The weak type $(1,1)$ estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems

Let $\mathcal{R}$ be the hyperfinite ${\mathrm{I}\mathrm{I}}_1$ factor. Considering the partial sum operators $({\mathcal{S}}_n{)}_{n\ge 1}$ of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant $c>0$ such that \begin{equation*} \sup_{n\geq1}|$\mathcal{S}$n(f)|{L_{1,\infty}($\mathcal{R}$)} \leq c|f|_{L_1($\mathcal{R}$)},\quad f \in L_1($\mathcal{R}$), \end{equation*} and, for every $1<p<\mathrm{\infty }$, $$\underset{n\ge 1}{\sup }\mathrm{\parallel }{\mathcal{S}}_n(f){\mathrm{\parallel }}_{L_p(\mathcal{R})}\le c\frac{p}{p-1}\mathrm{\parallel }f{\mathrm{\parallel }}_{L_p(\mathcal{R})},f\in L_p(\mathcal{R})\mathrm{.}$$ Besides the transference technique, the main novel ingredient is a modified version of noncommutative Calderón-Zygmund decomposition established in \cite{CCP2022}. Consequently, we resolve the problem of weak type $(1,1)$ estimate communicated to the authors by Fedor Sukochev, and substantially improve the strong type (p,p) estimates obtained in \cite{DFdePS2001} by achieving the optimal order $\frac{p}{p-1}$.