{"id":"1d712ba2-a63d-4a62-b593-67e3d6d0fbca","arxiv_id":"2507.09587","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every singular solution to a nonlocal linear equation with measurable kernel that is one-sided bounded near zero and infinity must equal a multiple of the fundamental solution plus a constant.","lead":"This paper proves that singular solutions to a broad class of nonlocal equations, including fractional Laplacians with measurable kernels, must be a constant multiple of a fundamental solution plus a regular term. It also establishes a Liouville theorem classifying such solutions that stay bounded on one side near the origin and near infinity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main Liouville/Bocher theorems rest on the unproved isolated-singularity theorem [44, Thm 1.3] and estimate [44, Eq (4.6)]; Theorem 1.3 also asserts the non-removable asymptotic (4.2) unconditionally, and Theorem 1.4's uniqueness claim is not derived.","rationale":"The paper's central claim is secured by a long dependency chain, and the least secure link is the quoted isolated-singularity theorem and Caccioppoli-type estimate from the authors' previous preprint [44]. The reader identified exactly this as the weakest assumption, and I agree. The paper gives no proof of Theorem 4.3 and only a sketch of Lemma 4.2, explicitly deferring the key integrability estimate to [44, Eq (4.6)]. Since Theorems 1.3, 1.4, 1.6, and 1.5 all use this input, a hidden hypothesis in [44] would invalidate the main classification. I also found two secondary internal gaps: Theorem 1.3's proof asserts the non-removable asymptotic (4.2) unconditionally, ignoring the removable case; and Theorem 1.4's uniqueness is merely promised from Theorem 1.5, which does not apply directly to the difference of two fundamental solutions. Both are repairable, but they reinforce the need for a full proof of the prior singularity theorem. Because the reader already gave a conditional verdict and this assessment does not change that judgment, I recommend keeping the verdict unchanged.","tokens_in":22941,"tokens_out":23043,"duration_ms":247206,"concrete_test":"Obtain the full proof of [44, Theorem 1.3] and of the estimate [44, Eq (4.6)], and verify both for the exact kernel class (1.2) with no hypotheses beyond L-harmonicity in Ω\\{0} and one-sided boundedness in Ω. If either proof invokes additional structural assumptions, such as nonnegativity of u in all of R^n, stronger kernel regularity, or a bound on the tail, the present paper must state them; otherwise Theorem 1.3 and hence Theorem 1.5 are unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.5 is load-bearing on [44, Theorem 1.3] (quoted as Theorem 4.3) and on [44, Eq (4.6)] inside Lemma 4.2. The proof chain is: Lemma 4.2 to Lemma 4.1 to Theorem 1.3 to Theorem 1.4 to Theorems 1.6, 7.2, and 7.3 to Theorem 1.5. If [44, Theorem 1.3] is valid only under assumptions not satisfied by the kernels in (1.2), or if Eq (4.6) is unavailable, then Lemma 4.1 and Theorem 1.3 fail, the fundamental-solution normalization in Theorem 1.4(i) is unproved, and the decomposition u = aΦ + v in Theorem 1.6 has no basis. Independently, the proof of Theorem 1.3 as written is incomplete: the sentence 'In view of Theorem 4.3, we can fix R ... so that (4.2) u ≂ |x|^{2s-n} in B_R' is false when the singularity is removable, and no separate argument showing E(u, φ) = 0 in that case is supplied. Also, Theorem 1.4's uniqueness is not established: 'The uniqueness will follow from Theorem 1.5' does not apply because the difference of two fundamental solutions need not be one-sided bounded near 0. These are gaps in the text; the decisive missing item is a complete proof of the quoted singularity theorem and estimate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Bocher-type and Liouville-type theorems for singular solutions of nonlocal linear equations with measurable kernels of fractional order, including the fractional Laplacian as a special case. The main results are a weak Bocher theorem (Theorem 1.3), the construction and normalization of a fundamental solution Φ for the operator L (Theorem 1.4), a Bocher decomposition u = aΦ + v near the singular point (Theorem 1.6), and a Liouville theorem for solutions that are one-sided bounded near the origin and at infinity (Theorem 1.5). The proofs introduce a localized maximum principle and comparison principle that do not require pointwise comparison on the complement of the domain, and they construct the fundamental solution as a limit of Green functions on balls. The paper relies heavily on the