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

Simon [12] proved that the maximal operator of (C, alpha)-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 + alpha). In this paper we prove that this boundedness result does not hold if p a parts per thousand broken vertical bar 1/(1 + alpha). However, in the endpoint case p = 1/(1 + alpha) the maximal operator sigma (*) (alpha,k) is bounded from the martingale Hardy space H (1/(1+alpha)) to the space weak-L (1/(1+alpha)).