Noncommutative weak type estimate of Vilenkin derivatives

Let $f\in L_1(\mathcal {N})$, where $\mathcal {N}=L_\infty (G_m)\mathbin{\bar{\otimes}}\mathcal {M}$, $G_m$ is a bounded Vilenkin group and $\mathcal {M}$ is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality $$\|(\mathbb {D}_n(f))_{n\geq 1}\|_{\Lambda _{1,\infty }(\mathcal {N},\ell _{\infty })}\leq C\|f\|_{L_1(\mathcal {N})},$$ where $\mathbb {D}_n(f)$ represents the Vilenkin derivative of the integral function $\mathbf {I}f$. The main strategy in the proof is to exploit the recent advances on the noncommutative Calderón–Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.