Multi-parameter setting (c,α) means with respect to one dimensional Vilenkin system

Abstract

We prove that the maximal operator of the (C,?n)-means of the one dimensional Vilenkin-Fourier series is of weak type(L1, L1). Moreover, we prove the almost everywhere convergence of the (C,?n) means of integrable functions (i.e. ? ?n n f ? f ), where n ? N?,q and n ? 1 for f ? L1(Gm), Gm is a bounded Vilenkin group, for every sequence ? = (?n), 0 < ?n < 1.