发表机构
Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学高级研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对无界非交换Vilenkin系统，利用修正的非交换Calderón-Zygmund分解，证明了Dirichlet均值算子的弱型(1,1)和最优阶强型(p,p)估计，解决了Fedor Sukochev提出的开放问题。
AI 中文摘要
设$\mathcal{R}$为超有限$\mathrm{II}_1$因子。考虑与任意容许Vilenkin群相关的非交换Vilenkin-Fourier级数的部分和算子$(\mathcal{S}_n)_{n\geq 1}$，我们证明存在一个普适常数$c>0$，使得对于所有$f \in L_1(\mathcal{R})$，有$\sup_{n\geq1}\\|\mathcal{S}_n(f)\\|_{L_{1,\infty}(\mathcal{R})} \leq c\\|f\\|_{L_1(\mathcal{R})}$；并且对于每个$1<p<\infty$，有$\sup_{n\geq1}\\|\mathcal{S}_n(f)\\|_{L_p(\mathcal{R})} \leq c\frac{p}{p-1}\\|f\\|_{L_p(\mathcal{R})}$。除了传递技术外，主要的新颖成分是建立在\cite{CCP2022}中的非交换Calderón-Zygmund分解的修正版本。因此，我们解决了由Fedor Sukochev向作者传达的弱型(1,1)估计问题，并通过达到最优阶$\frac{p}{p-1}$，实质性地改进了在\cite{DFdePS2001}中获得的强型(p,p)估计。
英文摘要
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 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 $\frac{p}{p-1}$.
Comments39 pages
