Dimension-Free $L^p$-Maximal Inequalities in $\mathbb{Z}_{m+1}^N$ (1406.7229v3)

Abstract: For $m \geq 2$, 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, [ \sigma_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} | \sigma_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 $N$.