Homogenization of the Lévy-type operators

In $L_2(\mathbb R^d)$, we consider a selfadjoint operator ${\mathbb A}_\varepsilon$, $\varepsilon >0$, of the form
$$ ({\mathbb A}_\varepsilon u) (\mathbf{x}) =\int_{\mathbb R^d} \mu\biggl(\frac{\mathbf{x}}{\varepsilon},\frac{\mathbf{y}}{\varepsilon}\biggr) \frac{(u(\mathbf{x}) -u(\mathbf{y}))}{| \mathbf{x}-\mathbf{y} |^{d+\alpha}}\,d \mathbf{y}, $$
where $0< \alpha < 2$. It is assumed that a function $\mu(\mathbf{x},\mathbf{y})$ is bounded, positive definite, periodic in each variable, and is such that $\mu(\mathbf{x},\mathbf{y})=\mu(\mathbf{y},\mathbf{x})$. A rigorous definition of the operator ${\mathbb A}_\varepsilon$ is given in terms of the corresponding quadratic form. It is proved that the resolvent $({\mathbb A}_\varepsilon+I)^{-1}$ converges in the operator norm on $L_2(\mathbb R^d)$ to the operator $({\mathbb A}^0+I)^{-1}$ as $\varepsilon\to 0$. Here, ${\mathbb A}^0$ is an effective operator of the same form with the constant coefficient $\mu^0$ equal to the mean value of $\mu(\mathbf{x},\mathbf{y})$. We obtain an error estimate of order $O(\varepsilon^\alpha)$ for $0< \alpha < 1$, $O(\varepsilon (1+| \operatorname{ln} \varepsilon|)^2)$ for $ \alpha=1$, and $O(\varepsilon^{2- \alpha})$ for $1< \alpha < 2$. In the case where $1< \alpha < 2$, the result is refined by taking the correctors into account.