Cesăro summability of the character system of the p-series field in the Kaczmarz rearrangement

Abstract

Let $G_p$ be the $p$-series field. In this paper we prove the a.e. convergence $\sigma_n f\to f$ $(n\to \infty)$ for an integrable function $f\in L^1(G_p)$, where $\sigma_nf$ is the $n$th $(C,1)$ mean of $f$ with respect to the character system in the Kaczmarz rearrangement. We define the maximal operator $\sigma^* $ by $\sigma^*f := \sup_n|\sigma_nf|$. We prove that $\sigma^*$ is of type $(q,q)$ for all $1<q\le \infty$ and of weak type $(1,1)$. Moreover, we prove that $\|\sigma^*f\|_1\le c\|f\|_{H}$, where $H$ is the Hardy space on the $G_p$.