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Integral Inequalities for Convex Functions of Operators on Martingales

  • Burgess DavisEmail author
  • Renming Song
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Let M be a family of martingales on a probability space (Ω, A, P) and let ф be a nonnegative function on [0, ∞]. The general question underlying both [2] and the present work may be stated as follows : If U and V are operators on M with values in the set of nonnegative A measurable functions on Ω, under what further conditions does
$${\lambda^{po}}{P(Vf{>}\lambda)}{\leqq}{\begin{array}{lllll} {po} \\ {po} \\ \end{array}} \,\lambda >o,f,\varepsilon M,$$
(1.1)
imply \(E{\Phi}(Vf)\leqq cE{\Phi}(Uf), f \ \in \ \mathcal{M}?\)

Keywords

Measurable Function Convex Function Probability Space Maximal Function Nonnegative Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    D. L. Burkholder"Martingale transforms," Ann. Math. Statist., Vol. 37 (1966), pp. 1494-1504.zbMATHCrossRefMathSciNetGoogle Scholar
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    D. L. Burkholder and R. F. Gundy, "Extrapolation and interpolation of quasi-linear operators on martingales," Acta Math., Vol. 124 (1970), pp. 249-304.zbMATHCrossRefMathSciNetGoogle Scholar
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    B. J. Davis"On the integrability of the martingale square function," Israel J. Math., Vol. 8 (1970) pp. 187-190.zbMATHCrossRefMathSciNetGoogle Scholar
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    C. Doleans"Variation quadratique des martingales continues à droite," Ann. Math. Statist., Vol. 40 (1969), pp. 284-289.zbMATHCrossRefMathSciNetGoogle Scholar
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    A. ZygmundTrigonometric Series, Vol. I, Cambridge, Cambridge University Press, 1959.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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