Seminar

A “Cauchy integral trick” that goes back to the seminal work of Coifman-Rochberg-Weiss, can be used to prove in a very easy way the boundedness of the commutator of classical singular integrals and point-wise multiplication by BMO functions from weighted estimates for singular integrals.  The approach has been already successfully applied in other situations as well. We develop a very general version of this technique, which allows us to obtain boundedness of commutators from very weak weighted estimates for the operators being commuted. The results work on general geometric or measure theoretic contexts and with a variety of differentiating family of sets used to define the corresponding Muckenhoupt classes of weights. With our approach we recover, in a unified way, most of the boundedness results for commutators already known and obtained several new ones for linear, linearizable, multiparameter, and multilinear operators, including some new ones for the commutators of the bilinear Hilbert transform.  Moreover, a partial converse going from commutator estimates to weighted estimates for a class of exponential weights is obtained too.  This is joint work with A. Benyi, J.M. Martell, K. Moen and E. Stachura.