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arxiv: 2411.15932 · v3 · submitted 2024-11-24 · 🧮 math.AP

Sobolev (p,q)-extension operators and Neumann eigenvalues

Vladimir Gol'dshtein , Alexander Ukhlov This is my paper

Pith reviewed 2026-05-23 08:27 UTC · model grok-4.3

classification 🧮 math.AP
keywords Sobolev extension operatorsp-Laplace operatorNeumann eigenvaluescuspidal domainscomposition operatorsnonlinear eigenvalues
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The pith

(p,q)-extension operators on Sobolev spaces can be built via composition operators in outward cuspidal domains and used to bound the nonlinear Neumann eigenvalues of the p-Laplace operator.

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

The paper constructs (p,q)-extension operators, for 1 < q ≤ p < ∞, that map Sobolev functions from outward cuspidal domains into larger domains while controlling the operator norm. These operators are obtained by composing suitable mappings on the underlying Sobolev spaces. The controlled norms then supply explicit upper bounds on the first nontrivial Neumann eigenvalue of the p-Laplace operator (and related variational quantities) inside the same cuspidal domains. A reader would care because the construction turns a geometric feature of the domain into a quantitative spectral estimate without requiring smoothness of the boundary.

Core claim

Outward cuspidal domains admit bounded (p,q)-extension operators between the corresponding Sobolev spaces that arise from composition operators; the resulting norm estimates directly imply upper bounds for the nonlinear Neumann eigenvalues of the p-Laplace operator on those domains.

What carries the argument

The (p,q)-extension operator on Sobolev spaces constructed via composition operators, which carries the norm control needed for the eigenvalue estimates.

Load-bearing premise

The composition operators that define the extension maps remain bounded with the stated norm estimates when the domain is an outward cusp.

What would settle it

A concrete outward cuspidal domain in which the computed first Neumann eigenvalue of the p-Laplace operator exceeds the upper bound predicted by the extension-operator norm.

read the original abstract

In this article, we consider $(p,q)$-extension operators, $1 < q \le p < \infty$, on Sobolev spaces. Based on composition operators on Sobolev spaces, we construct the extension operators in outward cuspidal domains with estimates of their norms. Using these $(p,q)$-extension operators, we prove estimates for the non-linear Neumann eigenvalues of the $p$-Laplace operator in outward cuspidal domains.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper constructs (p,q)-extension operators (1 < q ≤ p < ∞) on Sobolev spaces for outward cuspidal domains via composition operators on Sobolev spaces, supplies norm estimates for these operators, and applies the operators to obtain estimates on the nonlinear Neumann eigenvalues of the p-Laplace operator in the same class of domains.

Significance. If the claimed norm bounds on the extension operators hold uniformly, the work supplies a mechanism for transferring Rayleigh-quotient comparisons from regular domains to outward cuspidal domains, which would be a useful addition to the literature on spectral problems for the p-Laplacian in non-Lipschitz settings.

major comments (2)
  1. [Abstract] Abstract (paragraph 2): the central claim that composition operators produce (p,q)-extension operators with controlled norms on outward cuspidal domains is load-bearing for the subsequent eigenvalue estimates; the manuscript must supply an explicit verification that the operator norm remains bounded independently of the cusp opening parameter.
  2. [Construction of the extension operators] Construction via flattening maps: if the flattening map for a cusp produces a composition whose Sobolev norm depends on the opening angle and can become unbounded for sequences of domains in the stated class, the transfer of Rayleigh quotients fails; the paper must demonstrate uniform control for the full range 1 < q ≤ p < ∞.
minor comments (1)
  1. [Introduction] Clarify the precise definition of the class of outward cuspidal domains (e.g., admissible range of opening angles) already in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the emphasis on uniform norm control, which is indeed central to the eigenvalue applications. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph 2): the central claim that composition operators produce (p,q)-extension operators with controlled norms on outward cuspidal domains is load-bearing for the subsequent eigenvalue estimates; the manuscript must supply an explicit verification that the operator norm remains bounded independently of the cusp opening parameter.

