Year: 2026
Type: preprint
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 $1\leq p<\infty$ there exists $c_{p}>0$ such that for every $n$, $m\in \mathbb{N}$ the following holds \[ \begin{split} &\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|E_{\{1,\cdots,n\}}f(x+\varepsilon)-E_{\{1,\cdots,n\}}f(x-\varepsilon)\right|^{p}\\ \leq& c^{p}_{p}\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|\varepsilon_{j}\left[E_{\{1,\cdots,n\}\setminus\{j\}}f(x+e_{j})-E_{\{1,\cdots,n\}\setminus\{j\}}f(x-e_{j})\right]\right|^{p}, \end{split} \] $E_{A}f(x)=\frac{1}{2^{n}}\sum_{δ\in \{-1,1\}^{n}}f\left(x+\sum_{j\in A}δ_{j}e_{j}\right)$ for every $A\subseteq \{1,\cdots,n\}$. The growth order we obtain here $c_{p}=O(p)$ as $p\to \infty$. For the endpoint $p=\infty$, we show by a counterexample that the strong convolution inequality fails for the $L_{\infty}$, and we prove that the appropriate endpoint for $p=\infty$ is the martingale $BMO$ space.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6573401646, G8054714833
OpenAlex ID: W7213351964