{"id":"c40e2f8c-845b-4f76-a94e-abdb026bc9f9","arxiv_id":"2502.04107","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For operators comparable to the fractional Laplacian of order 2s, solutions with zero exterior data on Reifenberg flat domains are C^{s-ε} up to the boundary.","lead":"Boundary regularity of solutions to nonlocal elliptic equations is extended from Lipschitz domains to Reifenberg flat domains, which can have fractal boundaries: solutions are shown to be nearly as regular as the fractional order of the operator. The result matters because it pins down the sharp boundary behavior for a broad class of nonlocal equations, including operators comparable to the fractional Laplacian.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The compactness upgrade in Theorem 4.3 hinges on the black-boxed half-space Liouville theorem [14, Thm. 6.2]; the paper never verifies that the blow-up limit v and the limiting operator L∞ satisfy that theorem's hypotheses, so the contradiction step is unsecured.","rationale":"The reader's weakest_assumption identifies exactly the same point: the half-space Liouville theorem [14, Theorem 6.2] is imported without reproducing hypotheses, and the compactness argument relies on it plus the stability and blow-up selection results. My reading of the proof confirms that this is the single most load-bearing step. The barrier/induction part of Theorem 4.1 is self-contained modulo standard estimates and appears internally sound; the scale-dependence of the constant in Theorem 1.2 is a real but repairable quantitative issue, since allowing C to depend on r0 and the covering of Ω would not change the qualitative C^{s−ε} conclusion. The Liouville application, by contrast, is essential: if the theorem does not cover the measurable-kernel, distributional-solution, growth-s−γ setting obtained after blow-up, the contradiction in Theorem 4.3 collapses and no other mechanism in the paper yields the sharp exponent. The proposed check is therefore the exact statement of the cited theorem, matched against the limit objects. Unless that check reveals a mismatch, the paper's main line of proof can stand; hence the reader's conditional verdict is not changed.","tokens_in":15671,"tokens_out":31856,"duration_ms":355863,"concrete_test":"Obtain the exact statement of [14, Theorem 6.2] and check each hypothesis against the limit triple (L∞, v, {x·e>0}) produced in Theorem 4.3: (i) L∞ is only known to satisfy (1.2) with measurable kernel; (ii) v is a distributional solution with v=0 on {x·e≤0} and |v(x)|≤2(1+|x|^{s−γ}); (iii) γ>0 is arbitrary. If the stated theorem requires homogeneous kernels, energy-class membership, or a different growth assumption, re-derive the half-space Liouville theorem for measurable kernels and distributional solutions and re-run the compactness argument; if no such theorem holds, construct a nonzero half-space solution with growth below s for some L∈L_n^s(λ,Λ), which would refute Theorem 1.2. The concern is settled once the hypotheses of [14, Thm. 6.2] are written out and matched one-by-one with the blow-up objects.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2 is proved by first obtaining C^σ boundary regularity (Theorem 4.1) and then upgrading to C^{s−γ} by compactness (Theorem 4.3). The upgrade has no independent growth argument: after blow-up it invokes [14, Theorem 6.2] to conclude that the limit v, satisfying L∞v=0 in {x·e>0}, v=0 on {x·e≤0}, and |v(x)|≤2(1+|x|^{s−γ}), must vanish. That conclusion is the only step ruling out a nonzero blow-up profile; without it, (4.16) and hence (4.15) do not follow. The manuscript quotes the theorem by number but does not reproduce its hypotheses or check them against the objects produced by [4, Prop. 2.2.36] and [4, Lemma 4.4.11]. In particular: (a) [14, Thm. 6.2] may be stated for weak/energy solutions, whereas the limit v is only constructed as a distributional solution with polynomial growth; (b) it may assume additional kernel regularity, while stability only gives an operator in the same measurable class L_n^s(λ,Λ); (c) the growth bound in