H\"older estimates for the noncommutative Mazur maps

TL;DR

This paper establishes H"older continuity estimates for the noncommutative Mazur maps between noncommutative Lp spaces associated with von Neumann algebras, extending classical results to the noncommutative setting.

Contribution

It provides the first H"older estimates for noncommutative Mazur maps, generalizing classical commutative results to the noncommutative framework.

Findings

Mazur maps are $ ext{min}rac{p}{q},1$-H"older on balls

Estimates extend classical commutative case to noncommutative setting

Results apply to all von Neumann algebras with $1 eq p,q<\infty$

Abstract

For any von Neumann algebra $\mathcal M$, the noncommutative Mazur map $M_{p,q}$ from $L_p(\mathcal M)$ to $L_q(\mathcal M)$ with $1\leq p,q<\infty$ is defined by $f\mapsto f|f|^{\frac {p-q}q}$. In analogy with the commutative case, we gather estimates showing that $M_{p,q}$ is $\min\{\frac pq,1\}$-H\"older on balls.