{"id":"75f0b27f-6b7d-486f-96a0-846c95e51586","arxiv_id":"2501.00969","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fully nonlinear nonlocal operators of order 2s satisfying maximum and minimum principles, a viscosity solution decaying like o(|x|^{-(N+2s)}) is identically zero.","lead":"A mathematical proof shows that for a large class of nonlocal elliptic equations, any solution that decays to zero faster than |x|^{-(N+2s)} at infinity must be identically zero. This pins down the exact Landis decay threshold for nonlocal operators as polynomial rather than exponential.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 11 does not establish the L1_ωs limit it needs: the bound ‖\\tilde u_j‖≤m uses m=1/u_j(x0), which is not shown to be uniform in j, so the positive supersolution ψ used in Theorem 1 may not exist in L1_ωs.","rationale":"The reader correctly identified the spectral-positivity hypothesis as fragile, and that is part of the issue. But the more concrete, load-bearing problem is that Proposition 11, which is supposed to construct the positive supersolution ψ under exactly that hypothesis, has an unjustified compactness and convergence step. The proof bounds the normalized sequence by m_j, but m_j is a j-dependent quantity that is never shown to be uniformly bounded; the subsequent L1_ωs convergence by dominated convergence is therefore not valid. Since V≡-1 appears to be admissible under the paper's definition (both maximum and minimum principles hold), this is not an empty concern. The central theorem may still be true, but the argument as it stands does not establish the existence of the required ψ∈L1_ωs. This warrants moving the verdict from CONDITIONAL to UNVERDICTED pending a repair of Proposition 11 or an additional argument that u_j(x0) is bounded below uniformly.","tokens_in":11626,"tokens_out":48192,"duration_ms":487015,"concrete_test":"Fix I=(-Δ)^s (kernel K(y)=|y|^{-N-2s}), V≡-1, and solve the radial Dirichlet problem Iu_R-u_R=0 in B_R, u_R=1 outside B_R, for R=2,4,8,... . Compute m_R=1/u_R(0) and the weighted norm ‖u_R/u_R(0)‖_{L1_ωs}. If m_R→∞ and the weighted L1 norm of the normalized solution diverges as R grows, the dominated-convergence step in Proposition 11 fails for an admissible potential, and the proof of Theorem 1 is incomplete. If instead m_R remains bounded and the L1 norms converge, the concern is resolved.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The proof of Proposition 11 constructs functions \\tilde u_j = u_j/u_j(x0) on an increasing sequence G_j, with \\tilde u_j = m_j := 1/u_j(x0) outside G_j. It then says the maximum principle gives ‖\\tilde u_j‖_{C(K)} ≤ m_j and later that 0≤ψ≤m_j, using m_j as if it were a uniform constant. But m_j is not proved bounded, and for admissible V it can blow up. Consequently the claimed convergence of \\tilde u_j to ψ in L1_ωs by dominated convergence is unjustified: on the exterior R^N\\G_j the functions take the value m_j, and no integrable dominating function exists if m_j→∞. This is not a cosmetic gap because Theorem 1 needs ψ∈L1_ωs with a positive weighted norm to run the comparison argument and to obtain (11). The hypotheses allow V≡-1 for I=(-Δ)^s: at an interior maximum a constant test function gives Iu+Vu<0, and at an interior minimum it gives Iu+Vu>0, so both the maximum and minimum principles hold on every bounded G. For this V, the local limit s→1 of the Dirichlet problem Iu-u=0 in B_R, u=1 outside, is Δu-u=0, whose normalized solutions converge to e^{x_1}, which is not in L1_ωs; the proof gives no reason the nonlocal problem behaves differently. Thus Proposition 11, and with it the comparison step of Theorem 1, is unsupported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Landis-type unique continuation theorem at infinity for fully nonlinear, nonlocal elliptic integro-differential operators of order 2s. Under a spectral-positivity assumption on I+V in every bounded subdomain, any viscosity solution u in L^1_{\\omega_s}(R^N) of Iu+Vu=0 that decays like o(|x|^{-(N+2s)}) must vanish identically. The proof combines a nonlocal weak Harnack inequality with a scaling argument, constructs a positive entire supersolution via Leray-Schauder theory, and then compares u against this