Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems

Deyu Chen; Guixiang Hong

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arXiv · 2609.34689

Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems

Deyu Chen·Guixiang Hong

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Abstract

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.

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BibTeX RIS

Deyu Chen, Guixiang Hong. 2026-09-28. Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems. https://arxiv.org/abs/2609.34689

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