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）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Vilenkin系统上等长Littlewood--Paley--Rubio de Francia不等式的两种表述，发现典范表述在Schatten类上失效，而Vilenkin表述等价于UMD加type 2性质。
AI 中文摘要
我们研究了任意Vilenkin系统\\(\Gm\\)上等长Littlewood--Paley--Rubio de Francia（LPR）不等式的两种自然表述。典范表述使用\\(\mathbb N\\)中基数相等的典范区间，而Vilenkin表述使用\\(\Gm\\)对偶上群运算下公共初始区间的平移。本文的主要结果表明，这两类区间的定义展现出截然不同的性质。更精确地说，对于典范表述，我们构造了一个反例，表明对于每个\\(2<p<\infty\\)，Schatten类\\(S^p\\)不满足等长LPR性质，这与环面上相应结果形成鲜明对比；对于Vilenkin表述，我们证明了对于每个\\(2\le p<\infty\\)，Banach空间\\(X\\)具有等长LPR性质当且仅当它是UMD且具有type~2。充分性证明将环面到有限循环群的转移论证与Vilenkin区间嵌入数字矩形包络相结合，所得界不依赖于生成序列。
英文摘要
We investigate two natural formulations of equal-length Littlewood--Paley--Rubio de Francia (LPR) inequalities on arbitrary Vilenkin systems \(\Gm\). The canonical formulation uses canonical intervals in \(\mathbb N\) of equal cardinality, while Vilenkin's formulation uses translates of a common initial interval under the group operation on the dual of \(\Gm\). In this paper, our main results show that the definitions of these two types of intervals exhibit dramatically different properties. More precisely, for the canonical formulation, we construct a counterexample showing that, for every \(2<p<\infty\), the Schatten class \(S^p\) fails the equal-length LPR property, in sharp contrast to the corresponding result on the torus; for Vilenkin's formulation, we prove that, for every \(2\le p<\infty\), a Banach space \(X\) has the equal-length LPR property if and only if it is UMD and has type~\(2\). The sufficiency proof combines a transference argument from the torus to finite cyclic groups with an embedding of Vilenkin intervals into digital rectangular envelopes, yielding bounds independent of the generating sequence.
Comments31 pages
