The weak type (1,1) estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems
Abstract
Let ℛ be the hyperfinite II1 factor. Considering the partial sum operators (𝒮n)n≥ 1 of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant c>0 such that supn≥1|𝒮n(f)|L1,∞(ℛ) ≤ c|f|L1(ℛ), f ∈ L1(ℛ), and, for every 1<p<∞, supn≥1|𝒮n(f)|Lp(ℛ) ≤ cp/(p−1)|f|Lp(ℛ), f ∈ Lp(ℛ). Besides the transference technique, the main novel ingredient is a modified version of noncommutative Calderón-Zygmund decomposition established in \cite{CCP2022}. Consequently, we resolve the problem of weak type (1,1) estimate communicated to the authors by Fedor Sukochev, and substantially improve the strong type (p,p) estimates obtained in \cite{DFdePS2001} by achieving the optimal order p/(p−1).
Related
- Vector-valued partial sums on unbounded Vilenkin systems
- A noncommutative weak L1-estimate for rough singular integrals
- Two-sided logarithmic estimates of the Lebesgue constants with respect to Vilenkin systems
- Complex Dirichlet problem with unbounded boundary values
- Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems
BibTeX
@misc{hong2026weak,
title = {{The weak type $(1,1)$ estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems}},
author = {Guixiang Hong and Tiantian Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30779},
eprint = {2609.30779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30779},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The weak type (1,1) estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems
AU - Guixiang Hong
AU - Tiantian Zhao
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30779
UR - https://arxiv.org/abs/2609.30779
AB - Let ℛ be the hyperfinite II_1 factor. Considering the partial sum operators (𝒮_n)_n≥ 1 of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant c>0 such that sup_n≥1|𝒮_n(f)|_L_1,∞(ℛ) ≤ c|f|_L_1(ℛ), f ∈ L_1(ℛ), and, for every 1<p<∞, sup_n≥1|𝒮_n(f)|_L_p(ℛ) ≤ cp/(p−1)|f|_L_p(ℛ), f ∈ L_p(ℛ). Besides the transference technique, the main novel ingredient is a modified version of noncommutative Calderón-Zygmund decomposition established in \cite{CCP2022}. Consequently, we resolve the problem of weak type (1,1) estimate communicated to the authors by Fedor Sukochev, and substantially improve the strong type (p,p) estimates obtained in \cite{DFdePS2001} by achieving the optimal order p/(p−1).
ER -