Noncommutative weak Orlicz spaces and martingale inequalities

This paper is devoted to the study of noncommutative weak Orlicz spaces and martingale inequalities. Marcinkiewicz interpolation theorem is extended to include noncommutative weak Orlicz spaces as interpolation classes. In particular, we prove the weak type $\Phi$-moment Burkholder-Gundy inequality for noncommutative martingales through establishing a weak type $\Phi$-moment noncommutative Khintchine's inequality for Rademacher's random variables.