Capacitary estimates for solutions to nonlocal Dirichlet problems

Marvin Weidner; Minhyun Kim; Se-Chan Lee

REVIEW 2 major objections 5 minor 16 references

Capacitary estimates for solutions to nonlocal Dirichlet problems

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nonlocal boundary regularity is equivalent to a capacity density condition.

desk verdict Strong paper with a real quantitative Wiener modulus, but Theorem 1.1's same-constant equivalence is overclaimed as written. read the letter →

arxiv 2608.09503 v1 pith:5WLLTX76 submitted 2026-08-10 math.AP

Minhyun Kim , Se-Chan Lee , Marvin Weidner This is my paper · ORCID

classification math.AP MSC 31B2531B1535R11

keywords boundaryregularitycapacitydensityconditionnonlocalellipticequationsWienercriterionHöldercontinuityfractionalSobolevmeasurablecoefficientsDirichletproblem

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 that for a broad class of nonlocal nonlinear elliptic equations with bounded measurable coefficients, the way solutions behave at a boundary point is completely controlled by one geometric quantity: the capacitary thickness of the complement. The central result is an equivalence: a domain satisfies the capacity density condition at a boundary point if and only if every weak solution with Hölder continuous exterior data is uniformly Hölder continuous up to that point. The paper further quantifies the inheritance of regularity, giving a Wiener-type modulus of continuity that shows exactly how the exterior datum's modulus and the domain's capacitary thickness combine to control the solution. If correct, this provides the first geometric characterization of boundary Hölder regularity for nonlocal nonlinear operators with measurable coefficients, and a quantitative refinement of the nonlocal Wiener criterion.

What carries the argument

The load-bearing object is the exterior capacitary thickness $\Theta_{x_0}(r) = \bigl(\operatorname{cap}_{s,p}(B_r(x_0)\setminus\Omega, B_{2r}(x_0))\, r^{sp-n}\bigr)^{1/(p-1)}$, computed with the fractional condenser capacity associated to the fractional Sobolev space $W^{s,p}$; the capacity density condition (CDC) is that $\Theta_{x_0}(r)\ge\theta_0$ on all small scales. The proof of the quantitative estimates is carried by a boundary capacitary estimate (Lemma 4.1), which controls the drop of the essential supremum of a nonnegative subsolution across a dyadic shell by $\Theta$, together with a new discrete iteration lemma (Lemma 4.2) that absorbs the nonlocal tail contributions and turns the scale-by-scale decay into an exponential modulus. For the equivalence, the necessity direction uses the $L$-potential, the nonlocal analogue of a condenser potential, to show that boundary Hölder continuity forces the capacitary thickness to stay bounded below; the sufficiency direction follows from the Wiener modulus estimate under CDC.

What would settle it

Construct the non-CDC domain of Example 6.2, where $\Theta_0(r)\sim 1/\log(1/r)$ and the Wiener integral diverges, and solve (1.3) numerically for the fractional Laplacian with $g\equiv 0$. Theorem 1.4 predicts a sub-polynomial modulus of continuity at the origin, faster than any power but slower than every $r^\alpha$; if the numerical solution instead obeys a power-law decay $r^\beta$, the quantitative Wiener estimate would be wrong. Conversely, checking a CDC domain with a very small $\theta_0$ and observing non-Hölder decay would falsify the sufficiency part of the equivalence.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for the operator $L$ in (1.1)–(1.2), the capacity density condition (CDC) at $x_0$, meaning that the exterior capacitary thickness $\Theta_{x_0}(r)\ge \theta_0>0$ for all $0<r\le R$, is equivalent to a uniform boundary Hölder estimate for weak solutions of the localized Dirichlet problem with exterior data $g\in C^\alpha$. The estimate takes the form $\operatorname{ess\,sup}_{\Omega\cap B_r(x_0)} |u-g(x_0)| \le C(LR^\alpha + A_{u-g(x_0)}(R;x_0))(r/R)^\alpha$ for every $\alpha\in(0,\alpha_0]$. Theorem 1.4 goes further and gives a quantitative Wiener modulus of continuity, an explicit estimate of the solution's oscillation at $x_0$ in terms of the integrated capacitary thickness $\int \Theta_{x_0}(\rho)\,d\rho/\rho$ and the modulus $\omega_g(\rho;x_0)$ of the exterior datum; Corollary 1.5 extracts a simpler exponential decay bound. Together these results characterize when boundary Hölder continuity holds and precisely quantify how regularity is inherited from the datum to the solution.

