Φ-moment inequalities for noncommutative differentially subordinate martingales

We establish some Φ Φ -moment inequalities for noncommutative differentially subordinate martingales. Let Φ Φ be a p p -convex and q q -concave Orlicz function with 1 > p ≤ q > 2 1>p≤ q>2 . Suppose that x x and y y are two self-adjoint martingales such that y y is weakly differentially subordinate to x x . We show that, for N ≥ 0 N≥ 0 , τ [ Φ ( | y N | ) ] ≤ c p , q τ [ Φ ( | x N | ) ] , {equation*} τ [Φ (|y_N|) ]≤ cp,qτ [Φ (|x_N|) ], {equation*} where the constant c p , q cp,q is of the best order when p = q p=q . The Φ Φ -moment estimates for square functions of noncommutative differentially subordinate martingales are also obtained in this article. Our approach provides constructive proofs of noncommutative Φ Φ -moment Burkholder–Gundy inequalities and Burkholder inequalities.