{"id":"39da92b0-104f-44e8-9240-cc60479725c9","arxiv_id":"2608.09503","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For nonlocal p-Laplace type equations with measurable coefficients, boundary Hölder regularity holds exactly when the exterior capacity density condition holds, with a quantitative modulus estimate.","lead":"This mathematics paper proves that a geometric capacity density condition on the boundary of a domain exactly controls how quickly solutions to a broad class of nonlocal nonlinear equations become Hölder continuous up to the boundary. It also gives explicit formulas for the boundary modulus of continuity, refining the classical Wiener criterion for these equations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reverse implication in Theorem 1.1 recovers only a different CDC constant: the same-constant equivalence is not proved and should be restated.","rationale":"The central contribution is the equivalence in Theorem 1.1 and the quantitative Wiener modulus in Theorem 1.4. The proof of Theorem 1.4 is structurally coherent: Lemma 4.2 is a plausible discrete Gronwall-type lemma, and the tail bookkeeping in Step 3 is standard for this literature. The most load-bearing defect I find is in the exact-constant formulation of Theorem 1.1. The reverse implication uses a capacity argument that necessarily loses control of the constant: the size of ε is tied to the Hölder constant C from (b), and the capacity comparison then yields a CDC constant θ* that is a function of ε, not of the original θ0. This is not merely cosmetic, because the theorem is the paper's headline statement; however, it is a statement-level flaw, not a collapse of the underlying ideas. The qualitative equivalence—some capacity density condition iff some uniform boundary Hölder estimate—remains supported. I therefore do not move the verdict from the reader's CONDITIONAL, but I would request a reformulation or a proof that the constants can be matched. The reader's weakest_assumption focused on the supercritical sp>n modification in Theorems 3.5 and 3.6. That is also a genuine soft spot: Theorem 3.6 for sp>n is justified by a sketched modification, and Lemma 4.1 depends on it. I did not make it primary because in the supercritical range the capacity density condition is automatic, so the qualitative central claim is less endangered, and the sketched modification—choosing σ<s and q<n/σ—appears plausible. Still, an independent check of that Moser-iteration step is advisable, and it reinforces the conditional verdict.","tokens_in":22579,"tokens_out":27725,"duration_ms":791570,"concrete_test":"Rerun the final paragraph of the proof of Theorem 1.1 with explicit bookkeeping: fix n>sp and a prescribed θ0, choose an admissible α and the corresponding constant C from (b), set ε by Cε^α = 1/2, and compute the recovered CDC constant θ* from the chain cap(B_{εr/2},B_{2r}) ≤ 2^pΛ^2 cap(B_r\\Ω,B_{2r}) together with Lemma 2.4. Then check whether θ* ≥ θ0 can hold for all θ0>0. If not, Theorem 1.1 must be restated with a possibly smaller constant in (a), or with an explicit bound θ* = θ*(n,s,p,Λ,θ0).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 asserts that (a) and (b) are equivalent with the same constant θ0. The forward direction (a)⇒(b) is consistent with Theorem 1.2, since α0 is chosen below cθ0. The reverse direction, however, does not recover θ0. In the proof of (b)⇒(a), one fixes any α∈(0,α0], obtains C=C(n,s,p,Λ,α,θ0), then chooses ε so small that Cε^α≤1/2. The subsequent capacity comparison gives cap(B_r\\Ω,B_{2r}) ≥ c(ε) cap(B_r,B_{2r}) with c(ε) depending on ε. For n>sp this gives a recovered CDC constant θ* of order ε^{(n-sp)/(p-1)} (up to constants in n,s,p,Λ). Since ε is chosen from C, θ* is a function of θ0 through C, but nothing in the argument forces θ* ≥ θ0; in fact, if C grows as θ0→0, θ* can be far smaller than θ0. Thus the statement 'if and only if' with the identical θ0 is overclaimed. The qualitative equivalence between some capacity density condition and some uniform boundary Hölder estimate is not undermined, but the theorem as written requires either a different proof or a modified statement allowing the CDC constant to change, e.g., 'there exists θ*'>0' or θ* = θ*(n,s,p,Λ,θ0).