A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

In [14] we investigated some Vilenkin—Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H 1/(1+ α ) to the space weak- L 1/(1+ α ) , (0 < α ≦ 1). In this paper we construct a martingale in the space H 1/(1+ α ) , which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin—Nörlund means with non-increasing coefficients are not bounded from the Hardy space H 1/(1+ α ) to the space L 1/(1+ α ) . In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.