题目：Noncommutative weak $(1,1)$ estimate of Dirichlet mean on unbounded Vilenkin system

报告人：赵甜甜博士后

报告时间：2025年5月13日16:00-17:00

报告地点：235

报告摘要：Let $\mathcal{R}$ be the hyperfinite $\mathrm{II}_1$ factor, and let $(\mathcal{S}_n(f))_{n\geq1}$ be the partial sums of the noncommutative Vilenkin-Fourier series associated with an unbounded Vilenkin group. Then, for every $f \in L_p(\mathcal{R})$, there exist universal constants $c, c_p>0$ such that

\begin{equation*}

\|\mathcal{S}_n(f)\|_{L_{1,\infty}(\mathcal{R})} \leq c\|f\|_{L_1(\mathcal{R})},\quad \forall n\in \mathbb{N},p=1

\end{equation*}

and

$$\|\mathcal{S}_n(f)\|_{L_p(\mathcal{R})} \leq c_p\|f\|_{L_p(\mathcal{R})},\quad \forall n\in \mathbb{N}, 1<p<\infty.$$

The transference argument enables one to reduce problems from noncommutative setting to operator-valued setting. To prove the operator-valued results,

We establish a modified form of the noncommutative Calder\'{o}n-Zygmund decomposition for martingale filtrations in von Neumann algebras established by Cadilhac, Conde-Alonso and Parcet. We improve the strong $(p,p)$-type results established by Dodds, Ferleger, Pagter and Sukochev and obtain the weak-type estimate.

报告人简介：赵甜甜，哈尔滨工业大学博士后，合作导师王斯萌教授，研究方向为泛函分析，非交换分析。