Φ-moment inequalities for noncommutative differentially subordinate martingales

Abstract

We establish some Φ \Phi -moment inequalities for noncommutative differentially subordinate martingales. Let Φ \Phi be a p p -convex and q q -concave Orlicz function with 1 > p ≤ q > 2 1>p\leq 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\geq 0 , τ [ Φ ( | y N | ) ] ≤ c p , q τ [ Φ ( | x N | ) ] , \begin{equation*} \tau \big [\Phi (|y_N|)\big ]\leq c_{p,q}\tau \big [\Phi (|x_N|)\big ], \end{equation*} where the constant c p , q c_{p,q} is of the best order when p = q p=q . The Φ \Phi -moment estimates for square functions of noncommutative differentially subordinate martingales are also obtained in this article. Our approach provides constructive proofs of noncommutative Φ \Phi -moment Burkholder–Gundy inequalities and Burkholder inequalities.