Calder\'on-Zygmund theory with noncommuting kernels via $H_1^c$

TL;DR

This paper develops a new approach to the $H_1$-space in noncommutative analysis, enabling endpoint estimates for Calderón-Zygmund operators with noncommuting kernels, advancing the understanding of noncommutative harmonic analysis.

Contribution

It introduces a novel description of atoms in $H_1$ and applies it to establish endpoint estimates for noncommuting Calderón-Zygmund operators.

Findings

New characterization of $H_1$ atoms in noncommutative setting

Endpoint $H_1^c$-$L_1$ estimates for noncommuting kernels

Enhanced tools for noncommutative harmonic analysis

Abstract

We study an alternative definition of the $H_1$-space associated to a semicommutative von Neumann algebra $L_\infty(\mathbb{R}) \overline{\otimes} \mathcal{M}$, first studied by Mei. We identify a "new" description for atoms in $H_1$. We then explain how they can be used to study $H_1^c$-$L_1$ endpoint estimates for Calder\'on-Zygmund operators with noncommuting kernels.