Martingale Hardy spaces and divergence of maximal operators associated with Walsh-Fej\'er means

Abstract

In this paper we review some present knowledge concerning some weighted maximal operators of Fej\'er means with respect to Walsh-Fourier series. We also prove the new result that the main results in the recent papers  \cite{BBTT}, \cite{BMLP} and \cite{BaT1} are in the special sense sharp. More exactly, we  prove that there exist martingales such that the rate of increase of the denominators in the weighted maximal operators investigated in these papers can not be improved.