### Abstract

The one-dimensional dyadic martingale Hardy spaces H_{p} are introduced and it is proved that the maximal operator of the (C, α) means of a Walsh-Fourier series is bounded from H_{p} to L_{p} (1/(α + 1) < p < ∞) and is of weak type (L_{1}, L_{1}). As a consequence, we obtain the summability result due to Fine; more exactly, the (C, α) means of the Walsh-Fourier series of a function f ∈ L_{1} converge a.e. to the function in question. Moreover, we prove that the (C, α) means are uniformly bounded on H_{p} whenever 1/(α + 1) < p < ∞. We define the two-dimensional dyadic hybrid Hardy space H_{1}^{#} and verify that the maximal operator of the (C, α, β) means of a two-dimensional function is of weak type (H_{1}^{#}, L_{1}). Consequence, the Walsh-Fourier series of every function of f ε H_{1}^{#} is (C, α, β) summable to the function f.

Original language | English |
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Pages (from-to) | 141-155 |

Number of pages | 15 |

Journal | Analysis Mathematica |

Volume | 27 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2001 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Analysis Mathematica*,

*27*(2), 141-155. https://doi.org/10.1023/A:1014364010470