Gundy's decomposition for non-commutative martingales and applications
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Autores: Javier Parcet
, Narcisse Randrianantoanina
- Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 93, Nº 1, 2006, págs. 227-252
- Idioma: inglés
- DOI: 10.1017/s0024611506015863
- Texto completo no disponible (Saber más ...)
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Resumen
We provide an analogue of Gundy's decomposition for $L_1$-bounded non-commutative martingales. An important difference from the classical case is that for any $L_1$-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 $(x_n)_{n \ge 1}$ in a normed space $\mathrm{X}$ is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of $(x_n)_{n \ge 1}$ to $l_2$ taking each $x_n$ to the $n$th vector basis of $l_2$. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in $L_1 (\mathcal{M}, \tau)$ whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative $L_1$-spaces.
