Homogenization of Lévy-type operators: operator estimates with correctors
Andrey Piatnitski, Vladimir Sloushch, Tatiana Suslina, Elena Zhizhina
TL;DR
The paper develops a quantitative homogenization theory for Lévy-type operators with periodic coefficients, focusing on the operator ${\mathbb A}_\varepsilon$ defined by a nonlocal kernel $\mu(\mathbf x/\varepsilon, \mathbf y/\varepsilon)$. Using a spectral (Floquet-Bloch) framework and a careful threshold analysis near the spectral edge, the authors construct correctors ${\mathbb K}_m$ and show that the resolvent $({\mathbb A}_\varepsilon+I)^{-1}$ can be approximated by the homogenized resolvent $({\mathbb A}^0+I)^{-1}$ plus a finite sum of higher-order correctors with ${\mathcal O}(\varepsilon)$ precision when $2-1/N<\alpha\le 2-1/(N+1)$. The results rely on decomposing the operator into a direct integral over fiber operators ${\mathbb A}(\boldsymbol \xi;\alpha,\mu)$, obtaining explicit representations for quadratic forms, and deriving sharp operator-norm estimates for the resolvent, both at the fiber level and globally via the Gelfand transform. The main findings advance nonlocal periodic homogenization by delivering order-sharp resolvent estimates and explicit correctors, with implications for a broad class of Lévy-type processes and related applications. The framework accommodates general periodic lattices beyond $\mathbb Z^d$, extending the reach of the operator-theoretic homogenization approach to nonlocal operators.
Abstract
The goal of the paper is to study in $L_2(\R^d)$ a self-adjoint operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} μ(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+α}}\,d\y $$ with $1< α< 2$; here the function $μ(\x,\y)$ is $\Z^d$-periodic in the both variables, satisfies the symmetry relation $μ(\x,\y) = μ(\y,\x)$ and the estimates $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. The rigorous definition of the operator ${\mathbb A}_\eps$ is given in terms of the corresponding quadratic form. In the previous work of the authors it was shown that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\mathbb R^d)$ to the resolvent of the effective operator $A^0$, and the estimate $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} \| = O(\eps^{2-α})$ holds. In the present work we achieve a more accurate approximation of the resolvent of ${\mathbb A}_\eps$ which takes into account the correctors. Namely, for $N\in\mathbb N$ such that $2-1/N < α\le 2-1/(N+1)$, we obtain $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-α)} \mathbb{K}_m \bigr\| = O(\eps). $$