The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups

In this paper, we use martingale methods to 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 $1<p<\mathrm{\infty }$ and every $n$, $m\in \mathbb{N}$ we have $$\begin{array}{rl}{\displaystyle }& {\displaystyle \sum _{\epsilon \in \{-1,1{\}}^n}\sum _{x\in \mathbb{Z}}{\mid E_{\{1,\cdots ,n\}}f(x+\epsilon )-E_{\{1,\cdots ,n\}}f(x-\epsilon )\mid }^p}\\ {\displaystyle \le }& {\displaystyle (p^{\ast }-1{)}^p\sum _{\epsilon \in \{-1,1{\}}^n}\sum _{x\in \mathbb{Z}}{\mid {\epsilon }_j[E_{\{1,\cdots ,n\}\setminus \{j\}}f(x+e_j)-E_{\{1,\cdots ,n\}\setminus \{j\}}f(x-e_j)]\mid }^p,}\end{array}$$ where $p^{\ast }=\max \{p,p\mathrm{/}(p-1)\}$ and $E_{A}f(x)=\frac{1}{2^n}{\sum }_{\delta \in \{-1,1{\}}^n}f(x+{\sum }_{j\in A}{\delta }_j{e}_j)$ for every $A\subseteq \{1,\cdots ,n\}$.