Weighted Hardy spaces and the Fejér means of Walsh–Fourier series

Abstract. In this paper we consider the \(A_p\) weights and weighted martingale Hardy spaces. We present Doob's maximal inequality and the atomic decomposition of the Hardy spaces. As application, we investigate the convergence of partial sums and Fejér summability of
Walsh–Fourier series. We prove that the maximal operator of the Fejér means is bounded from the weighted dyadic Hardy space \(H_p(w)\) to \(L_p(w)\). This implies almost everywhere convergence of the Fejér means.