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

Abstract

Let $\mathcal{R}$ be the hyperfinite $\mathrm{II}_1$ factor. Considering the partial sum operators $(\mathcal{S}_n)_{n\geq 1}$ of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant $c>0$ 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 $1<p<\infty$, $$\sup_{n\geq1}\|\mathcal{S}_n(f)\|_{L_p(\mathcal{R})} \leq c\frac{p}{p-1}\|f\|_{L_p(\mathcal{R})},\quad f \in L_p(\mathcal{R}).$$ Besides the transference technique, the main novel ingredient is a modified version of noncommutative Calder\'{o}n-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 $\frac{p}{p-1}$.