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 $1\le p,q<\mathrm{\infty }$ is defined by $f\mapsto f\mathrm{\mid }f{\mathrm{\mid }}^{\frac{p-q}{q}}$. In analogy with the commutative case, we gather estimates showing that $M_{p,q}$ is $\min{\frac pq,1}$-H"older on balls.