A Weak Type Inequality for Non‐Commutative Martingales and Applications

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( M n ) n = 1 ∞is an increasing filtration of von Neumann subalgebras of M then for any given martingale x = ( x n ) n = 1 ∞that is bounded in L 2 ( M ) ∩ L 1 ( M ), adapted to( M n ) n = 1 ∞ , there exist two martingale difference sequences, a = ( a n ) n = 1 ∞ and b = ( b n ) n = 1 ∞ , with dx n = a n + b n for every n ⩾ 1, ∥ ( ∑ n = 1 ∞ a n ∗ a n ) 1 / 2∥ 2 + ∥ ( ∑ n = 1 ∞ b n b n ∗ ) 1 / 2∥ 2 ⩽ 2 ∥ x ∥ 2 ,and ∥ ( ∑ n = 1 ∞ a n ∗ a n ) 1 / 2∥ 1 , ∞ + ∥ ( ∑ n = 1 ∞ b n b n ∗ ) 1 / 2∥ 1 , ∞ ⩽ 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. 2000 Mathematics Subject Classification 46L53, 46L52 (primary); 46L51, 60G42 (secondary)