|Location:||MSRI: Simons Auditorium|
After describing some motivations for weighted estimates of singular integral operators with matrix weights, I'll discuss the method of estimating vector-valued singular integral operators by the so called sparse operators. In the scalar case the sparse domination is a recently introduced powerful tool of harmonic analysis, and generalizing it to the vector valued case helps in many problems.
Of course, trivial generalization does not work: the target space of our sparse operator is the set of convex-body-valued functions. It looks complicated, but the weighted estimated of such operators is an easy task. As an application we get a new easy proof of the weighted estimates of the vector Calderon-Zygmund operators with matrix Muckenhoupt weights.
Some intriguing open problems will also be discussed.
The talk is partially based on a joint work with F. Nazarov, S. Petermichl and A. Volberg.