On the maximal operator of (C, α)-means of Walsh-Kaczmarz-Fourier series

Simon [J. Approxim. Theory,127, 39-60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh-Kaczmarz-Fourier series is bounded from the martingale Hardy space H_p to the space L_p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved 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,κ,* is bounded from the martingale Hardy space H_1/(1+α) to the space weak- L_1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H_p such that the maximal operator σ^α,κ,*f does not belong to the space L_p.