Sergei Treil (Brown)

Title: Convex body domination and weighted estimates with matrix-valued weights

Abstract:

Main motivations for the weighted estimates with scalar and matrix weights come from the theory of stationary random processes, unconditional wavelet bases, and invertibility of Toeplitz operators. Recent developments in harmonic analysis, especially the powerful technique of domination by sparse operators allow us to obtain new results and to simplify proofs of the known ones. At the first glance it looks like the technique of sparse domination is specifically scalar-valued, and is not applicable to the estimates with matrix weights. However, the idea of convex body domination, introduced by F.Nazarov, S.Petermichl, S.Treil and A.Volberg allows one to generalize the sparse domination technique to the weighted settings. In the talk I discuss the classical (scalar) sparse domination, and its generalization to the vector-valued case, the sparse convex body domination. I'll show how it allows to simplify the weighted estimates, and discuss some open problems.

 The talk is based on a joint work with F.Nazarov, S.Petermichl and A.Volberg.