## Abstract

The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh- Fourier series is of type (H^{#},L_{1}) provided that the supremum in the maximal operator is taken over some special indices. The set of Walsh polynomials is dense in H^{#}, so by the well-known density argument we have that t_{2n,2m} f (x^{1}, x^{2}) → f (x^{1}, x^{2}) a. e. as m, n → ∞ for all f ∈ H^{#}(⊃ L log^{+} L). We also prove the sharpness of this result. Namely, For all measurable function δ: [0,+∞) → [0,+∞), lim_{t→∞} δ(t) = 0 we have a function f such as f ∈ Llog^{+} Lδ(L) and the two-dimensional Nörlund logarithmic means do not converge to f a.e. (in the Pringsheim sense) on I^{2}.

Original language | English |
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Pages (from-to) | 447-464 |

Number of pages | 18 |

Journal | Real Analysis Exchange |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2006 |

Externally published | Yes |

## Keywords

- A. e. convergence and divergence
- Double Walsh-Fourier series
- Logarithmic means

## ASJC Scopus subject areas

- Analysis
- Geometry and Topology