Studia Math. 144 (2) (2001), 101-120. ON (C, 1) SUMMABILITY OF INTEGRABLE FUNCTIONS ON COMPACT TOTALLY DISCONNECTED SPACES G. G´ at Abstract. In this paper we give a common generalization of the Walsh, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normed coordinate functions for continuous irreducible unitary represen- tations 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 σ n f → f (n →∞) a.e., where σ n f is the n-th (C, 1) mean of f . (This with respect to the character system of the group of m-adic integers was a more than 20 years old conjecture of M.H. Taibleson [24, p. 114].) Define the maximal operator σ * f := sup n |σ n f |. We prove that σ * is of type (p, p) for all 1 <p ≤∞ and of weak type (1, 1). Moreover, kσ * f k 1 ≤ ckf k H , where H is the Hardy space. Introduction, Examples Denote by N the set of natural numbers, P the set of positive integers, respec- tively. Denote m := (m k : k ∈ N) a sequence of positive integers such that m k ≥ 2, k ∈ N and G m k a set of cardinality m k . Suppose that each (coordinate) set has the discrete topology and measure μ k which maps every singleton of G m k to 1 m k (μ k (G m k ) = 1), k ∈ N. Let G m be the compact set formed by the complete direct product of G m k with the product of the topologies and measures (μ). Thus each x ∈ G m is a sequence x := (x 0 ,x 1 , ...), where x k ∈ G m k , k ∈ N. G m is called a Vilenkin space. G m is a compact totally disconnected space, with normalized reg- ular Borel measure μ, μ(G m ) = 1. The Vilenkin space G m is said to be bounded if the generating system m is a bounded one. Throughout this paper the boundedness of G m is supposed. In this paper c, c p denote absolute constants, the latter can depend (only) on p. A base for the neighborhoods of G m can be given as follows I 0 (x) := G m , I n (x) := {y =(y i ,i ∈ N) ∈ G m : y i = x i for i<n} for x ∈ G m ,n ∈ P. I := {I n (x): n ∈ N,x ∈ G m } is the set of intervals on G m . Denote by L p (G m ) the usual Lebesgue spaces (k.k p the corresponding norms) (1 ≤ p ≤∞), A n the σ algebra generated by the sets I n (x)(x ∈ G m ) and E n the conditional expectation operator with respect to A n (n ∈ N). 1991 Mathematics Subject Classification. primary: 42C10, secondary: 42C15, 43A75, 40G05. Research supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. F020334 and by the Hungarian “Mveldsi s Kzoktatsi Minisztrium”, grant no. FKFP 0710/1997 Typeset by A M S-T E X 1