Annali di Matematica Pura ed Applicata (1923 -)

, Volume 116, Issue 1, pp 143–150 | Cite as

Inequalities of Paley type for noncommutative martingales

  • Alberto Alesina
  • Leonede de-Michele


The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let\(\Gamma = (\underline {H,\mathcal{A},} {\text{ }}m)\) be a regular gage space, and\(\mathcal{A}_n \) a sequence of von-Neumann algebras such that\(\bigcup\limits_{n = 1}^\infty {\mathcal{A}_n } = \mathcal{A};\) we prove that for every\(F \in L^4 (\Gamma );m\left( {\sum\limits_{n = 1}^\infty {|\varepsilon _n (F) - \varepsilon _{n - 1} (F)|^2 } } \right)^2 \leqq \beta ||F||_4^4 \), where ɛn(F) is the conditional expectation of F with respect to the subalgebra\(\mathcal{A}_n \): We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series.


Harmonic Analysis Conditional Expectation Discrete Group Gage Space Noncommutative Martingale 


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata (1923 -) 1978

Authors and Affiliations

  • Alberto Alesina
    • 1
  • Leonede de-Michele
    • 1
  1. 1.Milano

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