# Weak boundedness of Calder\'on-Zygmund operators on noncommutative $L_1$-spaces

## TL;DR

This paper simplifies the proof of weak boundedness of Calderón-Zygmund operators on noncommutative L1 spaces, enabling broader boundedness results for operator-valued functions.

## Contribution

It provides a streamlined proof approach using Cuculescu's projections and pseudo-localisation, extending boundedness results to a wider class of noncommutative Lp spaces.

## Key findings

- Simplified proof of weak (1,1) boundedness for noncommutative Calderón-Zygmund operators.
- Extension of Lp-boundedness results for Hilbert valued kernels.
- Application of pseudo-localisation techniques to noncommutative harmonic analysis.

## Abstract

In 2008, J. Parcet showed the $(1,1)$ weak-boundedness of Calder\'on-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 Calder\'on-Zygmund operators with Hilbert valued kernels acting on operator valued functions for $1 < p < \infty$ and an $L_p$-pseudo-localisation result of P. Hyt\"onen.