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Abstract In a previous paper we introduced a new concept, the notion of -martingales and we extended the well-known Doob inequality (for 1 p + ) and the BurkholderDavisGundy inequalities (for p = 2) to -martingales. After showing new Fefferman-type inequalities that involve sharp brackets as well as the space bmoq, we extend the BurkholderDavisGundy inequalities (for 1 p + ) to -martingales. By means...