Inequalities of Paley type for noncommutative martingales

Summary

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.

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Entrata in Redazione il 26 gennaio 1977.

Partially sponsored by C.N.R.

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Alesina, A., de-Michele, L. Inequalities of Paley type for noncommutative martingales. Annali di Matematica 116, 143–150 (1978). https://doi.org/10.1007/BF02413871

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Keywords

  • Harmonic Analysis
  • Conditional Expectation
  • Discrete Group
  • Gage Space
  • Noncommutative Martingale