# Weighted maximal operators of Fej\'er means of Walsh-Fourier series in the martingale Hardy space $H_{1/2}$

## TL;DR

This paper establishes the boundedness of a weighted maximal operator of Fejér means of Walsh-Fourier series from the martingale Hardy space to Lebesgue space, including sharpness and new related results.

## Contribution

It introduces a restricted weighted maximal operator for Walsh-Fourier Fejér means and proves its boundedness from $H_{1/2}$ to $L_{1/2}$, demonstrating sharpness and deriving new results.

## Key findings

- Boundedness of the weighted maximal operator from $H_{1/2}$ to $L_{1/2}$.
- Proof of the sharpness of the boundedness result.
- Derivation of new and known results as corollaries.

## Abstract

In this paper we derive the restricted weighted maximal operator, defined by ${\sup }_{k\in \mathbb{N}}\left(\left\vert \sigma _{k}F\right\vert/A^2_k\right)$ of Fej\'er means of Walsh-Fourier series and prove that the it is bounded from the martingale Hardy space $H_{1/2}(G)$ to the Lebesgue space $L_{1/2}(G).$ The sharpness of this result is also proved. As a consequence we obtain some new and and well-know results.