Calderón–Zygmund theory with noncommuting kernels via $\mathrm H_1^c$

We study an alternative definition of the $\mathrm {H}_1$-space associated to a semicommutative von Neumann algebra $L_\infty (\mathbb {R}) \mathbin {\overline {\otimes }} \mathcal {M}$, first studied by Mei. We identify a “new” description for atoms in $\mathrm {H}_1$. We then explain how they can be used to study $\mathrm {H}_1^c$-$L_1$ endpoint estimates for Calderón–Zygmund operators with noncommuting kernels.