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Boundary estimates and a Wiener criterion for the fractional Laplacian

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read A boundary point is regular for the fractional Laplacian equation exactly when the lifted point is regular for the extended weighted equation.

desk verdict The paper shows regularity for the fractional Laplacian at a boundary point is equivalent to regularity for the extended weighted problem, and derives a Besov-capacity Wiener criterion from that. read the letter →

arxiv 2107.04364 v2 submitted 2021-07-09 math.AP

Jana Björn This is my paper
classification math.AP
keywords fractionalLaplacianboundaryregularityWienercriterionBesovcapacityCaffarelli-SilvestreextensionKelloggpropertyweightedequations
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

The paper proves that for any open set in R^n, regularity of a boundary point x0 for the fractional equation (-Δ)^s u = 0 is equivalent to regularity of the point (x0, 0) for the extended weighted equation in one higher dimension. This equivalence is obtained by applying the Caffarelli-Silvestre extension to the fractional operator. The result immediately produces a Wiener criterion that identifies regular points through the Besov capacity of the complement. It further yields decay estimates for solutions near regular points and establishes the Kellogg property.

What carries the argument

The Caffarelli-Silvestre extension, which converts the nonlocal fractional Laplacian into a local weighted divergence-form equation in one higher dimension.

What would settle it

An explicit open set and boundary point where regularity holds for one of the two equations but fails for the other would disprove the claimed equivalence.

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Extended reading notes

Core claim

Using the Caffarelli-Silvestre extension, a boundary point x0 is regular for (−Δ)^s u = 0 in an open set Ω ⊂ R^n if and only if (x0, 0) is regular for the corresponding weighted equation in a subset of R^{n+1}. This yields a Wiener criterion for regularity in terms of Besov capacity, along with boundary decay estimates and the Kellogg property.

Load-bearing premise

The Caffarelli-Silvestre extension can be applied directly to relate the fractional equation on arbitrary open sets to the weighted problem without requiring extra boundary regularity.

Editorial extensions

If this is right

  • Regular boundary points admit a Wiener-type integral test involving the Besov capacity of the complement.
  • Solutions exhibit a specific decay rate near regular boundary points.
  • The Kellogg property holds: almost every boundary point with respect to the capacity is regular.

Reading between the lines

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

  • The equivalence opens the possibility of importing other classical results on weighted equations to the fractional setting.
  • Numerical schemes for fractional problems could be built by solving the local extended equation instead.
  • Because the result requires no extra assumptions on the domain, it may apply to irregular geometries where direct fractional analysis is difficult.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper uses the Caffarelli-Silvestre extension to prove that, for an arbitrary open set Ω ⊂ R^n, a boundary point x0 is regular for the fractional equation (-Δ)^s u = 0 (0 < s < 1) if and only if the point (x0, 0) is regular for the corresponding extended weighted equation in R^{n+1}. As a direct consequence it obtains a Wiener criterion phrased in terms of Besov capacity, together with a decay estimate near regular points and the Kellogg property.

Significance. The equivalence transfers classical local regularity theory to the nonlocal setting without additional restrictions on Ω, and the resulting Besov-capacity Wiener criterion supplies a concrete, testable characterization that is likely to be adopted in subsequent work on boundary behavior for fractional operators.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'in a subset of R^{n+1}' is imprecise; the precise domain (upper half-space or the extension cylinder) should be stated explicitly.
  2. [Abstract] The statement that the result holds for 'general open sets' would benefit from a single sentence clarifying that no regularity or thickness assumption on ∂Ω is imposed beyond openness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The report does not list any specific major comments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The derivation applies the standard Caffarelli-Silvestre extension as an external tool to obtain an if-and-only-if equivalence between regularity for (−Δ)^s u = 0 at x0 and regularity for the weighted extension at (x0,0). The Wiener criterion in Besov capacity is then derived as a consequence. No step reduces by definition to its own inputs, no parameter is fitted and relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The argument is self-contained against the cited extension and holds for arbitrary open sets without hidden regularity assumptions on the boundary.

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

The central claim rests on the applicability of the Caffarelli-Silvestre extension to general open sets and standard properties of Besov capacity; these are domain assumptions drawn from prior literature rather than new postulates.

assumptions (1)
  • domain assumption Caffarelli-Silvestre extension relates regularity of the fractional Laplacian to a weighted local equation
    Invoked directly in the abstract to obtain the equivalence.

