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Martingale inequalities in noncommutative symmetric spaces

We provide generalizations of Burkholder's inequalities involving conditioned square functions of martingales to the general context of martingales in noncommutative symmetric spaces. More precisely, we prove that Burkholder's inequalities are valid for any martingale in noncommutative space constructed from a symmetric space defined on the interval $(0,\infty)$ with Fatou property and whose Boyd indices are strictly between 1 and 2. This answers positively a question raised by Jiao and may be viewed as a conditioned version of similar inequalities for square functions of noncommutative martingales. Using duality, we also recover the previously known case where the Boyd indices are finite and are strictly larger than 2.