Let $f$ be a function on ${ \Bbb R}^2$ in the inhomogeneous Besov space $B_{ \infty,1}^1({ \Bbb R}^2)$. 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 $\boldsymbol{ S}_p$ and $f$ is in the above inhomogeneous Besov space, then the following Lipschitz type estimate holds: $$ \|f(A_1,B_1)-f(A_2,B_2)\|_{ \boldsymbol{ S}_p} \le\operatorname{ const}\max\big\{ \|A_1-A_2\|_{ \boldsymbol{ S}_p},\|B_1-B_2\|_{ \boldsymbol{ S}_p}\big\}. $$
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