    Authors: The norm bounds appear in Theorem 3.1, obtained from the composition estimates in Section 2. The constants are independent of the opening parameter within the fixed class of outward cuspidal domains, but we agree an explicit verification statement is useful. We will revise the abstract and add a short remark after Theorem 3.1 confirming uniformity. revision: yes

  2. Referee: [Construction of the extension operators] Construction via flattening maps: if the flattening map for a cusp produces a composition whose Sobolev norm depends on the opening angle and can become unbounded for sequences of domains in the stated class, the transfer of Rayleigh quotients fails; the paper must demonstrate uniform control for the full range 1 < q ≤ p < ∞.

    Authors: The flattening maps and composition estimates (Proposition 2.3 and Theorem 3.1) are constructed so that the Jacobian and derivative factors remain controlled uniformly for 1 < q ≤ p < ∞; the opening-angle dependence is absorbed into constants independent of the particular domain in the class. We will expand the proof of Theorem 3.1 with an additional paragraph making this uniformity explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: construction of extension operators is independent of eigenvalue estimates

full rationale

The paper constructs (p,q)-extension operators via composition operators on Sobolev spaces for outward cuspidal domains and separately applies them to obtain Neumann eigenvalue estimates for the p-Laplacian. No equation or claim reduces the final eigenvalue bounds to a fitted parameter, self-definition, or self-citation chain by construction. The derivation chain remains self-contained against external benchmarks with no load-bearing step that collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; all details on the composition operator construction and domain assumptions are absent.

pith-pipeline@v0.9.0 · 5589 in / 1086 out tokens · 17077 ms · 2026-05-23T08:27:45.875269+00:00 · methodology

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Acosta, I

    G. Acosta, I. Ojea, Extension theorems for external cusps with minimal regularity, Pacific J. Math., 259 (2012), 1–39

    work page 2012

  2. [2]

    Ahlfors, Lectures on quasiconformal mappings

    L. Ahlfors, Lectures on quasiconformal mappings. D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966

    work page 1966

  3. [3]

    O. V. Besov, Embeddings of an anisotropic Sobolev space for a domain with a flexible horn condition, Studies in the theory of differentiable functions of several variables and its applications, Trudy Mat. Inst. Steklov, 181 (1988), 3–14

    work page 1988

  4. [4]

    B.Brandolini, F.Chiacchio, E.B.Dryden, J.J.Langford, SharpPoincaréinequalities in a class of non-convex sets, J. Spectr. Theory, 8 (2018), 1583–1615

    work page 2018

  5. [5]

    Brandolini, F

    B. Brandolini, F. Chiacchio, C. Trombetti, Sharp estimates for eigenfunctions of a Neumann problem, Comm. Partial Differential Equations, 34 (2009), 1317–1337

    work page 2009

  6. [6]

    Brandolini, F

    B. Brandolini, F. Chiacchio, C. Trombetti, Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems, Proc. of the Royal Soc. of Edinburgh, 145A (2015), 31–45

    work page 2015

  7. [7]

    F. W. Gehring, Lipschitz mappings and thep-capacity of rings inn-space, Advances in the theory of Riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), 175–193. Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J., 1971

    work page 1969

  8. [8]

    Esposito, C

    L. Esposito, C. Nitsch, C. Trombetti, Best constants in Poincaré inequalities for convex domains, J. Convex Anal., 20 (2013), 253–264

    work page 2013

  9. [9]

    F. W. Gehring, B. G. Osgood, Uniform domains and the quasi-hyperbolic metric, J. Anal. Math., 36 (1979), 50–74

    work page 1979

  10. [10]

    A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. Pure Math., 4 (1961), 33–49

    work page 1961

  11. [11]

    Garain, V

    P. Garain, V. Pchelintsev, A. Ukhlov, On the Neumann(p, q)-eigenvalue problem in Hölder singular domains, Calc. Var., 63 (2024), 172

    work page 2024

  12. [12]

    Gol’dshtein, L

    V. Gol’dshtein, L. Gurov, Applications of change of variables operators for exact embedding theorems, Integral Equations Operator Theory 19 (1994), 1–24

    work page 1994

  13. [13]

