### Abstract

The aim of this paper is to prove Paley type inequalities for two-parameter Vilenkin system. Our main result is the following estimate: (*) (formula omitted) for martingales f ∈ H^{p} (G_{p} × G_{q}) (0 <p ≤ 1). Here G_{p} and G_{q} are Vilenkin groups generated by the sequences p = (p_{n}) and q = (q_{n}), respectively, and f̂(u, v) (u, v ∈ N) is the (u, v)th (two-parameter) Vilenkin-Fourier coefficient of f. The Hardy space H^{p} (G_{p} × G_{q}) is defined by means of a usual martingal maximal function. We get the inequality (*) from its dual version, especially it follows from a BMO-result in the case p = 1. Furthermore, interpolation leads to an L^{p}-variant of (*) for 1 <p ≤ 2. We also formulate an analogous statement for another Hardy space. In the so-called unbounded case, i.e. when p or q is not bounded, we shall investigate whether (*) can be improved. Our results hold also in the case of higher dimensions.

Original language | English |
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Pages (from-to) | 187-199 |

Number of pages | 13 |

Journal | Analysis Mathematica |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2001 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Analysis

### Cite this

*Analysis Mathematica*,

*27*(3), 187-199. https://doi.org/10.1023/A:1014378715256

**Paley type inequalities for several parameter Vilenkin systems.** / Simon, P.; Weisz, F.

Research output: Contribution to journal › Article

*Analysis Mathematica*, vol. 27, no. 3, pp. 187-199. https://doi.org/10.1023/A:1014378715256

}

TY - JOUR

T1 - Paley type inequalities for several parameter Vilenkin systems

AU - Simon, P.

AU - Weisz, F.

PY - 2001

Y1 - 2001

N2 - The aim of this paper is to prove Paley type inequalities for two-parameter Vilenkin system. Our main result is the following estimate: (*) (formula omitted) for martingales f ∈ Hp (Gp × Gq) (0 p and Gq are Vilenkin groups generated by the sequences p = (pn) and q = (qn), respectively, and f̂(u, v) (u, v ∈ N) is the (u, v)th (two-parameter) Vilenkin-Fourier coefficient of f. The Hardy space Hp (Gp × Gq) is defined by means of a usual martingal maximal function. We get the inequality (*) from its dual version, especially it follows from a BMO-result in the case p = 1. Furthermore, interpolation leads to an Lp-variant of (*) for 1

AB - The aim of this paper is to prove Paley type inequalities for two-parameter Vilenkin system. Our main result is the following estimate: (*) (formula omitted) for martingales f ∈ Hp (Gp × Gq) (0 p and Gq are Vilenkin groups generated by the sequences p = (pn) and q = (qn), respectively, and f̂(u, v) (u, v ∈ N) is the (u, v)th (two-parameter) Vilenkin-Fourier coefficient of f. The Hardy space Hp (Gp × Gq) is defined by means of a usual martingal maximal function. We get the inequality (*) from its dual version, especially it follows from a BMO-result in the case p = 1. Furthermore, interpolation leads to an Lp-variant of (*) for 1

UR - http://www.scopus.com/inward/record.url?scp=0034771704&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034771704&partnerID=8YFLogxK

U2 - 10.1023/A:1014378715256

DO - 10.1023/A:1014378715256

M3 - Article

AN - SCOPUS:0034771704

VL - 27

SP - 187

EP - 199

JO - Analysis Mathematica

JF - Analysis Mathematica

SN - 0133-3852

IS - 3

ER -