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.
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