A decomposition for Hardy martingales (Q2866930)

It is proved that every Hardy martingale \(F=(F_k)\) can be written as \(F=G+B\), where \(G=(G_k)\) and \(B=(B_k)\) are again Hardy martingales such that NEWLINE\[NEWLINE \operatorname E \left(\sum_{k=1}^{n} \operatorname E_{k-1}\left|\Delta G_k\right|^2\right)^{1/2} + \operatorname E \left(\sum_{k=1}^{n} \left|\Delta B_k\right|\right) \leq C \operatorname E(\left|F_n\right|) NEWLINE\]NEWLINE and \(\left|\Delta G_k\right|\leq A_0\left|F_{k-1}\right|\), \(k\leq n\). As applications, some well-known martingale inequalities are proved, such as inequalities due to Burkholder, Gundy, Davis and Garsia.