Homogenization of L\'evy-type operators: operator estimates with correctors

Andrey Piatnitski; Elena Zhizhina; Tatiana Suslina; Vladimir Sloushch

REVIEW 3 major objections 4 minor 1 cited by

Homogenization of L\'evy-type operators: operator estimates with correctors

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Adding corrector terms makes resolvents of periodic Lévy operators converge to the effective resolvent with operator-norm error of order ε for every α in (1,2).

desk verdict Solid corrector expansion for Lévy-type homogenization; main theorem is right, but fix the α-range typos and don't claim sharpness without proving K1≠0. read the letter →

arxiv 2601.06832 v3 pith:FGJOETBX submitted 2026-01-11 math.AP math.FA

Andrey Piatnitski , Vladimir Sloushch , Tatiana Suslina , Elena Zhizhina This is my paper · ORCID

classification math.APmath.FA MSC 35B2745K0547A5560J75

keywords Lévy-typeoperatorsperiodichomogenizationoperator-normresolventestimatescorrectorsspectralmethodFloquet–BlochdecompositionfractionalLaplacianstable-likejumpprocesses

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a sharper homogenization theorem for a family of nonlocal Lévy-type operators with rapidly oscillating periodic coefficients. Previous work showed that the resolvent of the oscillating operator converges to the resolvent of a constant-coefficient effective operator with error O(ε^(2−α)), a rate that deteriorates as the jump index α approaches 2. Here the authors add explicit corrector terms—differential operators built from a matrix that solves auxiliary cell problems—and show that after subtracting the first N correctors, the operator-norm error becomes O(ε), for every α in (1,2). The significance is that even though the underlying jump processes have infinite second moments and the operators are nonlocal, quantitative homogenization works at the same linear rate familiar from elliptic equations. A byproduct is that the earlier O(ε^(2−α)) bound is order-sharp.

What carries the argument

The proof uses the operator-theoretic spectral method: after a scaling transformation and a Gelfand transform, the operator A is decomposed into a direct integral of fiber operators A(ξ) acting on periodic functions, with discrete spectra. Homogenization is a threshold effect at the bottom of the spectrum: for small quasi-momentum ξ, the spectral projector F(ξ) and the operator A(ξ)F(ξ) are approximated by (μ_0 c_0 |ξ|^α + ⟨g^0 ξ,ξ⟩) P plus a remainder of order |ξ|^{1+α}. This threshold approximation, derived via contour integration of the resolvent and a resolvent identity for quadratic forms with a common domain, yields the corrector expansion at the operator level.

What would settle it

Take d = 1, α = 3/2, and a periodic coefficient with nontrivial oscillation, for instance μ(x,y) = 2 + sin(2πx) + sin(2πy). For ε small, compute the operator norm of (A_ε + I)^{-1} − (A^0 + I)^{-1} − ε^{1/2} K_1. The theorem predicts this norm decays as O(ε). If the measured decay is only O(ε^{1/2}) (i.e., subtracting the first corrector produces no improvement), the leading corrector term is wrong. A direct spectral computation on the one-dimensional cell could compare the actual constant with the explicit bound from the proof.

Watch

Extended reading notes

Core claim

The central result (Theorem 5.2) is that for any integer N with 2 − 1/N < α < 2, the resolvent of the ε-periodic Lévy operator A_ε satisfies ‖(A_ε + I)^{-1} − (A^0 + I)^{-1} − Σ_{m=1}^{N} ε^{m(2−α)} K_m‖ ≤ C ε when α ≤ 2 − 1/(N+1), and ≤ C ε^{(N+1)(2−α)} otherwise. Here A^0 is the effective operator, a multiple of the fractional Laplacian (−Δ)^{α/2}, and the correctors are K_m = (div g^0 ∇)^m (A^0 + I)^{-m−1}, with g^0 an explicit (not necessarily sign-definite) symmetric matrix defined through periodic cell problems. Thus, by choosing N so that 2 − 1/N < α ≤ 2 − 1/(N+1), one obtains an operator-norm approximation of order ε for every α in (1,2).

Load-bearing premise

The coefficient μ must stay bounded below by a positive constant (uniform ellipticity); if that lower bound were zero, the spectral gap d_0 would close, and the contour Γ, the spectral projection F(ξ), and the whole corrector expansion would collapse.

Editorial extensions

If this is right

For any fixed α in (1,2), choosing N with 2 − 1/N < α ≤ 2 − 1/(N+1) gives an operator-norm approximation of the resolvent with error of order ε, independent of how close α is to 2.
Taking N larger than needed gives remainders of order ε^{(N+1)(2−α)}, so for each fixed α the accuracy can be made arbitrarily high by including more correctors.
The earlier O(ε^{2−α}) bound from the authors' previous work is order-sharp: no better rate is possible without correctors.
The correctors are explicit and computed from the effective operator and the matrix g^0, which is defined by solving periodic cell problems; the approximation is therefore implementable in practice.
The main estimate remains valid on any periodic lattice, with constants depending on the lattice parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same threshold-expansion strategy may extend to nonsymmetric Lévy kernels or stable-like processes with variable characteristics, where only strong-convergence rates are currently known; the spectral machinery here suggests operator-norm rates are plausible but would require a new treatment of non-self-adjoint fibers.
Because g^0 need not be sign-definite (Remark 3.12), the corrector terms are not positivity-preserving, so a probabilistic interpretation of the correction as an additional drift or diffusion is not straightforward.
The constants in the estimates diverge as α → 1 and α → 2, so the O(ε) accuracy is genuine for each fixed α but not uniform in α; tracking this divergence might yield a two-parameter asymptotic description near the endpoints.
The structure of the expansion—powers ε^{m(2−α)} with coefficients built from powers of div(g^0∇) applied to resolvents—suggests a full asymptotic series in ε^{2−α} may exist, with each coefficient determined by g^0 and the cell functions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real result. It upgrades the resolvent approximation for periodic Lévy-type operators from O(ε^{2−α}) to O(ε) by including explicit correctors, and the core proof holds up under scrutiny. The construction of g0 and the correctors K_m is a genuine extension of the authors' earlier leading-order work [25], and the distinction from convolution-type kernels with integrable moments is legitimate: the singular stable-like kernel requires the contour/spectral-projector machinery, not just moment expansions.

What's new: the N-term expansion (5.4) with operator-norm remainder O(ε) for α∈(1,2) by choosing N with 2−1/N<α≤2−1/(N+1). The correctors are explicit, the error terms are tracked, and Proposition 3.11 plus Section 4 gives a coherent derivation. I checked the threshold condition in the proof; the right threshold is 2−1/N, and that is what the abstract and Theorem 5.2 use.

Soft spots: 1. Theorems 4.4, 4.6, and 4.8 state the hypothesis as 1−1/N<α<2 instead of 2−1/N<α<2. This is a typo—the constants and later estimates clearly require the stronger bound—but it appears in three theorem statements and should be fixed. 2. Concluding remark 1 claims the O(ε^{2−α}) estimate (5.3) is order-sharp. That does not follow from Theorem 5.2 unless the first corrector K1 is nonzero. No nondegeneracy proof is given. If K1=0 or if g0=0, the approximation could be better. The claim needs a proof or should be weakened. 3. The paper imports a substantial amount from [25], an unpublished preprint: several lemmas and Theorem 4.5 are taken from it. I did not find a flaw there, but the dependence is real and should be highlighted so referees know what is being assumed. 4. Uniform ellipticity μ_->0 is genuinely load-bearing: it creates the spectral gap d0>0 that underpins the contour and spectral projector. That is not a flaw—it is the stated operator class—but the result does not touch degenerate coefficients.

This paper is for people working on quantitative homogenization, nonlocal operators, or spectral methods for periodic operators. The main theorem is likely correct, the typos are cosmetic, and the sharpness claim is the only substantive gap. I would send it to peer review and ask for the sharpness remark to be proved or removed.

Referee Report

3 major / 4 minor

Summary. The paper studies homogenization in L2(R^d) of a symmetric purely nonlocal Lévy-type operator A_ε with periodic coefficient μ(x/ε,y/ε) and kernel |x−y|^{-d−α}, 1<α<2. Following the authors' earlier work, the resolvent (A_ε+I)^{-1} converges in operator norm to (A0+I)^{-1} at rate ε^{2−α}. The present paper constructs a higher-order corrector expansion: for N∈N with 2−1/N<α<2, it claims the estimate (5.4), namely that (A_ε+I)^{-1}−(A0+I)^{-1}−Σ_{m=1}^N ε^{m(2−α)}K_m has operator norm O(ε) in the range 2−1/N<α≤2−1/(N+1), and O(ε^{(N+1)(2−α)}) in the complementary subrange. The correctors are explicit: K_m=(div g0∇)^m(A0+I)^{-m−1}, with g0 defined through auxiliary periodic problems. The proof combines the scaling relation (5.1)–(5.2), a Gelfand-transform decomposition, a threshold analysis of the fiber operators A(ξ), and a spectral resolvent expansion with explicit remainders.

Significance. If the main theorem is correct, this is a substantial contribution: it gives, for every α∈(1,2), an operator-norm approximation of the resolvent of a nonlocal periodic operator with error of order ε, by a finite sum of explicit correctors. This mirrors the Birman–Suslina corrector theory for elliptic operators and extends it to fractional-order jump processes. The proof is long but structured, with explicit constants and parameter-free corrector formulas; there are no fitted coefficients. The main residual risk is that several load-bearing estimates (notably Theorem 4.5, the base ε^{2−2α} fiber estimate) are imported from the unpublished preprint [25], so the current paper is not fully self-contained. I see no evidence of circularity: the corrector expansion is derived from a fixed spectral asymptotic expansion, not fitted to the final estimate.

major comments (3)

[§4, Theorems 4.4, 4.6, 4.8] The hypotheses of Theorems 4.4, 4.6, and 4.8 are stated as '1−1/N < α < 2', but all subsequent case distinctions and the proof rely on the condition '2−1/N < α < 2'. For example, equation (4.12) distinguishes cases according to 2−1/N and 2−1/(N+1), and the derivation of the remainder estimate (4.10)–(4.11) requires α>2−1/N. The stated weaker condition is inconsistent with the displayed estimate. This appears to be a systematic typo, but since it occurs in three central theorem statements, it must be corrected before publication.
[§5.2, Concluding Remark 1] The remark asserts that 'It follows from Theorem 5.2 that the precision O(ε^{2−α}) in estimate (5.3) is order-sharp.' This does not follow from the upper bound in Theorem 5.2. Sharpness would require a matching lower bound or, at minimum, a proof that the first corrector K_1 is nonzero. The manuscript provides no such argument. Either the remark should be removed or substantiated with a concrete example or a nonvanishing condition for K_1.
[§3–§4, dependence on [25]] The argument is heavily reliant on the unpublished preprint [25]. In particular, Theorem 4.1 (the base ε^{2−2α} fiber approximation), Theorem 4.5 (the base resolvent estimate), Theorem 4.7, and several lemmas in Sections 1–3 are quoted from [25] without proofs. Since Theorem 4.5 is a load-bearing input for the new corrector expansion, the paper would be easier to evaluate if the authors included a proof of this theorem in an appendix or cited a published version of [25]. This is a transparency concern rather than an identified error.

minor comments (4)

[Throughout] The constants C1, C2, and C3 are reused with different meanings in different sections (e.g., Lemma 3.2, Lemma 3.3, Theorems 4.5–4.8). This makes cross-referencing unnecessarily difficult; a consistent numbering system for constants would help.
[§1.1] In (1.1), the phrase 'bounded positive definite function' is imprecise; the condition given is boundedness and a two-sided positivity bound. The wording could be aligned with the displayed inequalities.
[§4.2, Theorem 4.8] The constant in the statement is denoted C2(α, μ), but the proof and Theorem 5.2 use a constant depending also on N. This is a minor notational inconsistency.
[§2.3, Lemma 2.5] In the proof of estimate (2.28), the parameter p is eventually set to 2, but the text says 'Letting p=2 in (2.30)'. This is fine, but the role of p in the auxiliary functions could be clarified, especially because the growth constants c(p,α) blow up as p→1.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: corrector expansion is derived, not fitted; heavy reliance on authors' prior preprint is dependency, not circularity.

full rationale

Derivation chain. The new content (Prop. 3.6, 3.8, 3.11; Thms 4.3, 4.4, 4.6, 4.8, 5.2) computes the threshold expansion of A(ξ)F(ξ) by contour integrals over Γ and the resolvent identity, defines g0 via the cell problem (3.48), then expands (μ0c0|ξ|^α+<g0ξ,ξ>+ε^α)^{-1} by a geometric series. The correctors K_m are the resulting terms, so they are produced by the expansion rather than fitted to the final estimate. No parameter is fitted to data, and no quantity called a prediction is defined in terms of the object it predicts. The main circularity risk is the repeated citation of [25] for Lemmas 1.2, 1.3, 2.1, 3.2, 3.9, Propositions 1.5, 3.5, and Theorems 4.1, 4.5, 4.7, 5.1. These are load-bearing in the sense that the proof imports them; however they are parameter-free estimates under the same assumptions, they do not contain the target corrector claim, and the proof of Thm 5.2 does not invoke Thm 5.1/4.5. Thus the dependence is prior-work reliance, not reduction to inputs. Non-circular presentation issues: Thms 4.4, 4.6, 4.8 state 1−1/N<α<2 where the proof uses 2−1/N<α<2; concluding remark 5.2.1 claims sharpness without a lower bound on K_1. These are correctness/rigor concerns, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No numerical fitting and no invented physical entities. The matrix g0 and the correctors K_m are derived from μ and the effective operator A0. The main unseen inputs are the cited lemmas from [25] and the coefficient assumptions above.

free parameters (1)

N (number of correctors)
Chosen by hand from α via 2−1/N < α ≤ 2−1/(N+1); not fitted, but the final rate and theorem constants depend on it.

assumptions (4)

domain assumption The coefficient μ is real-valued, symmetric, periodic, and satisfies 0<μ−≤μ≤μ+<∞ (conditions (1.1)–(1.2)).
This is the standing hypothesis of the paper; it provides self-adjointness, form bounds, and the spectral gap used in Proposition 3.1.
domain assumption The fractional exponent satisfies 1<α<2.
The whole analysis is restricted to this range; the estimates use α−1, 2−α, and 1+α in denominators that degenerate at α=1 or α=2.
domain assumption The lemmas and theorems imported from the authors' prior preprint [25] are correct (Lemmas 1.2, 1.3, 1.4, Proposition 1.5, Lemma 2.1, Lemma 3.2, Proposition 3.5, Lemma 3.9, Theorems 4.1, 4.5, 4.7, 5.1).
The paper does not reprove these results; the final estimate inherits their validity.
standard math The cell problem (3.48) has a unique mean-zero solution v_k∈eH^γ(Ω) for each k.
Existence follows from Lax–Milgram via the coercive form a(0) on the mean-zero subspace; Proposition 3.8 uses this to define g0.

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Pith. "Pith review of Homogenization of L\'evy-type operators: operator estimates with correctors." pith.science (2026). https://pith.science/paper/FGJOETBX

@misc{pith2026260106832,
  author       = {Pith},
  title        = {Pith review of: Homogenization of L\'evy-type operators: operator estimates with correctors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGJOETBX}},
  note         = {Machine review of arXiv:2601.06832}
}

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 $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} \mu(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+\alpha}}\,d\y $$ with $1< \alpha < 2$; here the function $\mu(\x,\y)$ is $\Z^d$-periodic in the both variables, satisfies the symmetry relation $\mu(\x,\y) = \mu(\y,\x)$ and the estimates $0< \mu_- \leqslant \mu(\x,\y) \leqslant \mu_+< \infty$. 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 $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} \| = O(\eps^{2-\alpha})$ 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 $2-1/N < \alpha \le 2-1/(N+1)$, we obtain $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-\alpha)} \mathbb{K}_m \bigr\| = O(\eps). $$

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

High-order convergence rates of periodic homogenization for symmetric L\'evy type operators
math.PR 2026-06 unverdicted novelty 7.0 of 10

Higher-order convergence rates established for periodic homogenization of symmetric Lévy-type operators via scale decomposition of the jumping kernel in multiple regimes.

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Works this paper leans on

39 extracted references · 2 canonical work pages · cited by 1 Pith paper

[25]

Piatnitski A., Sloushch V., Suslina T., Zhizhina E.,On operator estimates in homogenization of nonlocal operators of convolution type, J. Diff. Equ.352(2023), 153–188

2023
[1]

Allaire G., Conca C.,Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures Appl. 77(1998), 153–208

1998
[2]

Then, taking into account firstNcorrectors, we obtain an approximation of the resolvent (Aε +I) −1 of orderε

For anyα∈(1,2)one can chooseN∈Nin such a way that2− 1 N < α⩽2− 1 N+1. Then, taking into account firstNcorrectors, we obtain an approximation of the resolvent (Aε +I) −1 of orderε
[3]

Moreover, both constants tend to infinity, asα→1orα→2

From the explicit formulae for the constantsC 1(α, µ)andC 2(N, α, µ)it is easy to deduce thatC 1(α, µ)depends only ond,α,µ − andµ +, whileC 2(N, α, µ)depends also on N. Moreover, both constants tend to infinity, asα→1orα→2
[4]

In this case the constants in estimates (5.3) and (5.4) will also depend on the parameters of the lattice

The statements of Theorems 5.1 and 5.2 remain valid if the periodicity latticeZd is replaced with an arbitrary periodic lattice inRd. In this case the constants in estimates (5.3) and (5.4) will also depend on the parameters of the lattice. Список литературы
[5]

Allaire G., Conca C., Vanninathan M.,The Bloch transform and applications, ESAIM: Proceedings3 (1998), 65–84

1998
[6]

Barlow M., Bass R., Chen Z.-Q., Kassmann M.,Nonlocal Dirichlet forms and symmetric jump processes, Transactions of the AMS361(4) (2009), 1963–1999

2009
[7]

Kluwer Academic Publishers, Dordrecht, 1989

Bakhvalov N., Panasenko G.,Homogenisation:averaging processes in periodic media, Mathematical Problems in the Mechanics of Composite Materials. Kluwer Academic Publishers, Dordrecht, 1989

1989

Show all 39 references

[8]

Bensoussan A., Lions J.-L., Papanicolaou G.,Asymptotic analysis for periodic structures, Stud. Math. Appl., vol. 5, North-Holland Publishing Co., Amsterdam–New York, 1978

1978
[9]

Sh., Suslina T

Birman M. Sh., Suslina T. A.,Second order periodic differential operators. Threshold properties and homogenization, St. Petersburg Math. J.15(5) (2004), 639–714

2004
[10]

Sh., Suslina T

Birman M. Sh., Suslina T. A.,Homogenization with corrector term for periodic elliptic differential operators, St. Petersburg Math. J.17(6) (2006), 897–973

2006
[11]

Sh., Suslina T

Birman M. Sh., Suslina T. A.,Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev classH1(Rd), St. Petersburg Math. J.18(6) (2007), 857–955

2007
[12]

Braides A., Piatnitski A.,Homogenization of quadratic convolution energies in periodically perforated domains, Adv. Calc. Var.15(2022), 351–368. https://doi.org/10.1515/acv-2019-0083

