Commutator estimates for functions of noncommuting self-adjoint operators

We study properties of the calculus $\phi \mapsto \phi (A,B)$ for self-adjoint operators with commutator $[A,B]$ in the Schaten–von Neumann class ${\mathit{S}}_p$. It turns out that this calculus defined on the Besov class $B_{\mathrm{\infty },1}^1({\mathbb{R}}^2)$ in the case $p\le 2$ admits a commutator Lipschitz estimate and is multiplicative modulo ${\mathit{S}}_p$. On the other hand in the case $p>2$ there is no commutator Lipschitz estimate. There is no commutator Lipschitz estimate in the operator norm as well.