A weak type inequality for the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system

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Abstract Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H p to the space L p for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ * α,k is bounded from the martingale Hardy space H 1/(1+α) to the space weak-L 1/(1+α).