Calderón-Zygmund theory with noncommuting kernels via $H_1^c$

We study an alternative definition of the $H_1$-space associated to a semicommutative von Neumann algebra $L_{\mathrm{\infty }}(\mathbb{R})\bar{\otimes }\mathcal{M}$, 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 Calderón-Zygmund operators with noncommuting kernels.