Interpolation and $\mathrm{\Phi }$-moment inequalities of noncommutative martingales

This paper is devoted to the study of $\mathrm{\Phi }$-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative $\mathrm{\Phi }$-moment analogues of martingale transformations, Stein’s inequalities, Khintchine’s inequalities for Rademacher’s random variables, and Burkholder-Gundy’s inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.