Integral Inequalities for Convex Functions of Operators on Martingales

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


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
$${\lambda^{po}}{P(Vf{>}\lambda)}{\leqq}{\begin{array}{lllll} {po} \\ {po} \\ \end{array}} \,\lambda >o,f,\varepsilon M,$$
imply \(E{\Phi}(Vf)\leqq cE{\Phi}(Uf), f \ \in \ \mathcal{M}?\)


Measurable Function Convex Function Probability Space Maximal Function Nonnegative Function 
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© 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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