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Cite this review

Pith. "Pith review of Boundary estimates and a Wiener criterion for the fractional Laplacian." pith.science (2026). https://pith.science/paper/2107.04364

@misc{pith2026210704364,
  author       = {Pith},
  title        = {Pith review of: Boundary estimates and a Wiener criterion for the fractional Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2107.04364}},
  note         = {Machine review of arXiv:2107.04364}
}
abstract

Using the Caffarelli--Silvestre extension, we show for a general open set $\Om\subset\R^n$ that a boundary point $x_0$ is regular for the fractional Laplace equation $(-\Delta)^su=0$, $0<s<1$, if and only if $(x_0,0)$ is regular for the extended weighted equation in a subset of $\R^{n+1}$. As a consequence, we characterize regular boundary points for $(-\Delta)^su=0$ by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained.

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Works this paper leans on

35 extracted references · 35 canonical work pages · cited by 1 Pith paper

  1. [1]

    and G´omez, I., Improvement of Besov regularity for solutions of the fractional Laplacian, Constr

    Aimar, H., Beltritti, G. and G´omez, I., Improvement of Besov regularity for solutions of the fractional Laplacian, Constr. Approx. 41 (2015), 219–229

    work page 2015

  2. [2]

    and Ros-Oton, X., Global regularity for the free boundary in the obstacle problem for the fraction Laplacian, Amer

    Barrios, B., Figalli, A. and Ros-Oton, X., Global regularity for the free boundary in the obstacle problem for the fraction Laplacian, Amer. J. Math 140 (2018), 415–447

    work page 2018

  3. [3]

    and Hansen, W., Potential theory.An analytic and probabilistic approach to balayage, Universitext, Springer, Berlin, 1986

    Bliedtner, J. and Hansen, W., Potential theory.An analytic and probabilistic approach to balayage, Universitext, Springer, Berlin, 1986

    work page 1986

  4. [4]

    123 (1997), 4–80

    Bogdan, K., The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), 4–80

    work page 1997

  5. [5]

    and Kwa´snicki, M., Estimates and structure of α-harmonic functions, Probab

    Bogdan, K., Kulczycki, T. and Kwa´snicki, M., Estimates and structure of α-harmonic functions, Probab. Theory Related Fields 140 (2008), 34–381

    work page 2008

  6. [6]

    and Sire, Y., Variational problems with free boundaries for the fractional Laplacian, J

    Caffarelli, L., Roquejoffre, J-M. and Sire, Y., Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc. 12 (2010), 1151–1179

    work page 2010

  7. [7]

    and Silvestre, L., Regularity estimates for the solution and the free boundary of the obstacle problem for the fra ctional Lapla- cian, Invent

    Caffarelli, L., Salsa, S. and Silvestre, L., Regularity estimates for the solution and the free boundary of the obstacle problem for the fra ctional Lapla- cian, Invent. math. 171 (2008), 425–461

    work page 2008

  8. [8]

    and Silvestre, L., An extension problem related to the frac- tional Laplacian, Comm

    Caffarelli, L. and Silvestre, L., An extension problem related to the frac- tional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260

    work page 2007

Show all 35 references
  1. [9]

    and Mosco, U., Wiener criteria and energy decay for relaxed Dirichlet problems, Arch

    Dal Maso, G. and Mosco, U., Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Ration. Mech. Anal. 95 (1986), 345–387

    1986

  2. [10]

    and Mosco, U., Wiener’s criterion and Γ-convergence, Appl

    Dal Maso, G. and Mosco, U., Wiener’s criterion and Γ-convergence, Appl. Math. Optim. 15 (1987), 15–63

    1987

  3. [11]

    Eilertsen, S., On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian, Ark. Mat. 38 (2000), 53–75

    2000

  4. [12]

    B., Jerison, D

    F abes, E. B., Jerison, D. and Kenig, C. E., The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32:3 (1982), 151–182

    1982

  5. [13]

    B., Kenig, C

    F abes, E. B., Kenig, C. E. and Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77–116

    1982

  6. [14]

    and Voigt, P., The Dirichlet problem for nonlocal operators, Math

    Felsinger, M., Kassmann, M. and Voigt, P., The Dirichlet problem for nonlocal operators, Math. Z. 279 (2015), 779–809

    2015

  7. [15]

    Gariepy, R. F. and Ziemer, W. P., Behavior at the boundary of solutions of quasilinear elliptic equations, Arch. Ration. Mech. Anal. 56 (1974/75), 372–384

    1974

  8. [16]

    and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006

    Heinonen, J., Kilpel ¨ainen, T. and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006

    2006

  9. [17]

    and Jacob, N., On the Dirichlet problem for pseudodifferential op- erators generating Feller semigroups, J

    Hoh, W. and Jacob, N., On the Dirichlet problem for pseudodifferential op- erators generating Feller semigroups, J. Funct. Anal. 137 (1996), 19–48