supersolution. The paper claims this polynomial decay is sharp and new even for the fractional Laplacian.","tokens_in":12001,"tokens_out":18600,"duration_ms":201923,"significance":"If the result and proof are correct, the paper establishes a sharp polynomial decay threshold for nonlocal Landis-type unique continuation, significantly relaxing the exponential integrability condition used by Ruland and Wang for the fractional Laplacian. The method is attractive: it uses standard tools (weak Harnack, Holder regularity, Leray-Schauder) rather than Carleman estimates, and the statement covers fully nonlinear sup/inf operators. The scaling argument in Theorem 8 and the comparison idea are transparent and, with the issues below repaired, the proof could be a valuable contribution to the nonlocal unique continuation literature.","major_comments":[{"comment":"The convergence of the normalized solutions \\tilde u_j in L^1_{\\omega_s}(R^N) is not established. The proof sets \\tilde u_j = u_j/u_j(x_0), but the quantity denoted by m is in fact m_j = 1/u_j(x_0), and no uniform bound on m_j is proved. On R^N \\setminus G_j one has \\tilde u_j = m_j, so the dominated convergence argument requires a common integrable majorant, which is absent if m_j is unbounded. The admissible potential V\\equiv -1 for I=(-\\Delta)^s is a concrete concern: the nonlocal Dirichlet problems with exterior value 1 can have normalized solutions whose local limit is exponential (e.g. e^{x_1} in the local limit s\\to 1), so an L^1_{\\omega_s} limit cannot be taken for granted. Since Theorem 1 needs the positive solution \\psi to lie in L^1_{\\omega_s} with positive weighted norm to obtain (11), this gap is load-bearing.","section":"Proposition 11"},{"comment":"The final contradiction in the sup-operator case is not valid as written. After deriving u\\le 0, the proof applies Theorem 8 to -u and obtains C^{-1}\\|u\\|_{L^1_{\\omega_s}} R^{-(N+2s)} \\le \\inf_{B_R}(-u)=\\inf_{B_R}|u|. This does not contradict (4), because the infimum over the ball may be attained at a fixed interior point and need not tend to zero as R\\to\\infty; the decay hypothesis (4) controls only points with |x| arbitrarily large. To conclude u\\ge 0 one must instead run the inf-operator comparison argument for -u and let the parameter \\delta go to zero, as is done for the inf case later. As written, the sup-case proof is incomplete.","section":"Theorem 1, first case"},{"comment":"The choice r_0 = (1+\\|V\\|_\\infty)^{-2s} does not in general satisfy the scaling condition (7). To apply Proposition 6 one needs r_0^{2s}\\|V\\|_{L^\\infty(B_{2r_0}(x_0))}\\le 1, which is ensured by the choice r_0=(1+\\|V\\|_\\infty)^{-1/(2s)}. For s<1/2, the exponent -2s gives r_0^{2s}=(1+\\|V\\|_\\infty)^{-4s^2}>(1+\\|V\\|_\\infty)^{-1}, so the normalized potential \\tilde V need not have sup-norm at most 1. This is probably a typo, but it affects the proof of the key estimate (6).","section":"Theorem 8, after (9)"}],"minor_comments":[{"comment":"The statement assumes V\\in C(B_1)\\cap L^\\infty(R^N), which is a strange combination; presumably V is continuous and bounded on B_1. The constant in the statement is said to depend on N,s,\\Lambda,\\lambda, but the proof uses only \\lambda and not \\Lambda; please clarify the correct dependence.","section":"Proposition 6, statement"},{"comment":"The notation in the normalization step is confusing: '\\tilde u_j = u_j/u_j(x_0) =: m u_j' should introduce m_j=1/u_j(x_0) and then use m_j consistently. The subscript is dropped in the maximum-principle estimate and in the convergence argument, which obscures the missing uniform bound.","section":"Proposition 11"},{"comment":"The phrase 'we can replace |x|=R in (11)' is imprecise; one should apply (11) with R replaced by |x| and then use \\inf_{B_{|x|}}\\psi\\le \\psi(x). The current wording caused a momentary confusion about which inequality is being used.","section":"Theorem 1, comparison step"},{"comment":"There is a typo in the statement: 'for very t \\in [0,1]' should read 'for every t \\in [0,1]'.","section":"Proposition 12"},{"comment":"The abstract and acknowledgments contain a typo: 'Aknowlegments' should be 'Acknowledgments'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an interesting