Noncommutative weak Orlicz spaces and martingale inequalities

TL;DR

This paper explores noncommutative weak Orlicz spaces, extending classical interpolation theorems and establishing new martingale inequalities, including a weak type $\Phi$-moment Burkholder-Gundy inequality, advancing the understanding of noncommutative probability spaces.

Contribution

It extends the Marcinkiewicz interpolation theorem to noncommutative weak Orlicz spaces and proves a new weak type $\Phi$-moment Burkholder-Gundy inequality for noncommutative martingales.

Findings

Extended Marcinkiewicz interpolation theorem to noncommutative weak Orlicz spaces

Proved weak type $\Phi$-moment Burkholder-Gundy inequality for noncommutative martingales

Established a weak type $\Phi$-moment noncommutative Khintchine's inequality

Abstract

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.