An Extrapolation of Operator Valued Dyadic Paraproducts

Abstract

We consider the dyadic paraproducts \pi_\f on \T associated with an \M-valued function \f. Here \T is the unit circle and \M is a tracial von Neumann algebra. We prove that their boundedness on L^p(\T,L^p(\M)) for some $1<p<\infty$ implies their boundedness on L^p(\T,L^p(\M)) for all $1<p<\infty$ provided \f is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\T,L^p(\M))