On $(C,1)$ summability for Vilenkin-like systems

G. Gát ¹

¹ Department of Mathematics Bessenyei College P.O. Box 166 H-4400 Nyíregyháza, Hungary

Studia Mathematica, Tome 144 (2001) no. 2, pp. 101-120

Résumé

We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of $2$-adic ($m$-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions $\sigma _n f\to f$ $(n\to \infty )$ a.e., where $\sigma _nf$ is the $n$th $(C,1)$ mean of $f$. (For the character system of the group of $m$-adic integers, this proves a more than 20 years old conjecture of M. H. Taibleson [24, p. 114].) Define the maximal operator $\sigma ^*f := \sup_n|\sigma _nf|$. We prove that $\sigma ^*$ is of type $(p,p)$ for all $1 p\le \infty $ and of weak type $(1,1)$. Moreover, $\| \sigma ^*f\| _1\le c\| f\| _{H}$, where $H$ is the Hardy space.

Détail

DOI : 10.4064/sm144-2-1

Keywords: common generalization walsh system vilenkin system character system group adic m adic integers product system normalized coordinate functions continuous irreducible unitary representations coordinate groups noncommutative vilenkin groups udmd product systems defined schipp other systems prove integrable functions sigma infty where sigma nth mean character system group m adic integers proves years old conjecture taibleson define maximal operator sigma *f sup sigma prove sigma * type infty weak type moreover sigma *f where hardy space