Revisiting the Marcinkiewicz theorem for non commutative maximal functions

TL;DR

This paper offers an alternative proof of a Marcinkiewicz interpolation theorem tailored for non-commutative maximal functions, refining previous results and providing near-optimal constants, with extensions to non-positive maps.

Contribution

It introduces a new proof approach for non-commutative Marcinkiewicz interpolation, improving constants and extending results to certain non-positive maps.

Findings

Refined interpolation theorem with better constants

Extension of results to non-positive maps with weakened norms

Provides a substitute for the maximal function of a martingale in $L_p$

Abstract

We give an alternative proof of a Marcinkiewicz interpolation theorem for non commutative maximal functions and positive maps, slightly refining earlier versions of the statement. The main novelty is that it provides a substitute for the maximal function of a martingale in $L_p$, $1<p\leq \infty$, losing very little on numerical constants. For non positive maps, the above mentioned theorem fails but we can still obtain some interpolation results by weakening the maximal norms that we consider.