the theorem may be of a different form or exclude the endpoint γ>0. None of these points is discussed in the paper. Since the compactness argument cannot supply additional information about the kernel or the solution beyond what these black boxes provide, the contradiction step is only as sound as the exact matching of [14, Thm. 6.2]'s hypotheses with the blow-up objects. This is the most load-bearing external dependency in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves boundary C^{s-ε} regularity for solutions of Lu = f in Ω, u = 0 outside Ω, where L is a symmetric nonlocal operator of order 2s with kernel comparable to the fractional Laplacian and Ω is an (η,r0)-Reifenberg flat domain with η sufficiently small (Theorem 1.2). The proof is in two stages. Section 3 constructs supersolutions δ^ε via the regularized distance and a measure-theoretic density lemma, yielding the barrier estimate Lδ^ε ≥ c d^{ε-2s} (Lemma 3.4). Theorem 4.1 then uses an induction with these barriers and the comparison principle to obtain C^σ boundary regularity. Theorem 4.3 upgrades this to C^{s-γ} by a compactness argument: assuming the sharp estimate fails, it produces a normalized blow-up sequence, passes to a limiting solution v in a half-space, and invokes a half-space Liouville theorem from [14] to rule out nonzero limits.","tokens_in":16039,"tokens_out":23575,"duration_ms":254409,"significance":"If the proof is completed, this is a natural and valuable result: it extends optimal boundary regularity for operators comparable to the fractional Laplacian from C^1/Lipschitz boundaries to Reifenberg flat domains, with the sharp s-ε exponent and L∞ right-hand sides. The barrier construction is largely self-contained and the dyadic induction in Theorem 4.1 is transparent and does not rely on the main theorem circularly. The main weakness is the final compactness step, which outsources the only exclusion of nontrivial blow-up profiles to an external Liouville theorem whose hypotheses are not verified against the objects produced by the blow-up argument.","major_comments":[{"comment":"The statement 'By Liouville theorem in the half-space [14, Theorem 6.2] and the second property in (4.21) we have v = 0' is the only step that excludes a nonzero blow-up profile; estimates (4.16) and (4.15) both depend on it. The hypotheses of [14, Theorem 6.2] are not reproduced, and the manuscript does not check that the objects produced here satisfy them. In particular, [4, Proposition 2.2.36] yields L∞ only as a limit in the measurable class L_n^s(λ,Λ), and v is only constructed as a distributional solution with growth |v| ≤ 2(1+|x|^{s-γ}); it is not shown that [14, Theorem 6.2] covers distributional solutions, non-homogeneous kernels of class L_n^s(λ,Λ), or this growth bound. Since the compactness argument by itself cannot add kernel or solution regularity, the contradiction step is unsecured as written. Please either quote [14, Theorem 6.2] in full and verify these points, or replace it by a self-contained half-space Liouville argument.","section":"Theorem 4.3, after Eq. (4.21)"},{"comment":"The compactness chain also imports several black-box conclusions without stating them: the existence of scales r_k and the uniform growth bound (4.19) are taken from [4, Lemma 4.4.11], and the existence of the limiting operator L∞ is taken from [4, Proposition 2.2.36]. These results are load-bearing because (4.19) is needed both for the local uniform convergence and for the growth hypothesis of the Liouville theorem. The manuscript should state exactly what these external results deliver and confirm that the normalized sequence v_k satisfies the hypotheses of the stability result after the truncation/normalization performed at the beginning of the proof.","section":"Theorem 4.3, Eqs. (4.18)-(4.21)"}],"minor_comments":[{"comment":"The implication from (4.7) to (4.6) is not literal: the dyadic bound with constant one gives |ũ(x)| ≤ ρ^{-σ}|x|^σ when ρ^{k+1} ≤ |x| ≤ ρ^k. The missing factor ρ^{-σ} is harmless for the final theorem because constants may be renamed, but the displayed estimate