authors' earlier preprint [44] for the isolated singularity classification and for a key integral estimate.","tokens_in":23286,"tokens_out":6351,"duration_ms":71761,"significance":"If the results are correct, they settle a natural nonlocal analogue of the classical Bocher-Liouville theory for a broad class of operators with merely measurable kernels, going well beyond the fractional Laplacian. The localized comparison principle (Theorem 1.7 and Corollary 1.8) is a genuinely useful new tool, since standard nonlocal maximum principles require pointwise control in the exterior. The fundamental-solution construction via Green functions on balls is also a clean approach that yields two-sided bounds. The paper is clearly written and the internal structure is mostly coherent. However, the central chain of implications depends on results from the authors' unpublished preprint [44], and two specific steps in the present text are either invalid as written (the use of (4.2) in Theorem 1.3 when the singularity is removable) or incomplete (the uniqueness claim in Theorem 1.4). These issues must be fixed before the main claims can be accepted.","major_comments":[{"comment":"The sentence 'In view of Theorem 4.3, we can fix R ... so that (4.2) u ≂ |x|^{2s-n} in BR' is not justified when the singularity at the origin is removable. Theorem 4.3 allows u to be bounded near 0, in which case the two-sided power growth (4.2) is false. The subsequent estimate of I2,2 uses the positivity of u and the lower bound u(x) ≳ |x|^{2s-n}; no separate argument is supplied for the removable case. Please either treat the removable case explicitly (e.g., extend u across 0 and use dominated convergence without the pointwise lower bound) or prove that the same estimate holds without (4.2).","section":"§4, proof of Theorem 1.3, near (4.2)"},{"comment":"The proof of Lemma 4.2 depends on the estimate [44, Equation (4.6)], which is only described as 'obtained by using the Caccioppoli estimate for weak supersolution vj instead of [44, Lemma 3.3]' and is not proved in the present paper. Since Lemma 4.2 underpins Lemma 4.1 and hence Theorem 1.3, and since [44] is an unpublished preprint, the authors should either include a complete proof of (4.6) or state it as a self-contained lemma with full justification.","section":"§4, Lemma 4.2"},{"comment":"The uniqueness statement 'The uniqueness will follow from Theorem 1.5' is not immediate. Two fundamental solutions Φ1 and Φ2 both blow up like |x|^{2s-n} near 0, so their difference need not be bounded from one side near 0; it could behave like (a1-a2)|x|^{2s-n}, which is unbounded both from above and below when a1≠a2. Theorem 1.5 is therefore not directly applicable. A valid argument would first use Theorem 1.3 to show that Φ1-Φ2 has zero Dirac mass at 0, hence a removable singularity, and then apply the Liouville theorem. Please supply this missing step.","section":"§5, proof of Theorem 1.4"},{"comment":"The main results rely essentially on the isolated singularity theorem quoted as Theorem 4.3 from [44, Theorem 1.3] and on [44, Eq (4.6)]. In particular, Theorem 1.3, Theorem 1.4(i), Theorem 1.6, Lemma 7.3, and Theorem 1.5 all invoke Theorem 4.3 or the estimate from [44]. If [44] is still a preprint, its results are not machine-checked and may not be available to the reader; the present manuscript should either include proofs of the quoted statements (or precise statements with complete hypotheses) or clearly indicate that the companion paper has been accepted for publication.","section":"Theorems 1.3–1.6 and §7"}],"minor_comments":[{"comment":"In the displayed estimate, the term C (r/θR)^n ∫_{Br} u^- dx appears; dimensionally this looks like it should involve the average of u^- over Br, namely C (r/θR)^n (1/|Br|) ∫_{Br} u^- dx, or alternatively the exponent n should be adjusted after the tail computation. Please check the constant and the normalization.","section":"§7, Lemma 7.1"},{"comment":"When u is bounded from above near 0 but not from below, Theorem 4.3 cannot be applied to u directly. The argument should apply Theorem 4.3 to -u and then convert the resulting asymptotic to the stated inequality u(x) ≥ -aΦ in B1. Please make this application explicit.","section":"§7, proof of Lemma 7.3"},{"comment":"The bilinear form E(u, φ) is used for u only in W^{s,2}_loc(Ω) ∩ L^1_{2s}(R^n); it may be helpful to add a remark on when E(u, φ) is finite for such u, for example by citing the estimate (2.3) from [43] or by stating a standard truncation argument.","section":"§2, Definition 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution, but the