IUMJ

Title: Notes on matrix valued paraproducts

Authors: Tao Mei

Issue: Volume 55 (2006), Issue 2, 747-760

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 $1 < p < q < \infty$.