A NOTE ON THE MAXIMAL OPERATORS OF VILENKIN-NORLUND MEANS WITH NON-INCREASING COEFFICIENTS

摘要

In [14] we investigated some Vilenkin-Norlund 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 H1/(1+ alpha) to the space weak-L1/(1+ alpha), (0 < alpha <= 1). In this paper we construct a martingale in the space H1/(1+ alpha), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin-Norlund means with non-increasing coefficients are not bounded from the Hardy space H1/(1+ alpha) to the space L1/(1+ alpha). 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.