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 σα,κ,∗ of the (C,α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space Hp to the space Lp 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 σα,κ,∗ is bounded from the martingale Hardy space H1/(1+α) to the space weak- L1/(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∈Hp such that the maximal operator σα,κ,∗f does not belong to the space Lp.Published
25.02.2010
Issue
Section
Research articles
How to Cite
Goginava, U., and Nagy К. “On the Maximal Operator of (C,α)-Means of Walsh–Kaczmarz–Fourier Series”. Ukrains’kyi Matematychnyi Zhurnal, vol. 62, no. 2, Feb. 2010, pp. 158–166, https://umj.imath.kiev.ua/index.php/umj/article/view/2852.