Some new $$(H_p-L_p)$$ type inequalities for weighted maximal operators of partial sums 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 $$H_{p}(G)$$ H p ( G ) to the Lebesgue space $$L_{p}(G),$$ L p ( G ) , for $$0<p<1.$$ 0 < p < 1 . Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results.