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In the first part of this paper we describe the status of the art of thissubject. In the second part we present and motivate some new results. Indeed,we introduce some new weighted maximal operators of the partial sums of theWalsh-Fourier series. We prove that for some "optimal" weights these newoperators indeed are bounded from the martingale Hardy space $H_{1}(G)$ to thespace $\text{weak}-L_{1}(G),$ but is not bounded from $H_{1}(G)$ to the space$L_{1}(G).$

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