referee report identifies two concrete gaps in the text (the misuse of (4.2) in Theorem 1.3 and the unsupported uniqueness in Theorem 1.4) and a heavy, partly unverified dependence on the authors' earlier preprint [44]. The editor may wish to verify whether [44] is accepted; if it is not, the authors should be asked to include the missing proofs of Theorem 4.3 and Eq (4.6), at least in summarized form. The overall program is sound in conception and likely repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a substantial paper. The main new results are the Liouville theorem (Thm 1.5) and the Böcher-type decomposition (Thm 1.6) for linear nonlocal operators with merely measurable kernels, plus the construction of the fundamental solution for that class via Green functions and the localized comparison principle (Thm 1.7) that avoids pointwise comparison outside the domain. The latter is a real technical novelty, and the proofs are detailed and mostly careful. I believe the results are likely correct.\n\nThe soft spots are concentrated in two places. First, the proof of Theorem 1.3 assumes (4.2), i.e. u ≍ |x|^{2s-n} in B_R, which fails when the singularity is removable. The removable case is easy (then u is harmonic in B_R and E(u,φ)=0), but as written the proof does not split it off. This is a fixable gap, not a fatal one.\n\nSecond, and more important, the paper leans heavily on [44, Thm 1.3] and [44, Eq (4.6)], both from the same authors' unreviewed preprint. Theorem 4.3 (the isolated-singularity classification) is load-bearing for Theorems 1.3, 1.4, and 1.6; Lemma 4.2 uses Eq (4.6) directly. If [44] has hidden assumptions or an error in that estimate, the main results here collapse. I would send this to a referee but insist the authors either include full proofs of the quoted results, or at least state very precisely what in [44] is used and why it applies to the kernels in (1.2). The dependency is legitimate for linear operators—the authors do state [44] handles more general nonlinear equations—but a referee cannot verify the chain without access to a proof.\n\nThe uniqueness claim in Theorem 1.4 is also stated as \"will follow from Theorem 1.5\" without details. The concern that the difference of two fundamental solutions is not one-sided bounded is not actually an obstacle, since Theorem 1.5 applies directly to each positive fundamental solution. But the forward reference should be cleaned up.\n\nOn the positive side, the paper is honest about its limitations (n ≥ 2, critical case n = 2s open), and the literature appears properly cited. The fractional-Laplacian Theorem 1.2 is a modest extension of earlier Böcher results, but that is not where the value is.\n\nWho should read it: anyone working on nonlocal regularity, singular solutions, or fundamental solutions for stable-like operators. It deserves serious refereeing, with the request to fill the two gaps and clarify the dependency on [44].\n\nRecommendation: send to peer review, conditional on the authors addressing the above.","headline":"A genuinely new Liouville/Böcher theory for measurable-kernel nonlocal operators, but the main theorems lean on an unrefereed companion paper and two text-level gaps need closing.","tokens_in":23793,"tokens_out":3870,"would_cite":true,"duration_ms":43388,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A21","35A08","35B53","35R09"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Liouville theorem for nonlocal equations: every singular solution bounded on one side near zero and at infinity is a multiple of the fundamental solution plus a constant.","keywords":["Bôcher theorem","Liouville theorem","fractional Laplacian","nonlocal linear operators","isolated singularity","fundamental solution","measurable kernel","singular solutions"],"falsifier":"Check whether the unproved estimate [44, Eq (4.6)] holds for every measurable kernel satisfying the ellipticity bounds; if it fails, Lemma 4.2 and the weak Bôcher theorem lose support. The direct falsifier of the main theorem is an $L$-harmonic function in $\\mathbb{R}^n\\setminus\\{0\\}$, bounded below near $0$ and above near infinity, that is not of the form $a\\Phi+b$.","tokens_in":22748,"feed_emoji":"","tokens_out":11370,"duration_ms":115438,"temperature":0.7,"pith_summary":"This paper establishes a nonlocal analogue of Bôcher's theorem and the associated Liouville theorem for the fractional Laplacian and for linear nonlocal operators $Lu(x)=2\\,\\mathrm{p.v.