On some weighted maximal operators of partial sums of Walsh-Fourier series in the space $H_1$

In the first part of this paper we describe the status of the art of this subject. In the second part we present and motivate some new results. Indeed, we introduce some new weighted maximal operators of the partial sums of the Walsh-Fourier series. We prove that for some “optimal” weights these new operators indeed are bounded from the martingale Hardy space $H_1(G)$ to the space $\text{weak}-L_1(G),$ but is not bounded from $H_1(G)$ to the space $L_1(G)\mathrm{.}$