### Abstract

The atomic decomposition of a vector-valued martingale Hardy space is given. A classical inequality of Marcinkiewicz is generalized for UMD lattice valued (bounded) Vilenkin-Fourier series. It is proved that the Vilenkin-Fourier series of f ∈ L _{P}(X) (1 < p < ∞) converges to f in L _{p}(X) norm if and only if X is a UMD space. Moreover, a lacunary sequence of the UMD lattice valued Vilenkin-Fourier series of f ∈ H _{1} (X) converges almost everywhere to f in X norm.

Original language | English |
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Pages (from-to) | 413-424 |

Number of pages | 12 |

Journal | Publicationes Mathematicae |

Volume | 71 |

Issue number | 3-4 |

Publication status | Published - Nov 22 2007 |

### Keywords

- Atomic decomposition
- UMD spaces
- Vector-valued Hardy spaces
- Vector-valued Vilenkin-Fourier series

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Weisz, F. (2007). Hardy spaces and convergence of vector-valued Vilenkin-Fourier series.

*Publicationes Mathematicae*,*71*(3-4), 413-424.