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

TL;DR

This paper establishes a sharp boundedness result for restricted maximal operators of Vilenkin-Fourier series on martingale Hardy spaces, identifying the precise range of p for which the operators are bounded.

Contribution

It determines the maximal subspace of p-values where the restricted maximal operators are bounded from Hardy spaces to Lebesgue spaces, and proves the result's sharpness.

Findings

Boundedness of the operators for 0<p≤1.

Identification of the maximal subspace of p-values.

Proof of the sharpness of the boundedness result.

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 $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.