Sharp H p $H_{p}$ - L p $L_{p}$ type inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications

Abstract

We prove and discuss some new H p -L p type inequalities of weighted maximal operators of Vilenkin-Nörlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Nörlund means. As applications, both some well-known and new results are pointed out. MSC: 42C10; 42B25

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

“…Móricz and Siddiqi [17] investigate approximation properties of some special Nörlund means of Walsh-Fourier series of L p functions in norm. Similar results for the two-dimensional case can be found in Nagy [18,19], Nagy and Tephnadze [20,21,22,23], Gogolashvili and Tephnadze [13,14] (see also [2]). Approximation properties of general summability methods can be found in [4,5].…”

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.…”

Some inequalities related to strong convergence of Riesz logarithmic means

“…Remark 1. Similar problems for Fejér means with respect to Walsh and Vilenkin systems can be found in [2], [4], [42] (see also [44] and [49]). Our method and estimations of Reisz and Fejér kernels (see Lemmas 1 and 2) do not give possibility to prove even similar strong convergence result as for the case of Fejer means.…”