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":22807,"tokens_out":14223,"duration_ms":148650,"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":[{"comment":"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":"Section 5, proof of Theorem 1.1, (b)⇒(a)"},{"comment":"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.","section":"Section 3, Theorems 3.5 and 3.6"}],"minor_comments":[{"comment":"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.","section":"Section 5, proof of Theorem 1.1, (b)⇒(a)"},{"comment":"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":"Theorem 1.4, displayed estimate (1.7)"},{"comment":"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.","section":"Section 4, proof of Theorem 1.4"},{"comment":"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).","section":"Example 6.2"},{"comment":"The title in the manuscript appears as 'CAP ACIT AR Y ESTIMA TES' and should be corrected throughout.","section":"Title and headings"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically substantial and the main iteration argument is well organized. The central obstacle is the overclaim in Theorem 1.1 about recovering the same CDC constant; this is a statement-level issue rather than a fundamental flaw. I would be satisfied by a revision that restates the equivalence with a possibly different constant and adds a complete treatment of the supercritical boundary estimates. The paper fits the scope of math.AP and, after these corrections, is likely acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nFor 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.","headline":"Strong paper with a real quantitative Wiener modulus, but Theorem 1.1's same-constant equivalence is overclaimed as written.","tokens_in":23398,"tokens_out":3902,"would_cite":true,"duration_ms":34974,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["31B25","31B15","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonlocal boundary regularity is equivalent to a capacity density condition.","keywords":["boundary regularity","capacity density condition","nonlocal elliptic equations","Wiener criterion","Hölder continuity","fractional Sobolev capacity","measurable coefficients","Dirichlet problem"],"falsifier":"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.","tokens_in":22306,"feed_emoji":"📐","tokens_out":10569,"duration_ms":95219,"temperature":0.7,"pith_summary":"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.","feed_headline":"A single capacity condition governs nonlocal boundary regularity","feed_subtitle":"The result gives the first sharp geometric criterion for boundary Hölder regularity of nonlocal equations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the nonlocal Wiener criterion and the boundary weak Harnack/local boundedness theorems whose modification powers the supercritical case and Lemma 4.1.","marker":"[KLL23]"},{"why":"Provides the weak boundary-value framework (the $V^{s,p}$-sense), comparison principles, and $L$-potential facts used in the CDC necessity proof.","marker":"[BBK24]"},{"why":"Gives capacity subadditivity, Sobolev-capacity comparisons, and examples of sets with zero capacity used in Theorem 1.6.","marker":"[BBK25]"},{"why":"Origin of the local boundary Hölder estimate under CDC, whose iteration strategy is adapted to the nonlocal setting.","marker":"[GZ77]"},{"why":"Prior capacitary potential estimates for the fractional Laplacian via extension, replaced here by a direct iteration with tail terms.","marker":"[Bjö24]"},{"why":"Prior CDC-based boundary Hölder result for the fractional Laplacian using Caffarelli-Silvestre, which Theorem 1.2 extends to measurable coefficients and $p\\ne 2$.","marker":"[Li25]"},{"why":"Supplies the fractional Sobolev embedding used in the modification for $sp>n$.","marker":"[Coz17]"},{"why":"Wolff-potential characterization of capacity used to prove the inclusion theorem for parameter changes (Theorem 1.6).","marker":"[HW83]"}],"fun_headline_variants":["Capacity density condition governs nonlocal boundary regularity","Nonlocal boundary regularity: capacity density is the key","Boundary Holder regularity: the capacity density condition decides","Wiener criterion for nonlocal Dirichlet problems","Capacity density: sharp boundary regularity for nonlocal equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Capacity density condition governs nonlocal boundary regularity","Nonlocal boundary regularity: capacity density is the key","Boundary Holder regularity: the capacity density condition decides","Wiener criterion for nonlocal Dirichlet problems","Capacity density: sharp boundary regularity for nonlocal equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000951,"raw_usage":{"total_tokens":4014,"prompt_tokens":862,"completion_tokens":3152,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":3079}},"tokens_in":478,"tokens_out":3152,"duration_ms":22453,"temperature":1.0,"reasoning_tokens":3079,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:01:28.829430+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}