Gundy’s Decomposition for Non-commutative Martingales and Applications

We provide an analogue of Gundy’s decomposition for L1-bounded non-commutative martingales. An important difference from the classical case is that for any L1-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1, 1) boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder’s weak type inequality for square functions. A sequence (xn)n≥1 in a normed space X is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of (xn)n≥1 to l2 taking each xn to the n-th vector basis of l2. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in L1(M, τ) whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative L1-spaces. Introduction The main motivation for this paper comes from a fundamental decomposition of martingales due to Gundy [13] which is generally referred to as the Gundy’s decomposition theorem. Gundy’s theorem has been very useful in establishing weak type (1, 1) boundedness of certain quasi-linear mappings such as square functions and Doob’s maximal functions. In particular, certain classical inequalities such as the weak type (1, 1) boundedness of martingale transforms and Burkholder’s weak type inequality for square functions can be deduced from Gundy’s theorem. We refer to [4, 14, 21] for some variations of Gundy’s result and more applications and to Garsia’s notes [12] for a complete discussion on classical martingale inequalities. Gundy’s decomposition theorem played a central role in classical martingale theory and it can be regarded as a probabilistic counterpart of the well known Calderón-Zygmund decomposition for integrable functions [5] in harmonic analysis. Due to its relevance in the classical theory, it is natural to consider whether or not such decomposition theorem can be generalized to the non-commutative setting. In this paper, we investigate possible analogues of Gundy’s theorem for non-commutative martingales. We first recall this classical result: ∗ Partially supported by BFM 2001/0189. † Partially supported by NSF DMS-0096696. 2000 Mathematics Subject Classification. Primary: 46L53, 46L52. Secondary: 46L51, 60G42