The main aim of this paper is to prove that there exists a martingale f ∈ H12/▭ such that the restricted maximal operators of Fejér means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier
series does not belong to the space weak-L1/2.
Simon  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 Hp to the space Lp 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 H1/(1+α) to the space weak-L1/(1+α).
The main aim of this paper is to prove that the maximal operator σ0k*:= supn ∣σn,nk∣ of the Fej�r means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H1/2 to the space weak-L1/2.