Square function inequalities for non-commutative martingales

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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.