in (4.6) should either carry the constant or be rephrased as a growth estimate with a universal constant.","section":"Section 4.1, Eqs. (4.6)-(4.7)"},{"comment":"The sentence 'up to decreasing κ we may assume R ≫ D without loss of generality' is not justified from assumption (P_{R,κ}), which is only assumed for 0 < r < R; decreasing κ does not extend the range of r. The argument is repairable because the only application in Section 4 uses complements of balls, for which the property holds with arbitrarily large R, but the lemmas as stated need a precise hypothesis such as R large relative to diam(Ω^c) or an explicit statement of the stronger property used in the application.","section":"Lemma 3.2 and Lemma 3.4"},{"comment":"The theorem states the conclusion for every γ > 0 and writes C^{s-γ}(B_{1/2}); this only makes sense for γ < s. The statement should be restricted to γ ∈ (0,s), which is also what Theorem 1.2 needs.","section":"Theorem 4.3, statement"},{"comment":"There are several small typos and imprecisions: 're lies' in the abstract, 'tha t that' in the proof of Theorem 4.3, 'the same result h olds' in Section 2, and the phrase 'uniformly bounded in C^σ(Ω_k ∩ B_{1/(2r_k)})' in Theorem 4.3 is ambiguous and should read Ω_k/r_k ∩ B_{1/(2r_k)} or be phrased locally on compact subsets of the limiting half-space. The integral limits in the second case of Lemma 3.2 involving '100D' should also be double-checked against the size of R.","section":"General presentation"}],"recommendation":"major_revision","confidential_remarks":"The main line of the proof is plausible and the barrier/induction part is well organized. The decisive issue is the final compactness upgrade, where the manuscript relies on [14, Theorem 6.2] without verifying its hypotheses. If the author supplies the exact statement and a verification, or gives a self-contained Liouville argument, I would support acceptance; as it stands, a load-bearing step is not checkable from the manuscript. I did not find circularity or scope problems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result and it deserves a serious referee. The paper proves sharp C^{s-epsilon} boundary regularity for operators comparable to the fractional Laplacian over Reifenberg flat domains with L^infinity right-hand side, which genuinely extends earlier work: [14] covered delta-Lipschitz domains, and [13] was restricted to the fractional Laplacian with continuous RHS. That is a natural and worthwhile completion of the picture.\n\nThe best part of the paper is Section 3 and Theorem 4.1. The barrier construction via the regularized distance, the measure estimate, and the induction argument are adapted carefully from [4] and [12]-[13], with the right modifications for the weaker Reifenberg geometry. The proof that the barrier is admissible in the weak form (the finite-energy check) is done conscientiously. I read the main chain as likely correct.\n\nThe soft spots are the ones the reader flagged, and they are real but not fatal. First, in the proof of Theorem 4.1, the passage from the dyadic estimate (4.7) to the pointwise Holder bound (4.6) as written gives constant 1 instead of the usual constant rho^{-sigma}. That is an easy fix, but the text should make it. Second, the role of eta is compressed at the end of the induction: setting eta = rho after having chosen rho small is legitimate only if rho is chosen after eta0 is fixed, and the exposition should say so explicitly. Both are minor.\n\nThe more serious concern is Theorem 4.3. The compactness upgrade is standard in outline, but it relies on the half-space Liouville theorem [14, Theorem 6.2] without reproducing its hypotheses or checking them against the blow-up limit v. The blow-up objects are distributional solutions with polynomial growth and the limiting operator is only known to lie in L_n^s(lambda, Lambda); if [14, Theorem 6.2] requires energy solutions, extra kernel regularity, or a different growth bound, the contradiction step collapses. This is a load-bearing external dependency, and citing it by number is not enough in a paper whose main novelty is precisely this upgrade. I would not desk-reject over this, but I would insist that the referee or the author verify the matching.