H\"older estimates for the noncommutative Mazur maps

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.