Load-bearing premise

The whole quantitative estimate rests on two boundary estimates: nonnegative supersolutions cannot drop too abruptly near the boundary, and subsolutions stay bounded up to the boundary; the paper extends these to the hardest parameter range $sp>n$ by a modification of an existing argument, and if that extension is wrong, the capacitary decay iteration in Theorem 1.4 collapses.

Editorial extensions

If this is right

The capacity density condition is both necessary and sufficient for boundary Hölder continuity of weak solutions with Hölder exterior data; earlier results only supplied sufficient conditions such as the measure density condition.
Under (CDC), the boundary Hölder exponent and constant are explicit in terms of $n,s,p,\Lambda,\theta_0$ and the datum's exponent $\alpha$ (Theorem 1.2), with a logarithmic correction in the critical case $\alpha=c\theta_0$.
Theorem 1.4 gives an explicit Wiener modulus of continuity: the solution's oscillation at $x_0$ decays exponentially in the integrated capacitary thickness $\int \Theta_{x_0}(\rho)\,d\rho/\rho$, with additional terms tracking the datum's modulus of continuity.
Corollary 1.5 recovers the sufficient part of the Wiener criterion as a limiting case, providing a quantitative nonlocal analogue of classical capacitary potential estimates for second-order quasilinear equations.
Theorem 1.6 describes how the class of domains satisfying (CDC) changes with $(s,p)$: if $s_2p_2 \le s_1p_1$ (or $s_2p_2 = s_1p_1$ with $p_1<p_2$), then every $(s_2,p_2)$-CDC domain is an $(s_1,p_1)$-CDC domain, and the inclusion is strict.

Reading between the lines

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

The equivalence suggests that boundary Hölder regularity for this class of operators is a purely geometric property of the domain, independent of the fine structure of the kernel beyond uniform ellipticity; an analogous characterization might be expected for other scales of boundary regularity, but the paper does not address that.
The explicit Wiener modulus offers a quantitative way to distinguish boundary points that are continuous from those that are Hölder, and could be used to compute sharp moduli for domains with self-similar or lacunary boundary geometries, for instance the domains of Example 6.2.
Theorem 1.6 implies that a domain's boundary regularity status is not invariant under changing the fractional order or the integrability: a domain can be regular for one nonlocal operator and irregular for another, so the geometric class picked by the model matters in applications.

Share X Bluesky LinkedIn Reddit HN

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

Bottom line: this is a substantial paper that should go to review, but the headline equivalence in Theorem 1.1 is not proved as stated. The stress-test note is right. In the proof of (b)⇒(a) in Section 5, the authors fix α, get C = C(n,s,p,Λ,α,θ0), then choose ε so small that C ε^α ≤ 1/2. The capacity comparison then recovers CDC only with a constant θ* of order ε^{(n-sp)/(p-1)}. Nothing forces θ* ≥ θ0, so the claimed same-constant iff is an overclaim. The qualitative equivalence between some capacity density condition and some uniform boundary Hölder estimate is not undermined, but the theorem must be restated with "there exists θ* = θ*(n,s,p,Λ,θ0)" instead of the same θ0, or the proof needs to recover the original constant.

