We prove that atomic decomposition for the Hardy spaces h 1 and H 1 is valid for noncommutative martingales. We also establish that the conditioned Hardy spaces of noncommutative martingales h p and bmo form interpolation scales with respect to both complex and real interpolations.

This paper is devoted to the study of -moment inequalities for noncommutative martingales. In particular, we prove the noncommutative -moment analogues of martingale transformations, Stein's inequalities, Khintchine's inequalities for Rademacher's random variables, and Burkholder-Gundy's inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.

We report recent advances on noncommutative martingale inequalities associated with convex functions. These include noncommutative Burkholder-Gundy inequalities associated with convex functions due to the present authors and Dirksen and Ricard, noncommutative maximal inequalities associated with convex functions due to Osȩkowski and the present authors, and noncommutative Burkholder and Junge-Xu inequalities associated with convex functions due to Randrianantoanina and Lian Wu. Some open problems for noncommutative martingales are also included.The aim of this paper is to give a survey on noncommutative martingale inequalities associated with convex functions. Recall that classical martingale inequalities associated with convex functions Φ or Φ-moment martingale inequalities were initiated by Burkholder and Gundy [10]. The above three classical foundational martingale inequalities associated with convex functions were obtained in 1970s' (see [9,22] for details), in which the martingale inequalities in L p -spaces correspond to the case of Φ(t) = t p . In 2012, the present authors [3] proved the noncommutative Burkholder-Gundy inequality associated with convex functions, within which a gap in the proof of noncommutative Khintchine's inequality associated with convex functions was fixed later by Dirksen and Ricard [17].Recently, Osȩkowski and the present authors [6] proved the noncommtative Doob and ergodic maximal inequalities associated with convex functions. More recently, noncommutative Burkholder and Junge-Xu inequalities associated with convex functions were proved by Randrianantoanina and Wu [60].For what follows, in the next section, we will collect notations and definitions for noncommutative L p and symmetric spaces, convex functions, and noncommutative martingales. In Sections 3, 4 and 5 respectively, we will give detailed descriptions of noncommutative Burkholder-Gundy, Doob, and Burkholder and Junge-Xu inequalities associated with convex functions. Finally, in Section 6, we will give some open questions and comments on noncommutative martingale inequalities associated with convex functions. PreliminariesIn what follows, by X A,B,... Y for two nonnegative (possibly infinite) quantities X and Y, we mean that there exists a constant C > 0 depending only on A, B, . . . such that X ≤ CY, and by2.1. Symmetric spaces and interpolation with Boyd indices. First of all, we recall some definitions and notations on symmetric spaces (see [43] for details). We denote by S the set of all measurable functions f on (0, ∞) such that d λ (f ) = |{t ∈ (0, ∞) : |f (t)| > λ}| < ∞ for some λ < 0, where |A| is the Lebesgue measure of a measurable subset A ⊂ (0, ∞). For f ∈ S, we denote by µ t (f ) the decreasing rearrangement of f in (0, ∞), defined by µ t (f ) = inf{λ > 0 : d λ (f ) ≤ t} for all t > 0. For f, g ∈ S, we say f is submajorized by g, and write f ≺≺ g, if t 0 µ s (f )ds ≤ t 0

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Let M be a von Neumann algebra with a faithful normal semi-finite trace τ , and let A be maximal subdiagonal algebra of M. Then for 0 < p < ∞, we define the noncommutative H p -space. We obtain that the conjugation and Herglotz maps are bounded linear maps from L p (M) into L p (M) for 1 < p < ∞, and continuous map from L 1 (M) into L 1,∞ (M). We also give the dual space of H p (A) for 1 ≤ p < ∞, and extend Pisier's interpolation theorem to this case.