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

Authors

  • U. Goginava Tbilisi State Univ., Georgia
  • К. Nagy Inst. Math, and Comput. Sci., Hungary

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<α1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p1/(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<p1/(1+α), there exists a martingale fHp 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.