What is genuinely new: Theorem 1.4, the quantitative Wiener modulus, is the first result of its kind for equations like (1.1)–(1.2) with measurable kernels and p≠2. Previous boundary Hölder results under CDC were only for the fractional Laplacian via Caffarelli–Silvestre, so Theorems 1.2 and 1.4 are real advances. The iteration lemma (Lemma 4.2) is a nice device for controlling the tail terms, and Theorem 1.6 gives a clean parameter comparison. The exposition is careful and the proofs are detailed.

Soft spots beyond Theorem 1.1: the paper leans heavily on KLL23, BBK24, and BBK25 for local boundedness, weak Harnack, and capacity facts. BBK24 and BBK25 are still preprints, which makes independent verification harder. The supercritical sp>n modification in Section 3 is plausible but terse; if the modification of [KLL23, Lemma 3.4] does not go through, Lemma 4.1 and Theorem 1.4 break. I did not find an error, but this needs a careful check by a referee. The use of [BBK24, Corollary 2.9] in the proof of Theorem 1.1 also relies on those preprints.

For anyone working in nonlocal boundary regularity, this is a must-read, and I would cite it for Theorem 1.4. It deserves a serious referee. My recommendation: send to peer review, with the instruction that Theorem 1.1's statement and proof be fixed—either by strengthening the argument or, more likely, by restating the equivalence with a possibly smaller constant.

Referee Report

2 major / 5 minor

Summary. The paper studies boundary regularity for weak solutions of nonlocal nonlinear elliptic equations with bounded measurable coefficients (1.1)-(1.2). The main result, Theorem 1.1, claims that a capacity density condition (CDC) at a boundary point is equivalent to a uniform boundary Hölder estimate for solutions with Hölder continuous exterior data. Theorem 1.2 gives a quantitative boundary Hölder estimate under the CDC, with the exponent depending on θ0. Theorem 1.4 provides a capacitary Wiener-type modulus of continuity estimate (1.7), and Corollary 1.5 derives a simpler version. In addition, Theorem 1.6 characterizes inclusions between the classes of domains satisfying the CDC for different values of (s,p). The proofs are built on boundary weak Harnack and local boundedness estimates up to the boundary (Theorems 3.5 and 3.6), a boundary capacitary estimate (Lemma 4.1), and a new iteration lemma (Lemma 4.2).

Significance. If the main result is correct, it provides the first geometric characterization of boundary Hölder regularity for nonlinear nonlocal operators with measurable coefficients, going beyond the qualitative Wiener criterion. Theorem 1.4 is a quantitative refinement of the nonlocal Wiener criterion and appears to be new even for the fractional Laplacian with measurable kernels and for p≠2. Theorem 1.6 gives a complete comparison of CDC classes across parameter ranges. The paper's strengths include a self-contained iteration lemma, explicit dependence of constants, and a transparent reliance on independently published results for the main local estimates. The overclaim in Theorem 1.1 concerning the recovery of the same CDC constant is fixable and does not destroy the qualitative equivalence, but it must be corrected before publication.

major comments (2)

[Section 5, proof of Theorem 1.1, (b)⇒(a)] The proof does not recover the same constant θ0. After choosing ε so that C ε^α ≤ 1/2, the capacity comparison gives cap(B_r\Ω, B_{2r}) ≥ C(n,s,p,ε) cap(B_r, B_{2r}); for n>sp this yields a CDC constant of order ε^{(n-sp)/(p-1)} up to constants. The parameter ε is chosen from C = C(n,s,p,Λ,α,θ0), and nothing in the argument forces the resulting constant to be at least θ0. Therefore the assertion that (b) implies (a) with the identical constant θ0 is not proved. The qualitative equivalence between some CDC and some uniform boundary Hölder estimate is not undermined, but the statement of Theorem 1.1 should be revised to allow a possibly different constant θ* = θ*(n,s,p,Λ,θ0), or the proof must be strengthened. This is load-bearing because the theorem is the main characterization result.
[Section 3, Theorems 3.5 and 3.6] The proof of the supercritical case sp>n for the boundary local boundedness and weak Harnack estimates is only sketched. The text states that the modification is obtained by choosing σ∈(0,s) and q satisfying np/(n+σp)<q<min{p,n/σ}, and then that the argument goes through as in the critical case. These estimates are load-bearing for Lemma 4.1 and hence for Theorem 1.4 and Corollary 1.5. Since the published results [KLL23, Theorems 3.5 and 3.7] are quoted only for sp≤n, the paper should either provide a complete proof of the supercritical case or cite a theorem that covers it explicitly. As written, the reader cannot fully verify the main quantitative estimates in the regime sp>n.

