Dimension-Free $L^p$-Maximal Inequalities in ${\mathbb{Z}}_{m+1}^N$

Abstract

For $m\ge 2$, let $({\mathbb{Z}}_{m+1}^N,|\cdot |)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the $L^1$-normalized indicator of the $r$-sphere, \[ σ_r := \frac{1}{|\{|x| = r\}|} 1_{\{|x| =r\}}.\] We study the $L^p\to L^p$ mapping properties of the maximal operator \[ M^{N} f (x) := \sup_{r \leq N} | σ_r*f| \] acting on functions defined on ${\mathbb{Z}}_{m+1}^N$. Specifically, we prove that for all $p>1$, there exist absolute constants $C_{m,p}$ so that \[ \| M^{N} f \|_{L^p(\mathbb{Z}_{m+1}^N)} \leq C_{m,p} \| f \|_{L^p(\mathbb{Z}_{m+1}^N)} \] for all $N$.

26 pages

References in corpus (1)

• On the Hardy-Littlewood maximal function for the cube