Noncommutative maximal inequalities associated with convex functions

TL;DR

This paper establishes new noncommutative maximal inequalities related to convex functions, extending classical results to a broader noncommutative setting with applications to martingales and ergodic theory.

Contribution

It introduces a Marcinkiewicz type interpolation theorem for convex functions of maximal operators in noncommutative spaces, generalizing previous $L^p$ results.

Findings

Proved a Doob type inequality for convex functions of noncommutative martingales.

Established noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for convex functions.

Developed a Marcinkiewicz interpolation theorem in the noncommutative setting.

Abstract

We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the $L^p$ case to the case of convex functions.