## 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}.

Original language | English |
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Pages (from-to) | 175-185 |

Number of pages | 11 |

Journal | Ukrainian Mathematical Journal |

Volume | 62 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics