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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.06832 |
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| _version_ | 1866911729085054976 |
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| author | Piatnitski, Andrey Sloushch, Vladimir Suslina, Tatiana Zhizhina, Elena |
| author_facet | Piatnitski, Andrey Sloushch, Vladimir Suslina, Tatiana Zhizhina, Elena |
| contents | 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 $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-α)} \mathbb{K}_m \bigr\| = O(\eps). $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Homogenization of Lévy-type operators: operator estimates with correctors Piatnitski, Andrey Sloushch, Vladimir Suslina, Tatiana Zhizhina, Elena Analysis of PDEs Functional Analysis 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 $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-α)} \mathbb{K}_m \bigr\| = O(\eps). $$ |
| title | Homogenization of Lévy-type operators: operator estimates with correctors |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2601.06832 |