摘要
It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L 1 converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H p whenever 2/3<p<∞. Thus, in case f∈H p , the Marczinkiewicz-Fejér means conv f in H p norm. The same results are proved for the conjugate means, too.
It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L 1 converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H p whenever 2/3<p<∞. Thus, in case f∈H p , the Marczinkiewicz-Fejér means conv f in H p norm. The same results are proved for the conjugate means, too.
基金
This paperwas written while theauthorwasresearching at Humboldt University in Berlin supported by Alexandervon Humboldt Foundation.This research was also supported by the Hungarian Scientific Research Funds (OTKA) NoF0 1 963 3 and by the Foundation
