilogy" of paraproducts (Coifman-Meyer paraproducts, paraproducts on polydiscs, paraproducts with flag singularities) and their role in non-linear PDE.
2. Calderón's commutators and the Cauchy integral on Lipschitz curves, revisited.
3. Bi-linear Hilbert transform.
4. Calderón's bi-linear maximal function and its connection to Bourgain's double recurrence theorem in ergodic theory.
5. Carleson operator and almost everywhere convergence of Fourier series.

The presentation should be essentially self contained.