Functions of perturbed noncommuting unbounded self-adjoint operators

Let $f$ be a function on $\mathbb{R}^2$ in the inhomogeneous Besov space $\text{Б}_{\infty,1}^1(\mathbb{R}^2)$. For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators, we define the function $f(A,B)$ of $A$ and $B$ as a densely defined linear operator. We show that if $1\le p\le2$, $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that both $A_1-A_2$ and $B_1-B_2$ belong to the Schatten–von Neumann class $\mathbf{S}_p$ and $f\in\text{Б}_{\infty,1}^1(\mathbb{R}^2)$, then the following Lipschitz type estimate holds:
$$ \|f(A_1,B_1)-f(A_2,B_2)\|_{\mathbf{S}_p} \le\mathrm{const}\,\|f\|_{\text{Б}_{\infty,1}^1}\max\big\{\|A_1-A_2\|_{\mathbf{S}_p},\|B_1-B_2\|_{\mathbf{S}_p}\big\}. $$