and timely question, and the overall strategy is attractive. However, the proof of Proposition 11 has a serious gap that affects the existence of the positive L^1_{\\omega_s} supersolution, and the final comparison step in Theorem 1 contains an invalid contradiction. These are not merely cosmetic issues; they affect the central claim. The authors should either repair the uniform-bound argument under the stated hypotheses, or modify the hypotheses (for example, by explicitly assuming the existence of a positive L^1_{\\omega_s} solution or a stronger spectral condition). I would encourage a major revision rather than rejection, because the core scaling and Harnack ideas are promising and the theorem may be salvageable with a corrected comparison argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here is my read. The paper's claim is genuinely new and the architecture is attractive: a scaled weak Harnack gives polynomial lower decay for positive supersolutions, and comparison then forces any o(|x|^{-(N+2s)}) solution to vanish. That is a clean idea, and it plausibly extends the exponential-decay results of Ruland-Wang and Kow-Wang to fully nonlinear operators. The polynomial threshold N+2s matches examples from fractional nonlinear Schrödinger, so the target is the right one.\n\nThe soft spot is Proposition 11, and it is not cosmetic. The proof builds approximations \\tilde u_j = u_j/u_j(x0) taking the constant value m_j = 1/u_j(x0) outside G_j. It then treats m_j as a uniform bound and uses dominated convergence to get a limit ψ in L1_ωs. Nothing in the argument bounds m_j. For the admissible choice I=(-Δ)^s, V≡-1, the hypotheses λ±_1>0 hold, and the normalized solutions to (-Δ)^s u_j - u_j=0 in B_j, u_j=1 outside have u_j(x0)→0, hence m_j→∞. Then the exterior values blow up and there is no integrable dominating function; the claimed L1 convergence fails. Since Theorem 1 needs ψ∈L1_ωs with a positive weighted norm to run the comparison argument and obtain (11), the main theorem is not proved as written.\n\nI do not think this kills the underlying result. The weak Harnack part (Theorem 8) checks out, the scaling is right, and the final comparison is standard once a positive L1 supersolution exists. The fix likely comes from a different normalization or a direct construction that forces the exterior values to stay bounded in L1_ωs. But as it stands, the gap is load-bearing.\n\nMinor: Proposition 11 has a notational slip (F[u] vs (F-σ)u) in the Leray-Schauder step, and Proposition 6's constant dependence is sloppy. These are easy.\n\nWho is this for? PDE analysts working on nonlocal unique continuation; they will want the missing step fixed before citing. It deserves a serious referee, but the referee should be told to focus on Proposition 11. I would send it out with a request for revision rather than desk-reject, and I would not cite it in its current form.","headline":"Polynomial Landis threshold is real and worth pursuing, but the positive supersolution construction has a load-bearing gap: the normalization constant m_j can blow up, so the L1_ωs limit is unjustified.","tokens_in":12518,"tokens_out":6314,"would_cite":false,"duration_ms":61714,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47G20","45K05","35B40","35D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Polynomial decay forces triviality for nonlocal elliptic equations.","keywords":["Landis conjecture","fractional Laplacian","unique continuation at infinity","polynomial decay","fully nonlinear integro-differential operators","viscosity solutions","weak Harnack inequality"],"falsifier":"Directly check Theorem 1 by constructing a bounded continuous $V$ that satisfies $\\lambda^\\pm_1(I+V,G)>0$ for every bounded subdomain $G$, together with a nonzero viscosity solution $u\\in L^1_{\\omega_s}(\\mathbb{R}^N)$ of $Iu+Vu=0$ satisfying $|u(x)||x|^{N+2s}\\to 0$; any such example would invalidate the theorem. For the fractional Laplacian, a numerical search for nonzero whole-space solutions decaying faster than $|x|^{-(N+2s)}$ under this spectral-positivity condition would provide a concrete test.","tokens_in":11457,"feed_emoji":"📉","tokens_out":8159,"duration_ms":75671,"temperature":0.7,"pith_summary":"The paper proves a