Commutator estimates for functions of noncommuting self-adjoint operators

Abstract

We study properties of the calculus $\varphi\mapsto\varphi(A,B)$ for self-adjoint operators with commutator $[A,B]$ in the Schaten--von Neumann class $\boldsymbol{S}_p$. It turns out that this calculus defined on the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ in the case $p\le2$ admits a commutator Lipschitz estimate and is multiplicative modulo $\boldsymbol{S}_p$. 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.