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

Abstract: Let $\mathcal{R}$ be the hyperfinite $\mathrm{II}<em>1$ factor. Considering the partial sum operators $(\mathcal{S}_n)</em>{n\geq 1}$ of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant $c&gt;0$ 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 $1&lt;p&lt;\infty$, $$\sup</em>{n\geq1}|\mathcal{S}<em>n(f)|</em>{L_p(\mathcal{R})} \leq c\frac{p}{p-1}|f|_{L_p(\mathcal{R})},\quad f \in L_p(\mathcal{R}).$$ 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 $\frac{p}{p-1}$.