### Abstract

The atomic decomposition of weak Hardy spaces consisting of Vilenkin martingales is formulated. Some sufficient conditions for a sublinear operator T to be bounded from the weak Hardy space wH_{p} to the weak wL_{p} space are given. As applications a weak version of the Hardy-Littlewood inequality is obtained and it is shown that the maximal operator of the Cesàro means of a Vilenkin-Fourier series is bounded from wH_{p} to wL_{p} and is of weak type (1, 1). This yields that the Cesàro means of a function f ∈ L_{1} converge a.e. to the function in question, provided that the Vilenkin system is bounded.

Original language | English |
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Pages (from-to) | 249-264 |

Number of pages | 16 |

Journal | Acta Mathematica Hungarica |

Volume | 80 |

Issue number | 3 |

Publication status | Published - Aug 1998 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

*80*(3), 249-264.

**Bounded operators on weak Hardy spaces and applications.** / Weisz, F.

Research output: Contribution to journal › Article

*Acta Mathematica Hungarica*, vol. 80, no. 3, pp. 249-264.

}

TY - JOUR

T1 - Bounded operators on weak Hardy spaces and applications

AU - Weisz, F.

PY - 1998/8

Y1 - 1998/8

N2 - The atomic decomposition of weak Hardy spaces consisting of Vilenkin martingales is formulated. Some sufficient conditions for a sublinear operator T to be bounded from the weak Hardy space wHp to the weak wLp space are given. As applications a weak version of the Hardy-Littlewood inequality is obtained and it is shown that the maximal operator of the Cesàro means of a Vilenkin-Fourier series is bounded from wHp to wLp and is of weak type (1, 1). This yields that the Cesàro means of a function f ∈ L1 converge a.e. to the function in question, provided that the Vilenkin system is bounded.

AB - The atomic decomposition of weak Hardy spaces consisting of Vilenkin martingales is formulated. Some sufficient conditions for a sublinear operator T to be bounded from the weak Hardy space wHp to the weak wLp space are given. As applications a weak version of the Hardy-Littlewood inequality is obtained and it is shown that the maximal operator of the Cesàro means of a Vilenkin-Fourier series is bounded from wHp to wLp and is of weak type (1, 1). This yields that the Cesàro means of a function f ∈ L1 converge a.e. to the function in question, provided that the Vilenkin system is bounded.

UR - http://www.scopus.com/inward/record.url?scp=0032221601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032221601&partnerID=8YFLogxK

M3 - Article

VL - 80

SP - 249

EP - 264

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 3

ER -