### Abstract

The two-parameter dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from Hp to Lp (1/(α + 1), 1/(β + 1) < p < ∞) and is of weak type (H_{1} ^{#}, L_{1}), where the Hardy space H_{1}^{#} is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f ∈ H_{1}^{#} converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_{p} whenever 1/(α + 1), 1/(β + 1) < p < ∞. Thus, in case f ∈ Hp, the (C, α, β) means converge to f in H_{p} norm. The same results are proved for the conjugate (C, α, β) means, too.

Original language | English |
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Pages (from-to) | 389-401 |

Number of pages | 13 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 6 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 2000 |

### Keywords

- (C, α, β) summability
- Atomic decomposition
- Interpolation
- Martingale Hardy spaces
- P-quasi-local operator
- Rectangle p-atom
- Walsh functions

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Applied Mathematics