Convergence of Double Walsh-Fourier Series and Hardy Spaces

Abstract

It is proved that the maximal operator of the Marczinkiewicz-Fejer meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space Hpto Lp (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejer means of a function f∈L1converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on Hpwhenever 2/3<p<∞. Thus, in case f∈Hp, the Marczinkiewicz-Fejer means conv f in Hpnorm. The same results are proved for the conjugate means, too.