An operator-valued T1 theory for symmetric CZOs

Abstract

We provide a natural BMO-criterion for the L2-boundedness of Calderón-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the L2-boundedness of the commutators [Rj,b] whenever b belongs to the Bourgain's vector-valued BMO space, where Rj is the j-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.