Vector-valued partial sums on unbounded Vilenkin systems
Abstract
Let Gm=∏k≥0ℤmk be a Vilenkin group that is not necessarily bounded, i.e., supk mk=∞. We prove that, for every UMD Banach space X and every 1<p<∞, the Vilenkin partial-sum operators are uniformly bounded on Lp(Gm;X), with a bound depending only on p and the UMD constant of X, and not on 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 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.
Related
- Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems
- The weak type (1,1) estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems
- An intrinsic finite element framework for scalar- and vector-valued partial differential equations on general manifolds
- Two-sided logarithmic estimates of the Lebesgue constants with respect to Vilenkin systems
- Backward stochastic partial differential equations with unbounded random coefficients
BibTeX
@misc{chen2026vectorvalued,
title = {{Vector-valued partial sums on unbounded Vilenkin systems}},
author = {Deyu Chen and Guixiang Hong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22714},
eprint = {2609.22714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22714},
}
RIS
TY - UNPB
TI - Vector-valued partial sums on unbounded Vilenkin systems
AU - Deyu Chen
AU - Guixiang Hong
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22714
UR - https://arxiv.org/abs/2609.22714
AB - Let Gm=∏_k≥0ℤ_m_k be a Vilenkin group that is not necessarily bounded, i.e., sup_k m_k=∞. We prove that, for every UMD Banach space X and every 1<p<∞, 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 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 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.
ER -