A sharp boundedness result concerning some maximal operators of Vilenkin-Fej\'er means

TL;DR

This paper identifies the precise range of p-values for which the maximal Fejér means operator is bounded from Hardy space to Lebesgue space, establishing sharpness of the result.

Contribution

It determines the exact subspace of positive numbers where the maximal Fejér means operator is bounded from H_p to L_p for 0<p≤1/2, and proves the sharpness of this range.

Findings

Boundedness of the maximal Fejér means operator for 0<p≤1/2.

Identification of the maximal subspace of positive numbers for boundedness.

Proof of the sharpness of the boundedness result.

Abstract

In this paper we derive the maximal subspace of positive numbers, for which the restricted maximal operator of Fej\'er means in this subspace is bounded from the Hardy space $H_{p}$ to the space $L_{p}$ for all $0<p\leq 1/2.$ Moreover, we prove that the result is in a sense sharp.