Integral Inequalities for Convex Functions of Operators on Martingales

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 $E{\Phi}(Vf)\leqq cE{\Phi}(Uf), f \ \in \ \mathcal{M}?$