A weak type inequality for non-commutative martingales and applications

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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 M is a semi-finite von Neumann algebra and (Mn) 1=1 is an increasing filtration of von Neumann subalgebras of M then for any given martingale x = (xn) 1=1 that is bounded in L 2 (M) ∩ L 1 (M), adapted to (Mn) 1=1, there exist two martingale difference sequences, a = (an) 1=1 and b = (bn) 1=1 , with dxn = an + bn for every n ≥ 1, 1 X n=1 aan ! 1/2