Martingale inequalities in noncommutative symmetric spaces

TL;DR

This paper extends Burkholder's inequalities to noncommutative symmetric spaces, broadening their applicability to a wider class of martingales with specific Boyd index conditions.

Contribution

It generalizes classical martingale inequalities to noncommutative symmetric spaces with Boyd indices between 1 and 2, answering a previously open question.

Findings

Burkholder's inequalities hold for noncommutative martingales in specified symmetric spaces.

The results include a positive answer to a question by Jiao.

Duality methods recover known cases with Boyd indices greater than 2.

Abstract

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.