Homogenization of non-symmetric convolution type operators

The paper studies homogenization problem for a bounded in $L_2({\mathbb{R}}^d)$ convolution type operator ${\mathbb{A}}_{\text{eps}}$, $\text{eps}>0$, of the form $$({\mathbb{A}}_{\text{eps}}u)(\text{x})={\text{eps}}^{-d-2}{\int }_{{\mathbb{R}}^d}a((\text{x}-\text{y})\mathrm{/}\text{eps})\mu (\text{x}\mathrm{/}\text{eps},\text{y}\mathrm{/}\text{eps})(u(\text{x})-u(\text{y}))d\text{y}\mathrm{.}$$ It is assumed that $a(\text{x})$ is a non-negative function from $L_1({\mathbb{R}}^d)$, and $\mu (\text{x},\text{y})$ is a periodic in $\text{x}$ and $\text{y}$ function such that $0<{\mu }_{-}⩽\mu (\text{x},\text{y})⩽{\mu }_{+}<\mathrm{\infty }$. No symmetry assumption on $a(\cdot )$ and $\mu (\cdot )$ is imposed, so the operator ${\mathbb{A}}_{\text{eps}}$ need not be self-adjoint. Under the assumption that the moments $M_k={\int }_{{\mathbb{R}}^d}\mathrm{\mid }\text{x}{\mathrm{\mid }}^{k}a(\text{x})d\text{x}$, $k=1,2,3$, are finite we obtain, for small $\text{eps}>0$, sharp in order approximation of the resolvent $({\mathbb{A}}_{\text{eps}}+I{)}^{-1}$ in the operator norm in $L_2({\mathbb{R}}^d)$, the discrepancy being of order $O(\text{eps})$. The approximation is given by an operator of the form $({\mathbb{A}}^0+{\text{eps}}^{-1}⟨\mathit{\alpha },\mathrm{\nabla }⟩+I{)}^{-1}$ multiplied on the right by a periodic function $q_0(\text{x}\mathrm{/}\text{eps})$; here ${\mathbb{A}}^0=-\mathrm{div}{g}^0\mathrm{\nabla }$ is the effective operator, and $\mathit{\alpha }$ is a constant vector.