Almost commuting functions of almost commuting self-adjoint operators

Let $A$ and $B$ be almost commuting (i.e, $AB-BA\in {\text{bS}}_1$) self-adjoint operators. We construct a functional calculus $\text{f}\mapsto \text{f}(A,B)$ for $\text{f}$ in the Besov class $B_{\text{be},1}^1({\mathbb{R}}^2)$. This functional calculus is linear, the operators $\text{f}(A,B)$ and $\psi (A,B)$ almost commute for $\f,,\psi\in B_{\be,1}^1(\R^2)$,$\f(A,B)=u(A)v(B)$whenever$\f(s,t)=u(s)v(t)$, and the Helton–Howe trace formula holds. The main tool is triple operator integrals.