On the maximal Fejér operator for double Fourier series of functions in Hardy spaces

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Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces

H^{(1, 0)} (^2)

,

H^{(0, 1)} (T^{2})

, or

H^{(1, 1)} (T^{2})

. We prove that the maximal Fejér operator is bounded from

H^{(1, 0)} (T^{2})

or

H^{(0, 1)} (T^{2})

into weak-

L^{1} (T^{2})

, and also bounded from

H^{(1, 1)} (T^{2})

into

L^{1} (T^{2})

. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces

L^{1} \log^{+} L (T^{2})

,

L^{1} {(\log^{+} L)}^{2} (T^{2})

, and

L^{μ} (T^{2})

with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures.

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