unique continuation principle at infinity for fully nonlinear nonlocal elliptic equations: under a spectral-positivity assumption on the operator plus potential, any viscosity solution that decays faster than polynomially, specifically with $|u(x)||x|^{N+2s}\\to 0$ as $|x|\\to\\infty$, must be identically zero. For the fractional Laplacian this is new, and it shows that the nonlocal analogue of Landis' conjecture has polynomial rather than exponential decay as its critical rate. The result matters because it separates the local and nonlocal worlds: nonlocal operators of order $2s$ impose a much slower decay threshold, consistent with known positive solutions that decay exactly like $|x|^{-(N+2s)}$. If correct, the theorem identifies a sharp, parameter-free decay condition that forces triviality.","feed_headline":"Polynomial decay forces zero for nonlocal equations","feed_subtitle":"A large-ball Harnack inequality turns o(|x|^{-(N+2s)}) decay into a proof that the only solution is zero.","key_machinery":"The load-bearing object is a weak Harnack inequality for viscosity supersolutions of $M^-u+Vu\\le C_0$ in balls of radius $R$ (Theorem 8), which yields $\\|u\\|_{L^1_{\\omega_s}} \\le C(1+R^{N+2s})(\\inf_{B_R} u + C_0)$. This estimate, obtained by rescaling a half-Harnack inequality, converts the global weighted norm into a polynomial lower bound on the infimum of any positive solution. Combined with a positive global solution $\\psi$ of $I\\psi+V\\psi=0$ (Proposition 11), the inequality lets the paper compare an arbitrary decaying solution against $\\psi$ and then apply the maximum principle, forcing the solution to be nonpositive; applying the same to $-u$ forces nonnegativity.","core_discovery":"The central claim is Theorem 1: for an operator $I$ of the form $\\sup_a L_a$ or $\\inf_a L_a$, with each $L_a$ a stable-like integro-differential operator of order $2s$, and a bounded continuous potential $V$, assume the spectral-positivity condition $\\lambda^\\pm_1(I+V,G)>0$ holds in every bounded subdomain $G$. Then any viscosity solution $u\\in L^1_{\\omega_s}(\\mathbb{R}^N)$ of $Iu+Vu=0$ whose decay satisfies $|u(x)||x|^{N+2s}\\to 0$ as $|x|\\to\\infty$ must vanish identically. The proof has two main parts: a weak Harnack inequality in arbitrarily large balls giving a polynomial lower bound on the infimum of positive supersolutions, and a positive global supersolution built via a Leray-Schauder fixed-point argument under the assumed maximum and minimum principles. Comparing the solution against small multiples of this supersolution and sending the multiple to zero forces both $u\\le 0$ and $-u\\le 0$, hence $u\\equiv 0$.","pith_inferences":["Editorial inference: The scale of the large-ball weak Harnack inequality suggests that if the spectral-positivity assumption were weakened to $\\lambda^\\pm_1\\ge 0$, the critical decay rate would likely change; testing this would clarify whether $N+2s$ is intrinsic or an artefact of the proof.","Editorial inference: The theorem implies that any nontrivial solution under the spectral-positivity hypothesis must decay no faster than $|x|^{-(N+2s)}$, so searching for counterexamples should focus on the borderline case rather than on faster decays.","Editorial inference: Connecting Theorem 1 to criticality theory, as in the local setting, may yield a Liouville-type theorem and potentially remove the need to check the maximum and minimum principles on every bounded subdomain individually.","Editorial inference: A natural testable extension is to replace the pointwise decay condition by a weighted $L^1$ condition; the space $L^1_{\\omega_s}$ already controls behaviour at infinity, and the current proof may extend to this weaker hypothesis."],"forward_implications":["For the fractional Laplacian, Theorem 1 gives a unique continuation at infinity with a polynomial decay threshold, without requiring any regularity of the potential beyond bounded continuity.","Combined with exterior-domain positive solutions that decay as $|x|^{-(N+2s)}$, the theorem indicates that the decay condition is sharp at the rate level.","The same proof covers both sup- and inf-type fully nonlinear operators, so the conclusion applies to a broad class of nonlocal equations beyond the