The main aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. We also use our results to prove approximation and strong convergence theorems on the martingale Hardy spaces Hp, when 0 < p ≤ 1.2010 Mathematics Subject Classification. 42C10.

In this paper we prove and discuss some new (H p , weak − L p ) type inequalities of maximal operators of Vilenkin-Nörlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such Vilenkin-Nörlund means. It is also proved that these results are the best possible in a special sense. As applications, both some wellknown and new results are pointed out.2000 Mathematics Subject Classification. 42C10, 42B25.

Abstract. The main aim of this paper is to prove that when 0 < p < 1/2 the maximal operator1/p−2 is bounded from the martingale Hardy space H p to the space L p , where σ n is n -th Fejér mean with respect to bounded Vilenkin system.Mathematics subject classification (2010): 42C10.

The main aim of this paper is to prove that there exist a martingale f ∈ H 1/2 such that Fejér means of Vilenkin-Fourier series of the martingale f is not uniformly bounded in the space L 1/2 .2000 Mathematics Subject Classification. 42C10.