Paley type inequalities for several parameter Vilenkin systems

P. Simon, F. Weisz

Research output: Contribution to journalArticle

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 ∈ Hp (Gp × Gq) (0 <p ≤ 1). Here Gp 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 <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 languageEnglish
Pages (from-to)187-199
Number of pages13
JournalAnalysis Mathematica
Volume27
Issue number3
DOIs
Publication statusPublished - 2001

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Interpolation
Hardy Space
Two Parameters
Maximal Function
Fourier coefficients
Martingale
Higher Dimensions
Interpolate
Estimate

ASJC Scopus subject areas

  • Applied Mathematics
  • Analysis

Cite this

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

In: Analysis Mathematica, Vol. 27, No. 3, 2001, p. 187-199.

Research output: Contribution to journalArticle

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