A Weak Type Inequality for Non-commutative Martingales and Applications
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چکیده
X iv :m at h/ 04 09 13 9v 1 [ m at h. FA ] 8 S ep 2 00 4 A WEAK TYPE INEQUALITY FOR NON-COMMUTATIVE MARTINGALES AND APPLICATIONS NARCISSE RANDRIANANTOANINA Abstract. We prove a weak-type (1,1) inequality for square functions of noncommutative martingales that are simultaneously bounded in L and L. 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)n=1 is an increasing filtration of von Neumann subalgebras of M then for any given martingale x = (xn)∞n=1 that is bounded in L2(M) ∩ L1(M), adapted to (Mn)n=1, there exist two martingale difference sequences, a = (an) ∞ n=1 and b = (bn) ∞ n=1, with dxn = an + bn for every n ≥ 1, ∥
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