Bounded Mean Oscillation and Regulated Martingales

In the martingale context, the dual Banach space to 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 -martingales which involve the notion of -regulated -martingales where <!-- MATH $1 < p \leq \infty$ --> <img width="99" height="37" align="MIDDLE" border="0" src="images/img7.gif" alt="$ 1 < p \leq \infty $">. The strongest decomposition theorem is for <!-- MATH $p = \infty$ --> , and this provides full information about BMO. The weaker decomposition is fundamental in the theory of martingale transforms.