Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems
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 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<∞, the Schatten class Sp 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≤ p<∞, 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.
Related
- Vector-valued partial sums on unbounded Vilenkin systems
- Polynomial Bohnenblust--Hille Bounds and Hardy--Littlewood Inequalities for Finite Local Systems
- Two-sided logarithmic estimates of the Lebesgue constants with respect to Vilenkin systems
- The weak type (1,1) estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems
- Counterexamples to Hardy--Littlewood maximal inequalities on solvable groups
BibTeX
@misc{chen2026vectorvalued,
title = {{Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems}},
author = {Deyu Chen and Guixiang Hong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34689},
eprint = {2609.34689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34689},
}
RIS
TY - UNPB
TI - Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems
AU - Deyu Chen
AU - Guixiang Hong
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34689
UR - https://arxiv.org/abs/2609.34689
AB - 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 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<∞, 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≤ p<∞, 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.
ER -