Necessary and sufficient conditions for mean oscillation inequalities of martingales.

What is it about?

Given a Banach function space X, one can define a quasi-Banach space w-X as in the same way to define the weak-Lp space of Lebesgue space Lp. In this paper, necessary and sufficient conditions for some inequalities involving mean oscillations to be valid in w-X are given. One of such condition is described by using of extended fundamental function of X.

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Why is it important?

The most important feature of this paper is that not only sufficient conditions for some martingale inequalities but also necessary conditions are given.