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

Abstract

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 $L_p$.