A weak-type inequality for non-commutative martingales and applications

Abstract

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 $\mathcal{M}$ is a semi-finite von Neumann algebra and $({\mathcal{M}}_n{)}_{n=1}^{\infty }$ is an increasing filtration of von Neumann subalgebras of $\mathcal{M}$ then for any given martingale $x=(x_n{)}_{n=1}^{\infty }$ that is bounded in $L^2(\mathcal{M})\cap {\mathcal{L}}^1(\mathcal{M})$, adapted to $({\mathcal{M}}_n{)}_{n=1}^{\infty }$, there exist two \underline{martingale difference} sequences, $a=(a_n{)}_{n=1}^{\infty }$ and $b=(b_n{)}_{n=1}^{\infty }$, with $d{x}_n=a_n+b_n$ for every $n\ge 1$, \[ | (\sum^\infty_{n=1} a_n^*a_n)^{{1}/{2}}|_{2} + | (\sum^\infty_{n=1} b_nb_n^*)^{1/2}|_{2} \leq 2| x |_2, \] 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.

38 pages

Cited by in corpus (2)

• Gundy's decomposition for non-commutative martingales and applications
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