Noncommutative maximal inequalities associated with convex functions

Abstract

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

Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

“…We refer to Theorem 4.1 and Theorem 4.4 for more detailed explanations of the notation used in the formulations of (1.5) and (1.6). These results complement the series of Φ-moment inequalities from [1,2,11,13]. We note that if Φ(t) = t p for 1 < p < ∞, then these results become exactly the Junge and Xu's noncommutative Burkholder inequalities.…”

“…For instance, Φ-moment versions of the noncommutative Burkholder-Gundy inequalities from [34] were considered in [1,13]. Various maximal type-inequalities for noncommutative martingales initially proved in [18] for the case of noncommutative L p -spaces are now known to be valid for a wider class of convex functions ( [2,11]). In this paper, we are mainly interested on inequalities involving conditioned square functions of noncommutative martingales.…”

Noncommutative Davis type decompositions and applications

“…The function satisfies the ‐condition if and only if it is ‐concave for some . Recall the so‐called Matuzewska–Orlicz indices and of : where The indices and are used in the previous papers instead of the convexity and concavity indices in the present one. It is easy to see that if is ‐convex and ‐concave.…”

“…For noncommutative martingales, this line of research was initiated by Bekjan and Chen in where they provided several ‐moment inequalities such as ‐moment versions of the noncommutative Khintchine inequalities and noncommutative Burkholder–Gundy inequalities among other closely related results. Subsequently, ‐moment analogues of other inequalities were also considered (see for instance, ). Recently, the sharpest result for the ‐moment analogue of the noncommutative Burkholder–Gundy inequalities was obtained by Jiao et al .…”

Modular inequalities in martingale Orlicz–Karamata spaces

“…Inequalities of type (1.1) are precisely the so-called modular martingale inequalities. Subsequently, such kind of inequalities was extensively treated in various situations; see for example [1,3,4,8,[18][19][20]27].…”