2022 doi
[13]

Chen, X.; Chen, Z.-Q.; Kumagai, T.; Wang, J.,Quantitative periodic homogenization for symmetric non-local stable-like operators, arXiv: 2409.08120v1 (12 Sep 2024)

2024 arXiv
[14]

Chen, X., Chen, Z.-Q., Kumagai, T., Wang, J.: Periodic homogenization of nonsymmetric L´ evy-type processes. Ann. Probab. 49(6), 2874–2921 (2021)

2021
[15]

Conca C., Orive R., Vanninathan M.,Bloch approximation in homogenization and applications, SIAM J. Math. Anal.33(5) (2002), 1166–1198. 42

2002
[16]

Cont R., Tankov P.,Financial modelling with jump processes, Chapman and Hall/CRC Financial Mathematics Series, FL, 2004

2004
[17]

M., Phillips R

Edwards A. M., Phillips R. A., Watkins N. W., Freeman M. P., Murphy E. J., Afanasyev V., Buldyrev S. V., Da Luz M. G. E., Raposo E. P., Stanley H. E., Viswanathan G. M.,Revisiting L´ evy flight search patterns of wandering albatrosses, bumblebees and deer, Nature449(2007), 1044–1048

2007
[18]

Grafakos L.,Modern Fourier analysis, Springer, 2014

2014
[19]

Humphries N., Sims D.,Optimal foraging strategies:L´ evy walks balance searching and patch exploitation under a very broad range of conditions, J. Theoret. Biol.358(2014), 179–193

2014
[20]

Kassmann M., Piatnitski A., Zhizhina E.,Homogenization of L´ evy-type operators with oscillating coefficients, SIAM J. Math. Anal.51(5) (2019), 3641–3665

2019
[21]

Anal.96(8) (2017), 1390–1400

Kondratiev Yu., Molchanov S., Pirogov S., Zhizhina E.,On ground state of some non local Schr¨ odinger operators, Appl. Anal.96(8) (2017), 1390–1400

2017
[22]

Nualart D., Schoutens W.,Backward stochastic differential equations and Feynman-Kac formula for L´ evy processes, with applications in finance, Bernoulli7(2001), 761–776

2001
[23]

Piatnitski A., Zhizhina E.,Periodic homogenization of nonlocal operators with a convolution-type kernel, SIAM J. Math. Anal.49(1) (2017), 64–81

2017
[24]

Piatnitski A., Zhizhina E.,Homogenization of biased convolution type operators, Asymptot. Anal. 115(3-4) (2019), 241–262

2019
[26]

The Nina N

Piatnitski A., Sloushch V., Suslina T., Zhizhina E.,Homogenization of nonlocal convolution type operators:Approximation for the resolvent with corrector, in: Friends in Partial Differential Equations. The Nina N. Uraltseva Anniversary Volume, EMS Press, 2025, p. 291–336. DOI: ...

2025 doi
[27]

L., Sloushch V

Piatnitski A. L., Sloushch V. A., Suslina T. A., Zhizhina E. A.,On the homogenization of nonlocal convolution type operators, Russian J. Math. Phys.31(1) (2024), 137–145

2024
[28]

L., Sloushch V

Piatnitski A. L., Sloushch V. A., Suslina T. A., Zhizhina E. A.,Operator estimates in homogenization of L´ evy-type operators with periodic coefficients, Preprint (2024), arXiv:2412.20408

2024 arXiv
[29]

I.,L´ evy processes and infinitely divisible distributions, Cambridge Stud

Sato K. I.,L´ evy processes and infinitely divisible distributions, Cambridge Stud. Adv. Math. 68, Cambridge University Press, Cambridge, 2013

2013
[30]

V.,An asymptotic expansion of the solution of a second order elliptic equation with periodic rapidly oscillating coefficients, Sb

Sevostyanova E. V.,An asymptotic expansion of the solution of a second order elliptic equation with periodic rapidly oscillating coefficients, Sb. Math.,43(1982), 181–198

