Noncommutative Davis type decompositions and applications

Abstract

We prove the noncommutative Davis decomposition for the column Hardy space Hpc for every 0<p⩽1. A new feature of our Davis decomposition is a simultaneous control of H1c and Hqc norms for any noncommutative martingale in scriptH1c∩scriptHqc when q⩾2. As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space E that is either an interpolation of the couple (Lp,L2) for some 1<p<2 or is an interpolation of the couple (L2,Lq) for some 2<q<∞. We also obtain the corresponding normalΦ‐moment Burkholder/Rosenthal inequality for Orlicz functions that are either p‐convex and 2‐concave for some 1<p<2 or are 2‐convex and q‐concave for some 2<q<∞.

Square Functions for Noncommutative Differentially Subordinate Martingales

“…Remark 4.14. From the noncommutative Davis decomposition ( [20,32,41]), we may also deduce from Theorem 4.11 that for 1 ≤ p < 2 and x ∈ H p (M), we have (4.27) x p ≤ c p x Hp with c p = (δ ′ p/(2−p) ) 1/2 + 1 2/p + 2/p where δ ′ p/(2−p) denotes the constant from the noncommutative dual Doob inequality relative to the index p/(2 − p) ( [19,22]). This should be compared with [21,Corollary 4.3].…”

“…In this case, Φ(|x|) ∈ L 0 (M, τ ) and its trace τ (Φ(|x|)) is referred to as the Φ-moment of the operator |x|. For more information and background on moment inequalities involving convex functions, we refer to [2,3,10,17,18,40,41]. Below, the following two basic facts will be used.…”

“…In that particular situation, the only relevant case is when Φ is such that sup{p : Φ is p-convex} = 1. Since when Φ is p-convex and q-concave for 1 < p ≤ q < ∞, stronger results on Φ-moment Burkholder/Rosenthal inequalities ( [40,41] For the proofs, we need two auxiliary lemmas about the Cuculescu projections when the associated martingale is bounded in a noncommutative Orlicz space. The next lemma may be viewed as a Φ-moment version of Proposition 2.4(iv).…”

Sums of free variables in fully symmetric spaces

“…We also explain how to use our main result to interpolate the conditioned Burkholder inequality in the spirit of [5,21]. After we submitted this article, Randrianantoanina informed us that he also obtained with Xu these martingale inequalities in [22] but with different techniques.…”

Interpolation and the John–Nirenberg inequality on symmetric spaces of noncommutative martingales

“…which is a quasi-Banach space under the quasi-norm • E(M, 2 c ) (see [RW15,RWX19]). The conditioned version of E(M, 2 c ), denoted by E cond (M, 2 c ), is defined to be the completion of the space of finite sequences a = (a n ) n≥1 in E(M) ∩ M under the quasi-norm (see [Jun02,RWX19])…”