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