无界Vilenkin系统上的向量值部分和
Vector-valued partial sums on unbounded Vilenkin systems
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中文总结 AI 辅助
本文证明了无界Vilenkin群上向量值部分和算子在UMD空间中的一致有界性，解决了相关开放问题，并进一步建立了更细块分解下的R-有界性。
中文摘要 AI 辅助
设 \\(\Gm=\prod_{k\ge0}\mathbb Z_{m_k}\\) 是一个不必有界的Vilenkin群，即 \\(\sup_k m_k=\infty\\)。我们证明，对于每个UMD Banach空间 \\(X\\) 和每个 \\(1<p<\infty\\)，Vilenkin部分和算子在 \\(L^p(\Gm;X)\\) 上一致有界，其界仅依赖于 \\(p\\) 和 \\(X\\) 的UMD常数，而不依赖于 \\(\mathbf m\\)。这解决了由Clément等人~\cite{ClementDePagterSukochevWitvieliet2000} 的工作中产生并后来在Hytönen等人~\cite[p.~362]{HNVWI} 的书中明确记录的一个开放问题。证明通过Paley共轭恒等式和切线序列解耦论证，将部分和估计归结为有限循环群上Fourier投影的一个解耦不等式，该不等式似乎是新的。同样的方法还给出了与更细的块分解相关的部分和算子族的 \\(\mathcal R\\)-有界性，从而解决了Fedor Sukochev向我们传达的另一个相关问题。
英文摘要
Let \(\Gm=\prod_{k\ge0}\mathbb Z_{m_k}\) be a Vilenkin group that is not necessarily bounded, i.e., \(\sup_k m_k=\infty\). We prove that, for every UMD Banach space \(X\) and every \(1<p<\infty\), the Vilenkin partial-sum operators are uniformly bounded on \(L^p(\Gm;X)\), with a bound depending only on \(p\) and the UMD constant of \(X\), and not on \(\mathbf m\). This resolves an open problem arising from the work of Clément et al.~\cite{ClementDePagterSukochevWitvliet2000} and later recorded explicitly in the book of Hytönen et al.~\cite[p.~362]{HNVWI}. The proof reduces the partial-sum estimate, via a Paley conjugation identity and a tangent-sequence decoupling argument, to a decoupling inequality for Fourier projections on finite cyclic groups, which appears to be new. The same approach also yields \(\mathcal R\)-boundedness for the family of partial-sum operators associated with the finer block decomposition, thereby resolving another related problem communicated to us by Fedor Sukochev.
发表机构
- Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）
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