### 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

Original language | English |
---|---|

Pages (from-to) | 135-146 |

Number of pages | 12 |

Journal | Acta Mathematica Hungarica |

Volume | 96 |

Issue number | 1-2 |

Publication status | Published - Jul 2002 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

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

**Marcinkiewicz-θ-summability of Fourier transforms.** / Weisz, F.

Research output: Contribution to journal › Article

*Acta Mathematica Hungarica*, vol. 96, no. 1-2, pp. 135-146.

}

TY - JOUR

T1 - Marcinkiewicz-θ-summability of Fourier transforms

AU - Weisz, F.

PY - 2002/7

Y1 - 2002/7

N2 - 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 Hp(R2) to Lp(R2) for all p0 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 ∈ L1(R2) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(R2) and so they converge in norm (p0

AB - 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 Hp(R2) to Lp(R2) for all p0 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 ∈ L1(R2) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(R2) and so they converge in norm (p0

KW - Fourier transforms

KW - Hardy spaces

KW - Interpolation

KW - Marcinkiewicz-θ-summation

KW - p-atom

UR - http://www.scopus.com/inward/record.url?scp=0036651121&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036651121&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036651121

VL - 96

SP - 135

EP - 146

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1-2

ER -