Weighted maximal operators of Fejér means of Walsh-Fourier series in the martingale Hardy space $H_{1/2}$

Abstract

In this paper we derive the restricted weighted maximal operator, defined by ${\sup }_{k\in \mathbb{N}}\left(\left|{\sigma }_{k}F\right|/A_k^2\right)$ of Fejér 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.