}\\int_{\\mathbb{R}^n}(u(x)-u(y))k(x,y)\\,dy$ with measurable, symmetric kernels comparable to $|x-y|^{-n-2s}$. The central claim is that any $L$-harmonic function in $\\mathbb{R}^n\\setminus\\{0\\}$ that is bounded from one side near the origin and from one side near infinity must have the form $u=a\\Phi+b$, where $\\Phi$ is the fundamental solution of $L$ and $a,b$ are constants. This completely classifies isolated singularities for this operator class: the singular part is always a multiple of the kernel's scale-invariant fundamental solution, never an intermediate or oscillatory term. Along the way the paper constructs the fundamental solution as a limit of Green functions on balls and proves a localized comparison principle that controls solutions without imposing pointwise comparison in the complement.","feed_headline":"Nonlocal singular solutions: fundamental solutions plus constants","feed_subtitle":"The classical Bôcher classification now covers nonlocal equations with measurable kernels.","key_machinery":"The argument rests on four tools. The fundamental solution $\\Phi$, defined by $\\mathcal{E}(\\Phi,\\varphi)=\\varphi(0)$ for all test functions $\\varphi$, is constructed as the limit of Green functions on balls and satisfies $\\Phi\\asymp|x|^{2s-n}$ near the origin. The isolated-singularity theorem of the authors' earlier work supplies the dichotomy that a bounded-below $L$-harmonic function near $0$ is either regular or comparable to $|x|^{2s-n}$. A localized maximum principle and comparison principle use the nonlocal tail $\\mathrm{Tail}(w;0,r)=r^{2s}\\int_{\\mathbb{R}^n\\setminus B_r}|w(y)||y|^{-n-2s}\\,dy$ in place of pointwise comparison on the complement of the domain. Finally, a Harnack inequality on annuli and a Liouville theorem for nonnegative entire solutions control the behaviour at infinity and yield the constant $b$.","core_discovery":"The paper's core discovery is rigidity of isolated singularities for nonlocal linear equations with measurable kernels. If $u$ solves $Lu=0$ in $\\mathbb{R}^n\\setminus\\{0\\}$ and is bounded from one side near $0$ and from one side near infinity, then $u=a\\Phi+b$; locally, an $L$-harmonic function in $\\Omega\\setminus\\{0\\}$ bounded below splits as $u=a\\Phi+v$ with $v$ harmonic across the origin, so any genuine singularity has exact growth $u\\asymp|x|^{2s-n}$. The singular source is identified through the bilinear form $\\mathcal{E}(u,\\varphi)=\\int_{\\mathbb{R}^n}\\int_{\\mathbb{R}^n}(u(x)-u(y))(\\varphi(x)-\\varphi(y))k(x,y)\\,dy\\,dx$ as $a\\delta_0$, and the fundamental solution $\\Phi$ is shown to be unique with two-sided estimates matching $|x|^{2s-n}$.","pith_inferences":["A natural next test is the critical dimension $n=2s$, left open by the paper; the dichotomy may fail or acquire logarithmic fundamental solutions, and the localized comparison principle would need adaptation.","The localized comparison principle, replacing pointwise exterior comparison by tail integrals, appears transferable to nonlinear nonlocal operators and to operators with Orlicz growth.","Defining the fundamental solution through the bilinear form rather than distributions suggests a way to treat non-translation-invariant kernels, where $L\\varphi$ need not be defined for smooth compactly supported test functions."],"forward_implications":["Every isolated singularity of a solution to $Lu=0$ with the stated one-sided bounds is of the form $a\\Phi+b$; there are no intermediate growth rates between regular and $|x|^{2s-n}$.","For the fractional Laplacian, the Kelvin transform turns the theorem into an exterior-domain classification: a solution harmonic near infinity either oscillates with $\\limsup u=\\infty$ and $\\liminf u=-\\infty$, or converges to a finite limit.","Any nonnegative weak solution of $Lu=0$ in all of $\\mathbb{R}^n$ is constant, extending the classical Liouville theorem to measurable-kernel nonlocal operators.","The fundamental solution of $L$ exists, is unique, and is comparable to $|x|^{2s-n}$, so the singular behaviour is universal across the whole ellipticity class."],"supporting_citations":[{"why":"Supplies the classical Bôcher theorem and Liouville corollary that the paper generalizes to nonlocal operators.","marker":"[9]"},{"why":"Provides the isolated-singularity dichotomy (regular or comparable to $|x|^{2s-n}$) on which Theorems 1.3, 1.6, and 1.5 rest; also supplies the unproved estimate (4.6) used in Lemma 4.2.","marker":"[44]"},{"why":"Gives existence and two-sided near-diagonal estimates for Green functions on balls, used to construct the fundamental solution $\\Phi$.","marker":"[38]"},{"why":"Provides the superharmonic machinery, truncation results, and convergence theorems used to prove $L$-superharmonicity