\n\nWho is this for? Researchers working on boundary regularity for nonlocal operators. It is a solid contribution, not a breakthrough, and the proof is mostly a competent adaptation of known techniques. The citation pattern is fine; self-citation is limited and appropriate.\n\nRecommendation: send it to peer review. A good referee should check [14, Thm. 6.2] against the blow-up objects and ask for the small constant/eta clean-ups. If the Liouville theorem applies, the paper should be accepted after minor revision.","headline":"A genuine, carefully written generalization of nonlocal boundary regularity to Reifenberg flat domains; the main barrier argument is sound, but the compactness upgrade leans on an unverified black-box Liouville theorem and a few constants need cleaning.","tokens_in":16575,"tokens_out":2714,"would_cite":true,"duration_ms":32142,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35B65","35D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that weak solutions to nonlocal elliptic equations with kernels comparable to the fractional Laplacian are $C^{s-\\varepsilon}$ up to the boundary on Reifenberg flat domains whose flatness parameter is small enough.","keywords":["fractional Laplacian","nonlocal elliptic equations","boundary regularity","Reifenberg flat domains","Hölder regularity","barrier method","compactness argument","weak comparison principle"],"falsifier":"Solve, numerically or analytically, the Dirichlet problem for an operator of the form (1.1)-(1.2) with a non-homogeneous kernel, for example $K(y)=a(y/|y|)|y|^{-n-2s}$ with a nonconstant angular factor, on a Reifenberg-flat domain built with a self-similar boundary, and measure the boundary growth of $u$. If for some arbitrarily small flatness $\\eta$ the maximum of $u$ in $B_r$ decays slower than $r^{s-\\varepsilon}$ for some $\\varepsilon>0$, say like $r^s\\log(1/r)$, then Theorem 1.2 is false. Reproducing the predicted $C^{s-\\varepsilon}$ bound at several scales would confirm it.","tokens_in":15468,"feed_emoji":"📐","tokens_out":7822,"duration_ms":75614,"temperature":0.7,"pith_summary":"This paper proves that weak solutions to nonlocal elliptic equations driven by operators comparable to the fractional Laplacian are Hölder continuous up to the boundary with the near-optimal exponent $s-\\varepsilon$, provided the domain is Reifenberg flat with small flatness parameter and the solution vanishes outside the domain. This matters because Reifenberg flat sets include boundaries with fractal behavior, much less smooth than the Lipschitz or $C^{1,\\alpha}$ domains covered before, and the conclusion matches the sharp regularity known for the fractional Laplacian itself. The proof first establishes a small Hölder exponent $\\sigma$ by an induction with explicit barriers, then upgrades to $C^{s-\\varepsilon}$ through a compactness argument and a half-space Liouville theorem. The estimate is linear in the $L^\\infty$ norm of the right-hand side, so the result is quantitative and applies with arbitrary $L^\\infty$ data.","feed_headline":"Boundary regularity up to C^{s−ε} for nonlocal equations","feed_subtitle":"Small flatness suffices for sharp fractional Hölder estimates with L∞ data on fractal-like boundaries.","key_machinery":"The argument is carried by two mechanisms. First, a family of barriers $v_k$ built from powers of the regularized distance $\\delta^\\varepsilon$ to the complement of a shrinking ball: Lemma 3.4 shows $L\\delta^\\varepsilon \\geq c\\,d^{\\varepsilon-2s}$ near the boundary for the full class of comparable kernels, and an induction using the weak comparison principle turns this supersolution growth into a $C^\\sigma$ estimate at every scale, exploiting Reifenberg flatness only to align the barrier with an approximating hyperplane. Second, Theorem 4.3 upgrades $\\sigma$ to $s-\\gamma$ by contradiction: rescaling around a boundary point where the claimed bound fails produces a blow-up sequence that converges, via stability of distributional solutions, to a solution of a limiting nonlocal equation in a half-space; the half-space Liouville theorem then forces the limit to vanish while the normalization forces it to be nonzero. The regularized distance bounds (2.1)-(2.2), the measure estimate of Lemma 3.1, and the scaling property of the operator class are the auxiliary identities that make both steps quantitative.","core_discovery":"The central assertion is Theorem 1.2: for any $s\\in(0,1)$, any $\\varepsilon\\in(0,s)$, and any operator $L$ with kernel $K$ satisfying $\\lambda|y|^{-n-2s}\\leq K(y)\\leq \\Lambda|y|^{-n-2s}$, if $\\Omega$ is $(\\eta,r_0)$-Reifenberg flat with $\\eta$ no larger than a constant $\\eta_0(n,s,\\lambda,\\Lambda,\\varepsilon)$, then the weak solution of $Lu=f$ in $\\Omega$, $u=0$ outside $\\Omega$, belongs to $C^{s-\\varepsilon}(\\Omega)$ with norm at most $C(n,s,\\lambda,\\Lambda,\\varepsilon)\\lVert f\\rVert_{L^\\infty(\\Omega)}$. The regularity is sharp in the sense that for kernels of order $2s$ one cannot expect more than $C^{s-\\varepsilon}$ in this generality, matching the $\\delta$-Lipschitz result in [14] that this paper extends to the less regular Reifenberg-flat framework. A distinguishing feature is that the condition on $\\Omega$ is purely geometric and scale-invariant: at every boundary point and every small scale the boundary is uniformly close to a hyperplane, so the conclusion holds uniformly across possibly fractal boundaries.","pith_inferences":["A natural next step would be to test whether the flatness threshold can be made quantitative in $r_0$ or whether $\\eta_0$ has an explicit power-type dependence on $\\varepsilon$; the paper only states the existence of $\\eta_0$ and $C$.","The same barrier-and-compactness scheme may adapt to operators with kernels that are only comparable at small scales or to systems, since the barriers use only the pointwise ellipticity bounds and scaling.","Because the blow-up step uses the half-space Liouville theorem as a black box, replacing it with a self-contained proof for the full kernel class would remove the most fragile imported ingredient and might yield the same conclusion for more general translation-invariant limits.","One could look for a constructive example at the threshold: a Reifenberg-flat domain with flatness exactly $\\eta_0$ where the boundary behavior is no better than $|x|^{s-\\varepsilon}$, which would locate the sharpness of the condition."],"forward_implications":["The $C^{s-\\varepsilon}$ boundary estimate holds for all operators comparable to the fractional Laplacian, not only for the fractional Laplacian itself, and for arbitrary $L^\\infty$ right-hand sides.","The flatness threshold $\\eta_0$ depends only on $n$, $s$, $\\lambda$, $\\Lambda$, and $\\varepsilon$, so the regularity is uniform over the whole Reifenberg-flat class once $\\eta\\leq\\eta_0$.","Combining the local boundary estimate with interior $C^{2s}$ estimates gives the global Hölder norm on $\\Omega$, and the same conclusion extends to continuous distributional solutions by Remark 4.4.","The result is optimal in the exponent scale: $C^{s-\\varepsilon}$ for every $\\varepsilon>0$ is the best one can expect in this setting, consistent with the known optimality for $\\delta$-Lipschitz domains.","The theorem reduces the regularity theory over Reifenberg-flat sets to the same quantitative statement known for much smoother boundaries, so existing applications relying on boundary Hölder bounds can be transferred to fractal domains."],"supporting_citations":[{"why":"It supplies the maximum principle, interior regularity, stability of distributional solutions, blow-up selection lemma, and regularized-distance bounds used throughout the proof.","marker":"[4]"},{"why":"It is the source of the compactness upgrade to $C^{s-\\gamma}$ and of the half-space Liouville theorem used to force the blow-up limit to vanish, and its optimality transfers to Theorem 1.2.","marker":"[14]"},{"why":"It is the previous boundary Hölder regularity result for the fractional Laplacian over Reifenberg-flat