Bounded mean oscillation and regulated martingales

Abstract

In the martingale context, the dual Banach space to H 1 {H_1} is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for H 1 {H_1} -martingales which involve the notion of L p {L_p} -regulated L 1 {L_1} -martingales where 1 > p ≤ ∞ 1 > p \leq \infty . The strongest decomposition theorem is for p = ∞ p = \infty , and this provides full information about BMO. The weaker p = 2 p = 2 decomposition is fundamental in the theory of martingale transforms.