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The main aim of this paper is to prove that the maximal operator σ# of the Marcinkiewicz-Fejér means of the two-dimensional Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H1/2 to the space weak-L1/2 and is not bounded from the martingale Hardy space H1/2 to the space L1/2 provided that the supremum in the maximal operator is taken over spatial indices. Mathematics subject classification (2000): 42C10.

The weak type inequality for the maximal operator of the Marcinkiewicz-Fejér means of the two-dimensional Walsh-Kaczmarz system