# Some new restricted maximal operators of Fej\'er means of Walsh-Fourier series in the space $H_{1/2}$

## TL;DR

This paper identifies the largest subset of natural numbers for which a restricted maximal operator of Fejér means of Walsh-Fourier series remains bounded from the Hardy space $H_{1/2}$ to $L_{1/2}$, and proves the result's optimality.

## Contribution

It determines the maximal subspace of natural numbers ensuring boundedness of the restricted maximal operator in Walsh-Fourier series, establishing the sharpness of this characterization.

## Key findings

- Identified the maximal subset of natural numbers for boundedness.
- Proved the boundedness of the restricted maximal operator.
- Established the sharpness of the main result.

## Abstract

In this paper we derive the maximal subspace of natural numbers $\left\{n_{k}:k\geq 0\right\}$, such that the restricted maximal operator, defined by $\sup_{k\in \mathbb{N}}\left\vert \sigma_{n_{k}}F \right\vert$ on this subspace of Fej\'er means of Walsh-Fourier series is bounded from the martingale Hardy space $H_{1/2}$ to the Lebesgue space $L_{1/2}$. The sharpness of this result is also proved.