Walsh–Fejér means and weighted Hardy spaces

Abstract. In this paper we consider the \(A_p\) weights and weighted dyadic Hardy spaces. We investigate the convergence of Fejér means of Walsh–Fourier series. We prove that, under some conditions, 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.

Key words and phrases. Weighted martingale Hardy spaces, weighted \(p\)-atom, atomic decomposition, interpolation, Walsh functions, Fejér summability.