Maximal operators of Walsh–Kaczmarz logarithmic means

On the maximal operators of Walsh-Kaczmarz-Fejér means

“…Remark 1. The sequence {q k : k ≥ 0} of Cesáro means σ α n satisfy conditions (11) q 0 Q n ≥ c n α , q n − q n+1 ≥ c n α−2 , 0 < α ≤ 1, as n → ∞, but they are not uniformly bounded from the martingale Hardy spaces H 1/(1+α) (G) to the space L 1/(1+α) (G) .…”

“…We mention, for instance, the papers by Simon [20], Gát [2] and Blahota, Gát [1], (see also [26]). In [11] Goginava and Nagy proved that the maximal operator R * ,κ of Riesz's means is bounded from the Hardy space H p to the space weak − L p , when p > 1/2, but is not bounded from the Hardy space H p to the space L p , when 0 < p ≤ 1/2. They also showed that there exists a martingale f ∈ H p , (0 < p ≤ 1), such that the maximal operator L * ,κ of Nörlund logarithmic means is not bounded in the space L p .…”

On the maximal operators of $T$ means with respect to Walsh-Kaczmarz system

“…In [16] Goginava and Nagy proved that the maximal operator R * ,κ of Riesz means with respect to Walsh-Kaczmarz system is bounded from the Hardy space H 1/2 to the space L 1/2,∞ , but is not bounded from the Hardy space H p to the space L p , when 0 < p ≤ 1/2. In [35] it was proved that there exists a martingale f ∈ H p , (0 < p ≤ 1), such that the maximal operator L * ,κ of Nörlund logarithmic means with respect to Walsh-Kaczmarz system is not bounded in the Lebesgue space L p .…”