The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups
Abstract
In this paper, we resolve the convolution inequality problem posed by Naor and Schechtman in \cite[Question 6.1]{N-S2016} (see also \cite{Na2016}). More precisely, we establish the following strong convolution inequality on products of finite cyclic groups: For each 1≤ p<∞ there exists cp>0 such that for every n, m∈ ℕ the following holds &∑ε∈ {−1,1}n∑x∈ Z|E{1,⋯,n}f(x+ε)−E{1,⋯,n}f(x−ε)|p ≤& cpp∑ε∈ {−1,1}n∑x∈ Z|εj[E{1,⋯,n}∖{j}f(x+ej)−E{1,⋯,n}∖{j}f(x−ej)]|p, EAf(x)=1/2n∑δ∈ {−1,1}nf(x+∑j∈ Aδjej) for every A⊆ {1,⋯,n}. The growth order we obtain here cp=O(p) as p→ ∞. For the endpoint p=∞, we show by a counterexample that the strong convolution inequality fails for the L∞, and we prove that the appropriate endpoint for p=∞ is the martingale BMO space.
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BibTeX
@misc{jiao2026strong,
title = {{The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups}},
author = {Yong Jiao and Sijie Luo and Dmitriy Zanin and Dejian Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15768},
eprint = {2609.15768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15768},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups
AU - Yong Jiao
AU - Sijie Luo
AU - Dmitriy Zanin
AU - Dejian Zhou
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.15768
UR - https://arxiv.org/abs/2609.15768
AB - In this paper, we resolve the convolution inequality problem posed by Naor and Schechtman in \cite[Question 6.1]{N-S2016} (see also \cite{Na2016}). More precisely, we establish the following strong convolution inequality on products of finite cyclic groups: For each 1≤ p<∞ there exists c_p>0 such that for every n, m∈ ℕ the following holds &∑_ε∈ {−1,1}^n∑_x∈ Z|E_{1,⋯,n}f(x+ε)−E_{1,⋯,n}f(x−ε)|^p ≤& c^p_p∑_ε∈ {−1,1}^n∑_x∈ Z|ε_j[E_{1,⋯,n}∖{j}f(x+e_j)−E_{1,⋯,n}∖{j}f(x−e_j)]|^p, E_Af(x)=1/2^n∑_δ∈ {−1,1}^nf(x+∑_j∈ Aδ_je_j) for every A⊆ {1,⋯,n}. The growth order we obtain here c_p=O(p) as p→ ∞. For the endpoint p=∞, we show by a counterexample that the strong convolution inequality fails for the L_∞, and we prove that the appropriate endpoint for p=∞ is the martingale BMO space.
ER -