### Abstract

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function θ. Under some conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from H_{p}(R^{2}) to L_{p}(R^{2}) for all p_{0} < p ≦ ∞ and, consequently, is of weak type (1, 1), where p_{0} < 1 depends only on θ. As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz-θ-means of a function f ∈ L_{1}(R^{2}) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces H_{p}(R^{2}) and so they converge in norm (p_{0} < p <). Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski and Riesz summations.

Original language | English |
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Pages (from-to) | 135-146 |

Number of pages | 12 |

Journal | Acta Mathematica Hungarica |

Volume | 96 |

Issue number | 1-2 |

Publication status | Published - Jul 1 2002 |

### Keywords

- Fourier transforms
- Hardy spaces
- Interpolation
- Marcinkiewicz-θ-summation
- p-atom

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Acta Mathematica Hungarica*,

*96*(1-2), 135-146.