Martingale inequalities in noncommutative symmetric spaces

Abstract

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, ∞) 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.

Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

“…As noted earlier, we heavily employ results from interpolation theory. As in the case of noncommutative symmetric spaces, a simultaneous decomposition version of (1.4) from [39] also plays a significant role in our argument. The proof of (1.5) is a duality type-argument.…”

“…We refer the reader to [1,39] for noncommutative Hardy spaces associated with square functions. Recall that if x = (x n ) n≥1 is an L 2 (M) + M-martingale, then we can formally define:…”

“…These were formally introduced by Junge in [18] for noncommutative L p -spaces and were extensively used by Junge and Xu in [20,22]. Recently, these ideas were adapted in [39] to the case of more general classes of noncommutative symmetric spaces. Below, we use the usual convention that E 0 = E 1 .…”

“…As shown in [20,22], the equivalences (1.3) and (1.4) have far reaching applications ranging from random matrices to operator space classifications of some classes of subspaces of noncomutative L p -spaces. Recently, equivalences (1.3) and (1.4) were extended to certain classes of noncommutative symmetric spaces for which we refer to [39] for details. Motivated by these various results, we consider Φ-moments of conditioned square functions of noncommutative martingales in the spirit of (1.2).…”

“…We note that if Φ(t) = t p for 1 < p < ∞, then these results become exactly the Junge and Xu's noncommutative Burkholder inequalities. It is also important to note that the case of noncommutative symmetric spaces treated in [39] does not imply the corresponding Φ-moment inequalities.…”