Operator estimates in homogenization of Lévy-type operators with periodic coefficients

Abstract

The paper deals with homogenization of self-adjoint operators in $L_2({\mathbb{R}}^d)$ of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} μ(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+α}}\,d\y, $$ where $0<\alpha <2$, and $\eps>0$ is a small parameter. It is assumed that the function $μ(\x,\y)$ is ${\mathbb{Z}}^d$-periodic in each variable, $μ(\x,\y)=μ(\y,\x)$ for all $\x$ and $\y$, and $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. Under these assumptions we show that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2({\mathbb{R}}^d)$ to the resolvent $({\mathbb{A}}^0+I{)}^{-1}$ of the limit operator ${\mathbb{A}}^0$ given by $$ ({\mathbb A}^0 u) (\x) = \int_{\R^d} μ^0 \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+α}}\,d\y, $$ where ${\mu }^0$ is the mean value of $μ(\x,\y)$. We also show that the operator norm of the discrepancy $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1}\|_{L_2(\mathbb R^d)\to L_2(\mathbb R^d)}$ can be estimated by $O(\eps^α)$, if $0<\alpha <1$, by $O(\eps (1 + | \operatorname{ln} \eps|)^2)$, if $\alpha =1$, and by $O(\eps^{2- α})$, if $1<\alpha <2$.