minor comments (5)

[Section 5, proof of Theorem 1.1, (b)⇒(a)] When applying (b) to the potential u with exterior data ψ_r, the A-term A_{u-ψ_r}(r;0) is asserted to be harmless, but this bound is not shown. Since 0≤ψ_r≤1 and 0≤u≤1 by comparison, the bound is easy and should be stated explicitly for completeness.
[Theorem 1.4, displayed estimate (1.7)] The integral limits in (1.7) are typeset ambiguously: the expression 'R 4r' should be read as ∫_{4r}^R, and the inner limit 'ρ 4r' should be ∫_{4r}^ρ. Please fix the display.
[Section 4, proof of Theorem 1.4] The final step converting the discrete iteration into the continuous integrals in (1.7) is summarized as 'a standard argument' in one sentence. For a quantitative Wiener modulus, a few more lines explaining the discrete-to-continuous passage would improve verifiability.
[Example 6.2] The formula for ρ_i in (6.2) is ambiguous: ρ_i := r_i / 16^{ i - (p-1)/(n-sp) } should be written with explicit parentheses, since the current typesetting can be misread as 16^{i} - (p-1)/(n-sp).
[Title and headings] The title in the manuscript appears as 'CAP ACIT AR Y ESTIMA TES' and should be corrected throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the boundary estimates are derived by iteration from capacity and Harnack tools, not assumed.

full rationale

The paper's derivation chain is self-contained in the relevant sense. Theorem 1.4 (the quantitative Wiener modulus) is proved from the boundary capacitary estimate Lemma 4.1, the boundary weak Harnack/local-boundedness estimates Theorems 3.5-3.6, and the iteration Lemma 4.2; none of these inputs is the target estimate. Theorems 3.5-3.6 are attributed to [KLL23] but with an explicit complementary proof covering the supercritical case, so the dependence is on prior proven lemmas rather than on the conclusion being established. Theorem 1.2 follows by substituting the capacity-density lower bound into Theorem 1.4, and Theorem 1.1(a)=> (b) follows from Theorem 1.2. The reverse direction (b)=> (a) constructs an explicit L-potential and compares capacities step by step; it assumes the Holder estimate and derives the capacity lower bound, so it is not circular. The most significant caveat is quantitative rather than circular: the proof of (b)=> (a) concludes cap_{s,p}(B_r\Omega,B_{2r}) <= C cap_{s,p}(B_r,B_{2r}) with C=C(n,s,p,epsilon)>0, which recovers a CDC constant that may differ from the theta_0 appearing in the theorem statement; this is an overclaim in the exposition but not a reduction of the result to its own hypothesis. The paper relies on prior work by the same research group, but those cited results are independently stated theorems about capacity, weak Harnack inequalities, and potential-theoretic properties, not restatements of the present theorems. No fitted parameter is disguised as a prediction, and no uniqueness theorem is imported to forbid alternatives. Therefore I find no significant circularity and assign score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no ad hoc parameters or fitted quantities; all constants are structural and depend only on n,s,p,Λ. It relies on standard potential-theoretic background and on several theorems from prior papers by the same group that are cited and used as black boxes.

