Square function inequalities for non-commutative martingales

Abstract

We prove a non-commutative version of the weak-type (1,1) boundedness of square functions of martingales. More precisely, we prove that there is an absolute constantK with the following property: ifM is a semifinite von Neumann algebra with a faithful normal traceτ and (M _n ) ^∞ _n=1 is an increasing filtration of von Neumann subalgebras of (M then for any martingalex= ^∞ _n=1 inL ¹(M,τ), adapted to (M _n ) ^∞ _n=1 , there is a decomposition into two sequences (x _n ) ^∞ _n=1 and (z _n ) ^∞ _n=1 withx _n=y _n+z _nfor everyn≥1 and such that$\left\| {\left( {\sum\limits_{n = 1}^\infty {\left| {dy_n } \right|^2 } } \right)^{1/2} } \right\|_{1,\infty } + \left\| {\left( {\sum\limits_{n = 1}^\infty {\left| {dz_n^ * } \right|^2 } } \right)^{1/2} } \right\|_{1,\infty } \leqslant K\left\| x \right\|_1$. This generalizes a result of Burkholder from classical martingale theory to non-commutative martingales. We also include some applications to martingale Hardy spaces.