Atomic decompositions and asymmetric Doob inequalities in noncommutative symmetric spaces

Abstract

We provide asymmetric forms of Doob maximal inequality in the contexts of moments associated with convex functions and symmetric spaces of measurable operators. More precisely, let (M,τ) be a noncommutative measurable space equipped with a filtration (Mn)n≥1. Let (En)n≥1 be the conditional expectations corresponding to (Mn)n≥1. A sample result for the case of convex functions states that if Φ is an Orlicz function that is p-convex and q-concave for 1≤p≤q<2 and 1−p/2<θ<1, then for every x in the column Hardy space HΦc(M), there exist positive operators a,b, and a sequence of contractions (un)n≥1 in M such that for every n≥1,En(x)=aunb andτ(Φ(a1/(1−θ)))+τ(Φ(b1/θ))≲τ(Φ(Sc(x))). For the case of symmetric space, as an illustration, we obtain that if E is a separable symmetric Banach function space which is an interpolation of (Lp,Lq) for 1<p≤q<2, then for every z∈E(M), there exist positive operators α,β, two sequences of contractions (υn)n≥1 and (ηn)n≥1 in M such that for every n≥1,En(z)=αυn+ηnβ and‖α‖E(M)+‖β‖E(M)≲‖z‖E(M).Our approach is based on new algebraic atomic decompositions and improved Davis type decompositions which we develop for both the case of Hardy spaces associated with separable noncommutative symmetric spaces that are interpolation of the couple (Lp,Lq) for 1<p≤q<2 as well as the case of Hardy spaces associated with convex functions that are p-convex and q-concave for 1≤p≤q<2. Some other inequalities are also provided which are of independent interest.