A sharp boundedness result for restricted maximal operators of Vilenkin–Fourier series on martingale Hardy spaces

Abstract

The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space Hp to the Lebesgue space Lp for all 0 < p ≤ 1. We also prove that the result is sharp in a particular sense.2010 Mathematics Subject Classification. 42C10, 42B25.

On some new weighted maximal operators of Fejér means of Walsh-Fourier series

“…Proof. Since S n /V (n) is bounded from L ∞ to L ∞ , by Lemma 1, the proof of Theorem 1 will be complete, if we prove that (10) tµ x ∈ I M : S * ,∇ a(x) ≥ t ≤ c p < ∞, t ≥ 0 for every p-atom a. In this parer c p (or C p ) denotes a positive constant depending only on p but which can be different in different places.…”

“…Gulicev [18] estimated the rate of uniform convergence of a Walsh-Fourier series by using Lebesgue constants and modulus of continuity. These problems for Vilenkin groups were investigated by Blatota, Nagy, Persson and Tephnadze [10] (see also [7,6,8,9,25]), Fridli [13], Gát [14], Simon [29] and Tephnadze [20,34,35].…”

An Analogy of the Carleson–Hunt Theorem with Respect to Vilenkin Systems

“…Recent proof of almost everywhere convergence of Walsh-Fourier series was given by Demeter [12] in 2015. By using some methods of martingale Hardy spaces, almost everywhere convergence of subsequences of Vilenkin-Fourier series was considered in [8]. Antonov [3] proved that for f ∈ L 1 (log + L)(log + log + log + L)(G m ) its Walsh-Fourier series converges a.e.…”

Convergence of 𝑇 means with respect to Vilenkin systems of integrable functions

“…The a.e. convergence of subsequences of Vilenkin-Fourier series was considered in [5], where the methods of martingale Hardy spaces were used.…”