A weak-type inequality for non-commutative martingales and applications
| dc.creator | Randrianantoanina, Narcisse | |
| dc.date | 2004-09-08 | |
| dc.date.accessioned | 2026-07-07T05:11:57Z | |
| dc.date.available | 2026-07-07T05:11:57Z | |
| dc.description | We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant $K>0$ such that if $\cal{M}$ is a semi-finite von Neumann algebra and $(\cal{M}_n)^{\infty}_{n=1}$ is an increasing filtration of von Neumann subalgebras of $\cal{M}$ then for any given martingale $x=(x_n)^{\infty}_{n=1}$ that is bounded in $L^2(\cal{M})\cap L^1(\cal{M})$, adapted to $(\cal{M}_n)^{\infty}_{n=1}$, there exist two \underline{martingale difference} sequences, $a=(a_n)_{n=1}^\infty$ and $b=(b_n)_{n=1}^\infty$, with $dx_n = a_n + b_n$ for every $n\geq 1$, \[ | (\sum^\infty_{n=1} a_n^*a_n)^{{1}/{2}}|_{2} + | (\sum^\infty_{n=1} b_nb_n^*)^{1/2}|_{2} \leq 2| x |_2, \] and \[ | (\sum^\infty_{n=1} a_n^*a_n)^{{1}/{2}}|_{1,\infty} + | (\sum^\infty_{n=1} b_nb_n^*)^{1/2}|_{1,\infty} \leq K| x |_1. \] As an application, we obtain the optimal orders of growth for the constants involved in the Pisier-Xu non-commutative analogue of the classical Burkholder-Gundy inequalities. | |
| dc.description | 38 pages | |
| dc.identifier | https://arxiv.org/abs/math/0409139 | |
| dc.identifier | http://arxiv.org/abs/math/0409139 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72416 | |
| dc.subject | Functional Analysis | |
| dc.subject | Operator Algebras | |
| dc.subject | 46L53, 46L52 | |
| dc.title | A weak-type inequality for non-commutative martingales and applications | |
| dc.type | text |