Homogenization of Lévy-type operators: operator estimates with correctors

The goal of the paper is to study in $L_2({\mathbb{R}}^d)$ a self-adjoint operator ${\mathbb{A}}_{\text{eps}}$, $\text{eps}>0$, of the form $$({\mathbb{A}}_{\text{eps}}u)(\text{x})={\int }_{{\mathbb{R}}^d}\mu (\text{x}\mathrm{/}\text{eps},\text{y}\mathrm{/}\text{eps})\frac{(u(\text{x})-u(\text{y}))}{\mathrm{\mid }\text{x}-\text{y}{\mathrm{\mid }}^{d+\alpha }}d\text{y}$$ with $1<\alpha <2$; here the function $\mu (\text{x},\text{y})$ is ${\mathbb{Z}}^d$-periodic in the both variables, satisfies the symmetry relation $\mu (\text{x},\text{y})=\mu (\text{y},\text{x})$ and the estimates $0<{\mu }_{-}⩽\mu (\text{x},\text{y})⩽{\mu }_{+}<\mathrm{\infty }$. The rigorous definition of the operator ${\mathbb{A}}_{\text{eps}}$ is given in terms of the corresponding quadratic form. In the previous work of the authors it was shown that the resolvent $({\mathbb{A}}_{\text{eps}}+I{)}^{-1}$ converges, as $\text{eps}\to 0$, in the operator norm in $L_2({\mathbb{R}}^d)$ to the resolvent of the effective operator $A^0$, and the estimate $\mathrm{\parallel }({\mathbb{A}}_{\text{eps}}+I{)}^{-1}-({\text{A}}^0+I{)}^{-1}\mathrm{\parallel }=O({\text{eps}}^{2-\alpha })$ holds. In the present work we achieve a more accurate approximation of the resolvent of ${\mathbb{A}}_{\text{eps}}$ which takes into account the correctors. Namely, for $N\in \mathbb{N}$ such that $2-1\mathrm{/}N<\alpha \le 2-1\mathrm{/}(N+1)$, we obtain $$\parallel ({\mathbb{A}}_{\text{eps}}+I{)}^{-1}-({\text{A}}^0+I{)}^{-1}-\sum _{m=1}^N{\text{eps}}^{m(2-\alpha )}{\mathbb{K}}_m\parallel =O(\text{eps})\mathrm{.}$$