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

Abstract

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 $1<p<\infty$ and every $n$, $m\in \mathbb{N}$ we have \[ \begin{split} &\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|E_{\{1,\cdots,n\}}f(x+\varepsilon)-E_{\{1,\cdots,n\}}f(x-\varepsilon)\right|^{p}\\ \leq& (p^{*}-1)^{p}\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|\varepsilon_{j}\left[E_{\{1,\cdots,n\}\setminus\{j\}}f(x+e_{j})-E_{\{1,\cdots,n\}\setminus\{j\}}f(x-e_{j})\right]\right|^{p}, \end{split} \] where $p^{\ast }=\max \{p,p/(p-1)\}$ and $E_{A}f(x)=\frac{1}{2^n}{\sum }_{\delta \in \{-1,1{\}}^n}f\left(x+{\sum }_{j\in A}{\delta }_j{e}_j\right)$ for every $A\subseteq \{1,\cdots ,n\}$.