On the maximal operator of $(C,\alpha)$-means of Walsh-Kaczmarz-Fourier series

Abstract

Simon [J. Approx. Theory, 127 (2004), 39-60] proved that the maximal operator $\sigma ^{\alpha,\kappa,*}$ of the $(C,\alpha)$-means of the Walsh-Kaczmarz-Fourier series is bounded from the martingale Hardy space $H_p$ to the space $L_p$ for $p>1/(1+\alpha), 0<\alpha\leq 1$.

Recently, Gát and Goginava [2] proved that this boundedness result does not hold if $p\leq 1/(1+\alpha)$. However, in the endpoint case $p=1/(1+\alpha)$ the maximal operator $\sigma ^{\alpha,\kappa,*}$ is bounded from the martingale Hardy space $H_{1/(1+\alpha)}$ to the space weak-$L_{1/(1+\alpha )}$.

The main aim of this paper is to prove a stronger result, that is for any $0<p\leq 1/(1+\alpha)$ there exists a martingale $f\in H_p$ such that the maximal operator $\sigma ^{\alpha,\kappa,*}f$ does not belong to the space $L_p$.