assumptions (6)

domain assumption Boundary weak Harnack inequality and local boundedness up to the boundary (Theorems 3.5 and 3.6), proven following [KLL23] with a complementary argument for sp>n.
These estimates are the engine of the capacitary iteration in Lemma 4.1 and Theorem 1.4. They are stated as 'essentially contained in [KLL23]' and extended to the supercritical case.
standard math Condenser capacity comparability for balls (Lemma 2.4), taken from [KLL23, Lemma 2.17].
Used throughout to compare capacity at different scales, including the proof of Theorem 1.1 and the construction of examples.
standard math Wolff potential characterization of Sobolev capacity (Lemma 6.4), taken from [HW83].
Basis for the capacity comparison Lemma 6.5 and the proof of Theorem 1.6.
domain assumption Existence and properties of L-potentials and the comparison principle for the operator class, taken from [BBK24, KLL23].
Used in the proof of (b)⇒(a) in Theorem 1.1 to transfer regularity information to a capacity lower bound.
standard math Interior De Giorgi-Nash-Moser regularity for nonlocal operators with measurable kernels, from [DCKP14, DCKP16, Kas09].
Provides the interior Hölder exponent β referenced in Remark 1.3 and is a background input for the whole regularity theory.
domain assumption Existence of a compact set with C_{s2,p2}(K)=0 < C_{s1,p1}(K) under the parameter condition (1.9), from [BBK25, Theorem 8.3].
This external construction is the key counterexample used to prove strict inclusion in Theorem 1.6.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Capacitary estimates for solutions to nonlocal Dirichlet problems." pith.science (2026). https://pith.science/paper/5WLLTX76

@misc{pith2026260809503,
  author       = {Pith},
  title        = {Pith review of: Capacitary estimates for solutions to nonlocal Dirichlet problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5WLLTX76}},
  note         = {Machine review of arXiv:2608.09503}
}

read the original abstract

We study the boundary regularity of weak solutions to nonlocal nonlinear elliptic equations with bounded measurable coefficients. Our main result establishes that a capacity density condition is equivalent to the validity of a uniform boundary H\"older estimate for solutions with H\"older continuous exterior data. More generally, we derive a fine capacitary estimate on the modulus of continuity that captures how regularity is inherited from the exterior datum to the solution.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 9 canonical work pages

[4]

[BKS25] S. Byun, K. Kim, and K. Song. Nonlinear nonlocal equations in Reifenberg flat domains.arXiv e-prints, page arXiv:2508.12595,

work page Pith review arXiv
[8]

[Gru25] F. Grube. The inhomogeneous fractional Dirichlet problem.arXiv e-prints, page arXiv:2510.01055,

arXiv
[9]

[HW83] L

Unabridged republication of the 1993 original. [HW83] L. I. Hedberg and Th. H. Wolff. Thin sets in nonlinear potential theory.Ann. Inst. Fourier (Greno- ble), 33(4):161–187,

work page 1993
[10]

On boundary regularity for the fractional p-Laplacian with unbounded reactions

[IM26] A. Iannizzotto and S. Mosconi. On boundary regularity for the fractionalp-Laplacian with un- bounded reactions.arXiv preprint arXiv:2607.28436,

work page Pith review arXiv
[13]

[KLL23] M. Kim, K. Lee, and S. Lee. The Wiener criterion for nonlocal Dirichlet problems.Comm. Math. Phys., 400(3):1961–2003,

work page 1961
[14]

[Li25] F

To appear. [Li25] F. Li. H¨ older extension for fractional Laplacian.arXiv preprint arXiv:2508.12134,

arXiv
[1998]

Approximate factorizations for non-symmetric jump processes

[CS25] S. Cho and R. Song. Approximate factorizations for non-symmetric jump processes.arXiv preprint arXiv:2504.14763,

work page Pith review arXiv
[2006]

Bj¨ orn, J

[BBK24] A. Bj¨ orn, J. Bj¨ orn, and M. Kim. Perron solutions and boundary regularity for nonlocal nonlinear Dirichlet problems.arXiv preprint arXiv:2406.05994,

arXiv

Show all 16 references

[2010]

Fern´ andez-Real and X

[FRRO24] X. Fern´ andez-Real and X. Ros-Oton.Integro-differential elliptic equations, volume 350 ofProgress in Mathematics. Birkh¨ auser/Springer, Cham, [2024]©2024. [FSV15] A. Fiscella, R. Servadei, and E. Valdinoci. Density properties for fractional Sobolev spaces.Ann. Acad....