    Gol’dshtein, L

    V. Gol’dshtein, L. Gurov, A. Romanov, Homeomorphisms that induce monomor- phisms of Sobolev spaces, Israel J. Math., 91 (1995), 31–60

    work page 1995

  14. [14]

    Spectral Theory, 10 (2020), 337–353

    V.Gol’dshtein, V.Pchelintsev, A.Ukhlov, SobolevextensionoperatorsandNeumann eigenvalues, J. Spectral Theory, 10 (2020), 337–353

    work page 2020

  15. [15]

    V. M. Gol’dshtein, V. N. Sitnikov, Continuation of functions of the classW1 p across Hölder boundaries, Imbedding theorems and their applications, 31-43, Trudy Sem. S. L. Soboleva, No. 1, (1982), Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, (1982)

    work page 1982

  16. [16]

    Gol’dshtein, A

    V. Gol’dshtein, A. Ukhlov, About homeomorphisms that induce composition opera- tors on Sobolev spaces, Complex Var. Elliptic Equ., 55 (2010), 833–845

    work page 2010

  17. [17]

    Gol’dshtein, A

    V. Gol’dshtein, A. Ukhlov, On functional properties of weak(p, q)-quasiconformal homeomorphisms, Ukrainian Math. Bull., 16 (2019), 329–344. SOBOLEV(p, q)-EXTENSION OPERATORS AND NEUMANN EIGENV ALUES 15

    work page 2019

  18. [18]

    Gol’dshtein, A

    V. Gol’dshtein, A. Ukhlov, On the first Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains, Arch. Rational Mech. Anal., 221 (2016), 893–915

    work page 2016

  19. [19]

    Gol’dshtein, A

    V. Gol’dshtein, A. Ukhlov, The spectral estimates for the Neumann-Laplace operator in space domains, Adv. Math., 315 (2017), 166–193

    work page 2017

  20. [20]

    Hajlasz, Change of variable formula under the minimal assumptions, Colloq

    P. Hajlasz, Change of variable formula under the minimal assumptions, Colloq. Math., 64 (1993), 93–101

    work page 1993

  21. [21]

    Hajlasz, P

    P. Hajlasz, P. Koskela, H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217–1234

    work page 2008

  22. [22]

    Heinonen, T

    J. Heinonen, T. Kilpelinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Clarendon Press. Oxford, New York, Tokio (1993)

    work page 1993

  23. [23]

    Koskela, A

    P. Koskela, A. Ukhlov, Z. Zhu, The volume of the boundary of a Sobolev(p, q)- extension domain, J. Funct. Anal., 283 (2022), No. 109703

    work page 2022

  24. [24]

    3, Problems in mathematical analysis

    P.Koskela, Z.Zhu, Sobolevextensionsviareflections, J.Math.Sci.(N.Y.)268(2022), no. 3, Problems in mathematical analysis. No. 118, 376–401

    work page 2022

  25. [25]

    Koskela, Z

    P. Koskela, Z. Zhu, The extension property for domains with one singular point, Pure Appl. Funct. Anal., 9 (2024), 211–229

    work page 2024

  26. [26]

    P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981) 71–78

    work page 1981

  27. [27]

    Maz’ya, Sobolev spaces: with applications to elliptic partial differential equations, Springer, Berlin/Heidelberg, 2010

    V. Maz’ya, Sobolev spaces: with applications to elliptic partial differential equations, Springer, Berlin/Heidelberg, 2010

    work page 2010

  28. [28]

    V. G. Maz’ya, V. P. Havin, Non-linear potential theory, Russian Math. Surveys, 27 (1972), 71–148

    work page 1972

  29. [29]

    G.Maz’ya, S

    V. G.Maz’ya, S. V.Poborchi, Extensionof functionsin Sobolev classesto theexterior of a domain with the vertex of a peak on the boundary I, Czech. Math. Journ., 36 (1986), 634–661

    work page 1986

  30. [30]

    G.Maz’ya, S

    V. G.Maz’ya, S. V.Poborchi, Extensionof functionsin Sobolev classesto theexterior of a domain with the vertex of a peak on the boundary II, Czech. Math. Journ., 37 (1987), 128–150

    work page 1987

  31. [31]