domains, which this paper extends to the full comparable-operator class with $L^\\infty$ data.","marker":"[13]"},{"why":"It provides the iterative barrier argument for elliptic equations on Reifenberg-flat domains whose induction idea Theorem 4.1 follows.","marker":"[12]"},{"why":"It establishes the Reifenberg-flat geometric framework and the boundary regularity approach for the Poisson equation that the nonlocal proof adapts.","marker":"[11]"},{"why":"It is the baseline regularity result under the exterior ball condition, which the sharper $C^{s-\\varepsilon}$ conclusion extends to rougher domains.","marker":"[2]"}],"fun_headline_variants":["Sharp Hölder regularity for nonlocal equations on fractal boundaries","Optimal C^{s−ε} regularity over Reifenberg flat domains","Reifenberg flatness yields sharp fractional Hölder bounds","Fractal boundaries still yield sharp nonlocal regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that the half-space Liouville theorem imported from [14]—that the only distributional solution of a limiting comparable nonlocal equation in a half-space with zero exterior data and growth at most $(1+|x|)^{s-\\gamma}$ is zero—holds for the full class of kernels allowed by (1.2); together with the stability and blow-up selection tools from [4], it is what turns the compactness limit into a contradiction.","fun_headline_variants_meta":{"raw":{"variants":["Sharp Hölder regularity for nonlocal equations on fractal boundaries","Optimal C^{s−ε} regularity over Reifenberg flat domains","Reifenberg flatness yields sharp fractional Hölder bounds","Fractal boundaries still yield sharp nonlocal regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1554,"prompt_tokens":885,"completion_tokens":669,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":599}},"tokens_in":501,"tokens_out":669,"duration_ms":6641,"temperature":1.0,"reasoning_tokens":599,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T23:31:50.791163+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve, numerically or analytically, the Dirichlet problem for an operator of the form (1.1)-(1.2) with a non-homogeneous kernel, for example $K(y)=a(y/|y|)|y|^{-n-2s}$ with a nonconstant angular factor, on a Reifenberg-flat domain built with a self-similar boundary, and measure the boundary growth of $u$. If for some arbitrarily small flatness $\\eta$ the maximum of $u$ in $B_r$ decays slower than $r^{s-\\varepsilon}$ for some $\\varepsilon>0$, say like $r^s\\log(1/r)$, then Theorem 1.2 is false. Reproducing the predicted $C^{s-\\varepsilon}$ bound at several scales would confirm it.","supporting_citations":[{"cited_title":"Integro-differential elliptic equations","cited_arxiv_id":null,"evidence_quote":"It supplies the maximum principle, interior regularity, stability of distributional solutions, blow-up selection lemma, and regularized-distance bounds used throughout the proof."},{"cited_title":"Boundary H\\\"older regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle","cited_arxiv_id":"2501.14639","evidence_quote":"It is the previous boundary Hölder regularity result for the fractional Laplacian over Reifenberg-flat domains, which this paper extends to the full comparable-operator class with $L^\\infty$ data."},{"cited_title":"Boundary H\\\"older Regularity for Elliptic Equations on Reifenberg Flat Domains","cited_arxiv_id":"1812.11354","evidence_quote":"It provides the iterative barrier argument for elliptic equations on Reifenberg-flat domains whose induction idea Theorem 4.1 follows."},{"cited_title":"Boundary regularity for the Poisson equation in Reifenberg -flat domains","cited_arxiv_id":null,"evidence_quote":"It establishes the Reifenberg-flat geometric framework and the boundary regularity approach for the Poisson equation that the nonlocal proof adapts."},{"cited_title":"Regularity results for nonlocal equations by approximatio n","cited_arxiv_id":null,"evidence_quote":"It is the baseline regularity result under the exterior ball condition, which the sharper $C^{s-\\varepsilon}$ conclusion extends to rougher domains."}],"review_version":1}