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

Abstract

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.

Lebesgue and Vilenkin-Lebesgue points and a. e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions

“…Similar results for the two-dimensional case can be found in Nagy [23,24], Nagy and Tephnadze [25,26,27,28]. Approximation properties of general summability methods can be found in [2,5,7,8,2,21] (see also [17,18]). Fridli, Manchanda and Siddiqi [12] improved and extended results of Móricz and Siddiqi [22] to Martingale Hardy spaces.…”

Bounded operators on Martingale Hardy spaces

“…In [84] it was proved that the maximal operator t * of the Nörlund summability method with non-increasing sequence {q k : k ∈ N}, satisfying the condition…”

The One-dimensional Martingale Hardy Spaces and Partial Sums and Fejér Means with respect to Walsh system

“…Such counterexamples first appear in the papers of Goginava [24] (see also [20,23]). Such constructions of martingales are also used in the papers [3], [4], [11], [12], [30], [29], [31], [32], [33], [34], [35], [37], [38], [39], [40], [52], [57], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [73]. For the one-dimensional case we use martingales which were used in [72].…”