    Pólya, G

    G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, 1951

    work page 1951

  32. [32]

    L. E. Payne, H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rat. Mech. Anal., 5 (1960), 286-292

    work page 1960

  33. [33]

    Shvartsman, On Sobolev extension domains inRn

    P. Shvartsman, On Sobolev extension domains inRn. J. Funct. Anal., 258 (2010), 2205–2245

    work page 2010

  34. [34]

    Shvartsman, N

    P. Shvartsman, N. Zobin, On planar SobolevLm p -extension domains, Adv. Math., 287, (2016), 237–346

    work page 2016

  35. [35]

    E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, New Jersey, 1970

    work page 1970

  36. [36]

    Tanaka, K

    K. Tanaka, K. Sekine, M. Mizuguchi1 and S. Oishi. Estimation of Sobolev-type em- bedding constant on domains with minimally smooth boundary using extension op- erator, Journal of Inequalities and Applications (2015) 2015:389

    work page 2015

  37. [37]

    J., 34 (1993), 185–192

    A.Ukhlov, Onmappings, whichinduceembeddingsofSobolevspaces, SiberianMath. J., 34 (1993), 185–192

    work page 1993

  38. [38]

    Ukhlov, Extension operators on Sobolev spaces with decreasing integrability, Trans

    A. Ukhlov, Extension operators on Sobolev spaces with decreasing integrability, Trans. A. Razmadze Math. Inst., 174 (2020), 381–388

    work page 2020

  39. [39]

    S. K. Vodop’yanov, Taylor Formula and Function Spaces, Novosibirsk, Novosibirsk Univ. Press., 1988

    work page 1988

  40. [40]

    S. K. Vodop’yanov, V. M. Gol’dstein, Lattice isomorphisms of the spacesW1 n and quasiconformal mappings, Siberian Math. J., 16 (1975), 224–246

    work page 1975

  41. [41]

    S. K. Vodop’yanov, V. M.Gol’dshtein, T. G. Latfullin, Criteria for extension of func- tions of the classL1 2 from unbounded plain domains, Siberian Math. J., 20 (1979), 298–301

    work page 1979

  42. [42]

    S. K. Vodop’yanov, V. M. Gol’dshtein, Yu. G. Reshetnyak, On geometric properties of functions with generalized first derivatives, Uspekhi Mat. Nauk 34 (1979), 17–65

    work page 1979

  43. [43]

    S. K. Vodop’yanov, A. D. Ukhlov, Sobolev spaces and(p, q)-quasiconformal mappings of Carnot groups. Siberian Math. J., 39 (1998), 776–795

    work page 1998

  44. [44]

    S. K. Vodop’yanov, A. D. Ukhlov, Superposition operators in Sobolev spaces, Russian Mathematics (Izvestiya VUZ) 46 (2002), 11–33. SOBOLEV(p, q)-EXTENSION OPERATORS AND NEUMANN EIGENV ALUES 16

    work page 2002

  45. [45]

    S. K. Vodop’yanov, A. D. Ukhlov, Set functions and their applications in the theory of Lebesgue and Sobolev spaces, Siberian Adv. Math, 14 (2004), 78–125

    work page 2004

  46. [46]

    S. K. Vodop’yanov, A. Ukhlov, Set functions and their applications in the theory of Lebesgue and Sobolev spaces, Siberian Adv. Math., 15 (2005), 91–125

    work page 2005

  47. [47]

    Zhu, Sobolev extension onLp-quasidisks, Potential Anal., 58 (2023), 529–544

    Z. Zhu, Sobolev extension onLp-quasidisks, Potential Anal., 58 (2023), 529–544. Vladimir Gol’dshtein Alexander Ukhlov Department of Mathematics Department of Mathematics Ben-Gurion University of the Negev Ben-Gurion University of the Negev P.O.Box 653, Beer Sheva, 84105, Israel P.O.Box 653, Beer Sheva, 84105, Israel E-mail: vladimir@bgu.ac.il E-mail: ukhl...

    work page 2023