1982
[31]

M.,Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proceedings of the Steklov Institute of Mathematics 171, AMS, Providence (1987)

Skriganov M. M.,Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proceedings of the Steklov Institute of Mathematics 171, AMS, Providence (1987)

1987
[32]

Surveys78(6) (2023), 1023–1154

Suslina T.,Operator-theoretic approach to the homogenization of Schr¨ odinger-type equations with periodic coefficients, Russian Math. Surveys78(6) (2023), 1023–1154

2023
[33]

V., Zolotarev V

Uchaikin V. V., Zolotarev V. M.,Chance and stability. Stable distributions and their applications, in Modern Probability and Statistics, VSP, Utrecht, 1999

1999
[34]

A.,L´ evy processes in the physical sciences, in: L´evy Processes, Birkh¨ auser, Boston, MA, 2001, pp

Woyczynski W. A.,L´ evy processes in the physical sciences, in: L´evy Processes, Birkh¨ auser, Boston, MA, 2001, pp. 241-266

2001
[35]

Equ.25(1) (1989), 33–39

Zhikov V.,Spectral approach to asymptotic diffusion problems, Differ. Equ.25(1) (1989), 33–39

1989
[36]

V.,On operator estimates in homogenization theory, Dokl

Zhikov V. V.,On operator estimates in homogenization theory, Dokl. Math.72(1) (2005), 534–538

2005
[37]

V., Kozlov S

Zhikov V. V., Kozlov S. M., Oleinik O. A.,Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994

1994
[38]

V., Pastukhova S

Zhikov V. V., Pastukhova S. E.,On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys.12(4) (2005), 515–524

2005
[39]

V., Pastukhova S

Zhikov V. V., Pastukhova S. E.,Operator estimates in homogenization theory, Russian Math. Surveys 71(3) (2016), 417–511

2016

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [25]

Piatnitski A., Sloushch V., Suslina T., Zhizhina E.,On operator estimates in homogenization of nonlocal operators of convolution type, J. Diff. Equ.352(2023), 153–188

2023

[2] [1]

Allaire G., Conca C.,Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures Appl. 77(1998), 153–208

1998

[3] [2]

Then, taking into account firstNcorrectors, we obtain an approximation of the resolvent (Aε +I) −1 of orderε

For anyα∈(1,2)one can chooseN∈Nin such a way that2− 1 N < α⩽2− 1 N+1. Then, taking into account firstNcorrectors, we obtain an approximation of the resolvent (Aε +I) −1 of orderε

[4] [3]

Moreover, both constants tend to infinity, asα→1orα→2

From the explicit formulae for the constantsC 1(α, µ)andC 2(N, α, µ)it is easy to deduce thatC 1(α, µ)depends only ond,α,µ − andµ +, whileC 2(N, α, µ)depends also on N. Moreover, both constants tend to infinity, asα→1orα→2

[5] [4]

In this case the constants in estimates (5.3) and (5.4) will also depend on the parameters of the lattice

The statements of Theorems 5.1 and 5.2 remain valid if the periodicity latticeZd is replaced with an arbitrary periodic lattice inRd. In this case the constants in estimates (5.3) and (5.4) will also depend on the parameters of the lattice. Список литературы

[6] [5]

Allaire G., Conca C., Vanninathan M.,The Bloch transform and applications, ESAIM: Proceedings3 (1998), 65–84

1998

[7] [6]

Barlow M., Bass R., Chen Z.-Q., Kassmann M.,Nonlocal Dirichlet forms and symmetric jump processes, Transactions of the AMS361(4) (2009), 1963–1999

2009

[8] [7]

Kluwer Academic Publishers, Dordrecht, 1989

Bakhvalov N., Panasenko G.,Homogenisation:averaging processes in periodic media, Mathematical Problems in the Mechanics of Composite Materials. Kluwer Academic Publishers, Dordrecht, 1989

1989