Square Functions for Noncommutative Differentially Subordinate Martingales

We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if x is a self-adjoint noncommutative martingale and y is weakly differentially subordinate to x then y admits a decomposition dy = a + b + c (resp. dy = z + w ) where a , b , and c are adapted sequences (resp. z and w are martingale difference sequences) such that: ‖ ( a n ) n ≥ 1 ‖ L 1 , ∞ ( M ⊗ ¯ ℓ ∞ ) + ‖ ( ∑ n ≥ 1 ε n - 1 | b n | 2 ) 1 / 2 ‖ 1 , ∞ + ‖ ( ∑ n ≥ 1 ε n - 1 | c n ∗ | 2 ) 1 / 2 ‖ 1 , ∞ ≤ C ‖ x ‖ 1 (resp. ‖ ( ∑ n ≥ 1 | z n | 2 ) 1 / 2 ‖ 1 , ∞ + ‖ ( ∑ n ≥ 1 | w n ∗ | 2 ) 1 / 2 ‖ 1 , ∞ ≤ C ‖ x ‖ 1 ) . We also prove strong-type ( p , p ) versions of the above weak-type results for 1