Weak boundedness of Calderón-Zygmund operators on noncommutative $L_1$-spaces

Abstract

In 2008, J. Parcet showed the $(1,1)$ weak-boundedness of Calderón-Zygmund operators acting on functions taking values in a von Neumann algebra. We propose a simplified version of his proof using the same tools : Cuculescu's projections and a pseudo-localisation theorem. This will unable us to recover the $L_p$-boundedness of Calderón-Zygmund operators with Hilbert valued kernels acting on operator valued functions for $1<p<\infty$ and an $L_p$-pseudo-localisation result of P. Hytönen.

arXiv comment29 pages, added a section on Lp pseudo-localisation