On The maximal operators of Vilenkin-Fej\'er means on Hardy spaces

The main aim of this paper is to prove that when $0<p<1/2$ the maximal operator $\overset{\sim }{\sigma }_{p}^{\ast }f:=\underset{n\in \mathbb{N}}{% \sup }\frac{\left\vert \sigma_{n}f\right\vert }{\left( n+1\right) ^{1/p-2}}$ is bounded from the martingale Hardy space $H_{p}$ to the space $L_{p},$ where $\sigma_{n}$ is $n$-th Fej\'er mean with respect to bounded Vilenkin system.