    1996

  10. [18]

    and Mal´y, J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math

    Kilpel¨ainen, T. and Mal´y, J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161. 12 Jana Bj¨ orn

    1994

  11. [19]

    and Shi, W., Higher regularity for the fractional thin obstacle problem, New York J

    Koch, H., R ¨uland, A. and Shi, W., Higher regularity for the fractional thin obstacle problem, New York J. Math. 25 (2019), 745–838

    2019

  12. [20]

    S., Foundations of modern potential theory, Grundlehren der mathematischen Wissenschaften 180, Springer, New York-Heidelberg, 1972

    Landkof, N. S., Foundations of modern potential theory, Grundlehren der mathematischen Wissenschaften 180, Springer, New York-Heidelberg, 1972

    1972

  13. [21]

    and Martio, O., Two theorems of N

    Lindqvist, P. and Martio, O., Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153–171

    1985

  14. [22]

    and Weinberger, H

    Littman, W., Stampacchia, G. and Weinberger, H. F., Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (1963), 43–77

    1963

  15. [23]

    G., Regularity at the boundary of solutions of elliptic equations and conformal mapping, Dokl

    Maz′ya, V. G., Regularity at the boundary of solutions of elliptic equations and conformal mapping, Dokl. Akad. Nauk SSSR 152 (1963), 1297–1300 (Russian). English transl.: Soviet Math. 4 (1963), 1547–1551

    1963

  16. [24]

    G., On the continuity at a boundary point of solutions of quasi- linear elliptic equations, Vestnik Leningrad

    Maz′ya, V. G., On the continuity at a boundary point of solutions of quasi- linear elliptic equations, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 25:13 (1970), 42–55 (Russian). English transl.: Vestnik Leningrad Univ. Math. 3 (1976), 225–242

    1970

  17. [25]

    G., Behaviour of solutions to the Dirichlet problem for the bi- harmonic operator at a boundary point, in Equadiff IV (Prague, 1977), pp

    Maz′ya, V. G., Behaviour of solutions to the Dirichlet problem for the bi- harmonic operator at a boundary point, in Equadiff IV (Prague, 1977), pp. 250–262, Lecture Notes in Math. 703, Springer, Berlin–Heidelberg, 1979

    1977

  18. [26]

    G., On the Wiener type regularity of a boundary point for the polyharmonic operator, Appl

    Maz′ya, V. G., On the Wiener type regularity of a boundary point for the polyharmonic operator, Appl. Anal. 71 (1999), 149–165

    1999

  19. [27]

    G., Sobolev Spaces with Applications to Elliptic Partial Differ en- tial Equations, Grundlehren der mathematischen Wissenschaften 342, Springer, Heidelberg, 2011

    Maz′ya, V. G., Sobolev Spaces with Applications to Elliptic Partial Differ en- tial Equations, Grundlehren der mathematischen Wissenschaften 342, Springer, Heidelberg, 2011

    2011

  20. [28]

    G., Boundary Behavior of Solutions to Elliptic Equations in Gen - eral Domains, EMS Tracts in Mathematics 30, European Math

    Maz′ya, V. G., Boundary Behavior of Solutions to Elliptic Equations in Gen - eral Domains, EMS Tracts in Mathematics 30, European Math. Soc., Z¨ urich, 2018

    2018

  21. [29]

    Maz′ya, V. G. and Donchev, T., Regularity in the sense of Wiener of a boundary point for a polyharmonic operator, C. R. Acad. Bulgare Sci. 36 (1983), 177–179 (Russian)

    1983

  22. [30]

    Mikkonen, P., On the Wolff Potential and Quasilinear Elliptic Equations In - volving Measures, Ann. Acad. Sci. Fenn. Math. Dissertationes 104 (1996)

    1996

  23. [31]

    Ros-Oton, X., Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat. 60 (2016), 3–26

    2016

  24. [32]

    and Serra, J., The Dirichlet problem for the fractional Lapla- cian: regularity up to the boundary, J

    Ros-Oton, X. and Serra, J., The Dirichlet problem for the fractional Lapla- cian: regularity up to the boundary, J. Math. Pures Appl. 101 (2014), 275–302

    2014

  25. [33]

    and Serra, J., The Pohozaev identity for the fractional Lapla- cian, Arch

    Ros-Oton, X. and Serra, J., The Pohozaev identity for the fractional Lapla- cian, Arch. Ration. Mech. Anal. 213 (2014), 587–628

    2014

  26. [34]

    Pure Appl

    Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112

    2007

  27. [35]

    Wiener, N., The Dirichlet problem, J. Math. Phys. 3 (1924), 127–146

    1924

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