Noncommutative weak type estimate of Vilenkin derivatives

Chengshu Tian; Tiantian Zhao; Dejian Zhou

doi:10.4064/sm250626-14-11

Noncommutative weak type estimate of Vilenkin derivatives

Chengshu Tian
, Tiantian Zhao^*
, Dejian Zhou

^*Corresponding author for this work

Harbin Institute of Technology
School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

Let f ∈ L₁(N ), where N = L_∞(G_m) ¯⊗ M, G_m is a bounded Vilenkin group and M is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality ∥(D_n(f))_n≥1 ∥_Λ1,∞ (N,ℓ_∞) ≤ C∥f∥_L1 (N ), where D_n(f) represents the Vilenkin derivative of the integral function If. The main strategy in the proof is to exploit the recent advances on the noncommutative Calderón– Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.

Original language	English
Pages (from-to)	181-203
Number of pages	23
Journal	Studia Mathematica
Volume	287
Issue number	2
DOIs	https://doi.org/10.4064/sm250626-14-11
State	Published - 2026
Externally published	Yes

Keywords

Fourier series
almost uniform convergence
noncommutative Calderón–Zygmund decomposition
noncommutative maximal inequality

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10.4064/sm250626-14-11

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title = "Noncommutative weak type estimate of Vilenkin derivatives",

abstract = "Let f ∈ L1(N ), where N = L∞(Gm) ¯⊗ M, Gm is a bounded Vilenkin group and M is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality ∥(Dn(f))n≥1 ∥Λ1,∞ (N,ℓ∞) ≤ C∥f∥L1 (N ), where Dn(f) represents the Vilenkin derivative of the integral function If. The main strategy in the proof is to exploit the recent advances on the noncommutative Calder{\'o}n– Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.",

keywords = "Fourier series, almost uniform convergence, noncommutative Calder{\'o}n–Zygmund decomposition, noncommutative maximal inequality",

author = "Chengshu Tian and Tiantian Zhao and Dejian Zhou",

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year = "2026",

doi = "10.4064/sm250626-14-11",

language = "英语",

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TY - JOUR

T1 - Noncommutative weak type estimate of Vilenkin derivatives

AU - Tian, Chengshu

AU - Zhao, Tiantian

AU - Zhou, Dejian

N1 - Publisher Copyright: © Instytut Matematyczny PAN, 2026.

PY - 2026

Y1 - 2026

N2 - Let f ∈ L1(N ), where N = L∞(Gm) ¯⊗ M, Gm is a bounded Vilenkin group and M is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality ∥(Dn(f))n≥1 ∥Λ1,∞ (N,ℓ∞) ≤ C∥f∥L1 (N ), where Dn(f) represents the Vilenkin derivative of the integral function If. The main strategy in the proof is to exploit the recent advances on the noncommutative Calderón– Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.

AB - Let f ∈ L1(N ), where N = L∞(Gm) ¯⊗ M, Gm is a bounded Vilenkin group and M is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality ∥(Dn(f))n≥1 ∥Λ1,∞ (N,ℓ∞) ≤ C∥f∥L1 (N ), where Dn(f) represents the Vilenkin derivative of the integral function If. The main strategy in the proof is to exploit the recent advances on the noncommutative Calderón– Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.

KW - Fourier series

KW - almost uniform convergence

KW - noncommutative Calderón–Zygmund decomposition

KW - noncommutative maximal inequality

UR - https://www.scopus.com/pages/publications/105040116984

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DO - 10.4064/sm250626-14-11

M3 - 文章

AN - SCOPUS:105040116984

SN - 0039-3223

VL - 287

SP - 181

EP - 203

JO - Studia Mathematica

JF - Studia Mathematica

IS - 2

ER -

Noncommutative weak type estimate of Vilenkin derivatives

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Keywords

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