A characterisation of the closure of H ^∞ in BMO

Abstract

We show that a continuous martingale M∈BMO has a ‖·‖BMO ₂ ⁻ distance to H ^∞ less than ε>0 iff M may be written as a finite sum $M = \sum\limits_{n = 0}^N {{}^{T_n }M^{T_{n + 1} } }$ such that, for each 0≤n≤N, we have $\parallel {}^{T_n }M^{T_{n + 1} } \parallel _{BMO_2 } < \varepsilon$. In particular, we obtain a characterisation of the BMO-closure of H ^∞.

This result was motivated by some problems posed in the survey of N. Kazamaki [K