On $(C,1)$ summability for Vilenkin-like systems
Tom 144 / 2001
Streszczenie
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.