2024
[2014]

Ros-Oton and J

[ROS16a] X. Ros-Oton and J. Serra. Boundary regularity for fully nonlinear integro-differential equations. Duke Math. J., 165(11):2079–2154,
[2015]

[Gru24] F. Grube. Boundary regularity and Hopf lemma for nondegenerate stable operators.arXiv e-prints, page arXiv:2410.00829,

arXiv
[2016]

Kim and S

[KL23] M. Kim and S. Lee. Supersolutions and superharmonic functions for nonlocal operators with Orlicz growth.arXiv:2311.01246,

arXiv
[2017]

Ros-Oton and M

[ROW24] X. Ros-Oton and M. Weidner. Optimal regularity for nonlocal elliptic equations and free boundary problems.arXiv:2403.07793; to appear in Amer. J. Math.,

arXiv
[2023]

Kim and S

[KL24] M. Kim and S. Lee. Singularities of solutions of nonlocal nonlinear equations.arXiv e-prints, page arXiv:2410.13292,

arXiv
[2024]

Bj¨ orn, J

24 MINHYUN KIM, SE-CHAN LEE, AND MAR VIN WEIDNER [BBK25] A. Bj¨ orn, J. Bj¨ orn, and M. Kim. Semiregular and strongly irregular boundary points for nonlocal Dirichlet problems.arXiv preprint arXiv:2506.23188,

arXiv
[2025]

Bogdan, T

[BGR10] K. Bogdan, T. Grzywny, and M. Ryznar. Heat kernel estimates for the fractional Laplacian with Dirichlet conditions.Ann. Probab., 38(5):1901–1923,

1901

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [4]

[BKS25] S. Byun, K. Kim, and K. Song. Nonlinear nonlocal equations in Reifenberg flat domains.arXiv e-prints, page arXiv:2508.12595,

work page Pith review arXiv

[2] [8]

[Gru25] F. Grube. The inhomogeneous fractional Dirichlet problem.arXiv e-prints, page arXiv:2510.01055,

arXiv

[3] [9]

[HW83] L

Unabridged republication of the 1993 original. [HW83] L. I. Hedberg and Th. H. Wolff. Thin sets in nonlinear potential theory.Ann. Inst. Fourier (Greno- ble), 33(4):161–187,

work page 1993

[4] [10]

On boundary regularity for the fractional p-Laplacian with unbounded reactions

[IM26] A. Iannizzotto and S. Mosconi. On boundary regularity for the fractionalp-Laplacian with un- bounded reactions.arXiv preprint arXiv:2607.28436,

work page Pith review arXiv

[5] [13]

[KLL23] M. Kim, K. Lee, and S. Lee. The Wiener criterion for nonlocal Dirichlet problems.Comm. Math. Phys., 400(3):1961–2003,

work page 1961

[6] [14]

[Li25] F

To appear. [Li25] F. Li. H¨ older extension for fractional Laplacian.arXiv preprint arXiv:2508.12134,

arXiv

[7] [1998]

Approximate factorizations for non-symmetric jump processes

[CS25] S. Cho and R. Song. Approximate factorizations for non-symmetric jump processes.arXiv preprint arXiv:2504.14763,

work page Pith review arXiv

[8] [2006]

Bj¨ orn, J

[BBK24] A. Bj¨ orn, J. Bj¨ orn, and M. Kim. Perron solutions and boundary regularity for nonlocal nonlinear Dirichlet problems.arXiv preprint arXiv:2406.05994,

arXiv