This is an announcement for the paper "A decomposition for Hardy martingales. Part II" by Paul F. X. Mueller. Abstract: We prove Davis and Garsia Inequalities for dyadic perturbations of Hardy Martingales. We apply those to estimate the $L^1 $ distance of a dyadic martingale to the class of Hardy martingales. We revisit Bourgains embedding of $L^1$ into the quotient space $ L^1 / H^1 . $ Archive classification: math.FA math.CV Submitted from: pfxm@bayou.uni-linz.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.3964 or http://arXiv.org/abs/1209.3964