On the convergence of partial sums with respect to Vilenkin system on the martingale Hardy spaces

Abstract

In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for the modulus of continuity of $f\in H_{p}$ martingales, which provide convergence of subsequences of partial sums on the martingale Hardy spaces. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.