Some new weak $(H_p-L_p)$ type inequality for weighted maximal operators of Walsh-Fourier series

Abstract

In this paper 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 $H_p$ to the Lebesgue space $\text{weak}-L_p,$ for $0<p<1.$ Moreover, we also prove sharpness of this result.