A noncommutative Davis’ decomposition for martingales

The theory of noncommutative martingale inequalities has been rapidly developed since the establishment of the noncommutative Burkholder-Gundy inequalities in [12]. Many of the classical martingale inequalities has been transferred to the noncommutative setting. These include, in particular, the Doob maximal inequality in [3], the Burkholder/Rosenthal inequality in [5], [8], several weak type (1, 1) inequalities in [15, 16, 17] and the Gundy decomposition in [11]. We would point out that the noncommutative Gundy’s decomposition in this last work is remarkable and powerful in the sense that it implies several previous inequalities. For instance, it yields quite easily Randrianantoanina’s weak type (1, 1) inequality on martingale transforms (see [11]). It is, however, an open problem weather there exist a noncommutative analogue of the classical Davis’ decomposition for martingales (see [17] and [10]). This is the main concern of our paper. We now recall the classical Davis’ decomposition for commutative martingales. Given a probability space (Ω, A, μ), let A1, A2, · · · be an increasing filtration of σ-subalgebras of A and let E1,E2, · · · denote the corresponding family of conditional expectations. Let f = (fn)n≥1 be a martingale adapted to this filtration and bounded in L1(Ω). Then M(f) = sup |fn| ∈ L1(Ω) iff we can decompose f as a sum f = g + h of two martingales adapted to the same filtration and satisfaying