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

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 --> f (n --> infinity) a.e., where rho (n)f is the nth (C, 1) mean of f. (For the character system of the group of m-adic integers, this proves a more than 20 years did conjecture of M. H. Taibleson [24, p. 114].) Define the maximal operator sigma *f := sup(n) /sigma (n)f/ we prove that sigma* is of type (p,p) for all 1 < p < infinity and of weak type (1, 1). Moreover, //sigma *f//(1) less than or equal to c//f//(H), where H is the Hardy space.