Notes on Matrix Valued Paraproducts

Abstract

Denote by $M_n$ the algebra of $n\times n$ matrices. We consider the dyadic paraproducts ${\pi }_b$ associated with $M_n$ valued functions $b$, and show that the $L^{\infty }(M_n)$ norm of $b$ does not dominate $||{\pi }_b|{|}_{L^2({\ell }_n^2)\to L^2({\ell }_n^2)}$ uniformly over $n$. We also consider paraproducts associated with noncommutative martingales and prove that their boundedness on bounded noncommutative $L^p-$% martingale spaces implies their boundedness on bounded noncommutative $L^q-$% martingale spaces for all $1<p<q<\infty$.

12 pages

References in corpus (1)

• Operator-valued dyadic BMO spaces