and to pass Green functions to the limit.","marker":"[47]"},{"why":"Supplies the nonlocal Harnack inequality used in the annulus Harnack lemma and hence in the infinity analysis.","marker":"[21]"},{"why":"Supplies local boundedness and Hölder regularity estimates used for the Green-function limit and for boundedness of solutions.","marker":"[22]"},{"why":"Provides the smoothness lemma for distributional fractional-harmonic functions and facts about the fractional fundamental solution used in Theorem 1.2.","marker":"[29]"},{"why":"Provides the Liouville theorem for nonnegative fractional-harmonic functions used to conclude that the regular part is constant in the fractional case.","marker":"[10]"},{"why":"Provides the uniform Hölder estimate used to prove that nonnegative entire solutions of $Lu=0$ are constant.","marker":"[36]"}],"fun_headline_variants":["Liouville rigidity for nonlocal singular solutions","Singular nonlocal solutions: only fundamental plus constants","Nonlocal Bôcher: singularities are fundamental solutions","Rigid singularities for measurable-kernel nonlocal equations","Bocher and Liouville for nonlocal singular solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the isolated-singularity theorem from the authors' earlier work: an $L$-harmonic function bounded below near an isolated point either extends regularly or grows exactly like $|x|^{2s-n}$; the proofs also quote without proof the estimate [44, Eq (4.6)] used in Lemma 4.2.","fun_headline_variants_meta":{"raw":{"variants":["Liouville rigidity for nonlocal singular solutions","Singular nonlocal solutions: only fundamental plus constants","Nonlocal Bôcher: singularities are fundamental solutions","Rigid singularities for measurable-kernel nonlocal equations","Bocher and Liouville for nonlocal singular solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000442,"raw_usage":{"total_tokens":2158,"prompt_tokens":782,"completion_tokens":1376,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":398,"completion_tokens_details":{"reasoning_tokens":1309}},"tokens_in":398,"tokens_out":1376,"duration_ms":13159,"temperature":1.0,"reasoning_tokens":1309,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:58:04.341127+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the unproved estimate [44, Eq (4.6)] holds for every measurable kernel satisfying the ellipticity bounds; if it fails, Lemma 4.2 and the weak Bôcher theorem lose support. The direct falsifier of the main theorem is an $L$-harmonic function in $\\mathbb{R}^n\\setminus\\{0\\}$, bounded below near $0$ and above near infinity, that is not of the form $a\\Phi+b$.","supporting_citations":[{"cited_title":"Bˆ ocher","cited_arxiv_id":null,"evidence_quote":"Supplies the classical Bôcher theorem and Liouville corollary that the paper generalizes to nonlocal operators."},{"cited_title":"Singularities of solutions of nonlocal nonlinear equations","cited_arxiv_id":"2410.13292","evidence_quote":"Provides the isolated-singularity dichotomy (regular or comparable to $|x|^{2s-n}$) on which Theorems 1.3, 1.6, and 1.5 rest; also supplies the unproved estimate (4.6) used in Lemma 4.2."},{"cited_title":"Kassmann, M","cited_arxiv_id":null,"evidence_quote":"Gives existence and two-sided near-diagonal estimates for Green functions on balls, used to construct the fundamental solution $\\Phi$."},{"cited_title":"Korvenp¨ a¨ a, T","cited_arxiv_id":null,"evidence_quote":"Provides the superharmonic machinery, truncation results, and convergence theorems used to prove $L$-superharmonicity and to pass Green functions to the limit."},{"cited_title":"Di Castro, T","cited_arxiv_id":null,"evidence_quote":"Supplies the nonlocal Harnack inequality used in the annulus Harnack lemma and hence in the infinity analysis."},{"cited_title":"Di Castro, T","cited_arxiv_id":null,"evidence_quote":"Supplies local boundedness and Hölder regularity estimates used for the Green-function limit and for boundedness of solutions."},{"cited_title":"Garofalo","cited_arxiv_id":null,"evidence_quote":"Provides the smoothness lemma for distributional fractional-harmonic functions and facts about the fractional fundamental solution used in Theorem 1.2."},{"cited_title":"Bogdan, T","cited_arxiv_id":null,"evidence_quote":"Provides the Liouville theorem for nonnegative fractional-harmonic functions used to conclude that the regular part is constant in the fractional case."},{"cited_title":"Kassmann","cited_arxiv_id":null,"evidence_quote":"Provides the uniform Hölder estimate used to prove that nonnegative entire solutions of $Lu=0$ are constant."}],"review_version":1}