Algebraic Davis Decomposition and Asymmetric Doob Inequalities

Abstract

In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let (M, τ ) be a noncommutative probability space equipped with a weak- * dense filtration of von Neumann subalgebras (Mn) n≥1 . Let En denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for 1 < p < 2 and x ∈ Lp(M, τ ) one can find a, b ∈ Lp(M, τ ) and contractions un, vn ∈ M such thatMoreover, it turns out that aun and vnb converge in the row/column Hardy spaces H r p (M) and H c p (M) respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case p = 1, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in L 1 , whose noncommutative form has remained open for quite some time. Given 1 ≤ p ≤ 2, we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of En(x) in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.

Noncommutative Davis type decompositions and applications

“…A constructive approach appeared in for the space for . The noncommutative Davis decomposition has proven to be a powerful tool in noncommutative martingale inequalities; for instance, it plays a prominent role in establishing various forms of Doob maximal inequalities in as well as in the study of continuous time noncommutative martingale inequalities in .…”

Asymmetric Doob inequalities in continuous time

“…The row spaces h r pw (M) and h 1r pw (M) are defined in a similar way. Moreover, according to [5,Lemma 2.5] we conclude that the family (x σ u ) σ∈Σ is uniformly bounded in the reflexive space L p (M); whence in L p (M) and we can define…”

“…Moreover, the self-adjointness and contractivity of conditional expectations allow us to commute them with the weak L p ultralimits below. Finally, emulating our argument in the discrete case [5] we may combine the mean zero property of X • 2 (M) with the adapted sequences in…”

“…Proof of Theorem Biii. Unlike the discrete case, the proof of this part becomes quite involved since it requires an algebraic atomic characterization of the hat Hardy spaces, which could be regarded as a continuous version of Lemma 2.1 in [5]. We shall need some preliminaries.…”

“…In our recent paper [5], we established new asymmetric Doob inequalities based on an algebraic form of the atomic decomposition which yield a very satisfactory and complete picture of the noncommutative maximal inequalities for martingales in discrete time. This included the right reformulation of Davis' martingale theorem [1] for p = 1, which escaped previous attempts for quite some time.…”