linear case.","Under minor modifications, the result also holds in exterior domains provided the solution has a sign in the unbounded complement, as stated in Remark 1."],"supporting_citations":[{"why":"Supplies the weak Harnack inequality (Proposition 6) that, after rescaling to balls of large radius, yields the polynomial lower bound in Theorem 8.","marker":"[19]"},{"why":"Provides the comparison-principle strategy and positive-supersolution approach that the proof adapts to the nonlocal setting.","marker":"[23]"},{"why":"Gives solvability of the Dirichlet problem for fully nonlinear nonlocal equations, used to construct the fixed-point solutions in Proposition 11.","marker":"[17]"},{"why":"Provides the pointwise evaluation lemma (Lemma 3) that lets viscosity supersolutions be tested at a touching point.","marker":"[5]"},{"why":"Supplies the viscosity framework, stability under limits, and the Pucci-type inequality between $Iu-Iv$, $M^-$, and $M^+$, used throughout.","marker":"[10]"},{"why":"Yields the regularity estimate for viscosity solutions and the equivalence between principal eigenvalues and maximum/minimum principles used in Definition 2.","marker":"[18]"},{"why":"Supplies the Leray-Schauder alternative used to obtain the positive global supersolution $\\psi$.","marker":"[1]"},{"why":"Provides explicit positive solutions of the fractional nonlinear Schr\\\"odinger equation with decay $|x|^{-(N+2s)}$, used to indicate sharpness.","marker":"[9]"},{"why":"Gives the lemma that radial functions with $|x|^{-(N+2s)}$ decay are mapped by the fractional Laplacian to comparable functions, supporting the sharpness discussion.","marker":"[2]"}],"fun_headline_variants":["Nonlocal Landis: polynomial decay implies only trivial solution","Polynomial decay at infinity forces zero in nonlocal Landis","For nonlocal elliptic operators, polynomial decay yields zero alone","Unique continuation at infinity: polynomial decay forces zero solution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the assumption that the operator $I+V$ satisfies both maximum and minimum principles in every bounded subdomain, since without $\\lambda^\\pm_1(I+V,G)>0$ the positive supersolution and the final comparison step collapse.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal Landis: polynomial decay implies only trivial solution","Polynomial decay at infinity forces zero in nonlocal Landis","For nonlocal elliptic operators, polynomial decay yields zero alone","Unique continuation at infinity: polynomial decay forces zero solution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000975,"raw_usage":{"total_tokens":4098,"prompt_tokens":856,"completion_tokens":3242,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":3175}},"tokens_in":472,"tokens_out":3242,"duration_ms":23008,"temperature":1.0,"reasoning_tokens":3175,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:42:54.675822+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly check Theorem 1 by constructing a bounded continuous $V$ that satisfies $\\lambda^\\pm_1(I+V,G)>0$ for every bounded subdomain $G$, together with a nonzero viscosity solution $u\\in L^1_{\\omega_s}(\\mathbb{R}^N)$ of $Iu+Vu=0$ satisfying $|u(x)||x|^{N+2s}\\to 0$; any such example would invalidate the theorem. For the fractional Laplacian, a numerical search for nonzero whole-space solutions decaying faster than $|x|^{-(N+2s)}$ under this spectral-positivity condition would provide a concrete test.","supporting_citations":[{"cited_title":"The Boundary Harnack Princip le for Nonlocal Elliptic Operators in Non-divergence Form","cited_arxiv_id":null,"evidence_quote":"Supplies the weak Harnack inequality (Proposition 6) that, after rescaling to balls of large radius, yields the polynomial lower bound in Theorem 8."},{"cited_title":"The V´ azquez maximum princi ple and the Landis conjec- ture for elliptic PDE with unbounded coefficients","cited_arxiv_id":null,"evidence_quote":"Provides the comparison-principle strategy and positive-supersolution approach that the proof adapts to the nonlocal setting."},{"cited_title":"Perron’s method for nonlocal fully nonlineare quations","cited_arxiv_id":null,"evidence_quote":"Gives solvability of the Dirichlet problem for fully nonlinear nonlocal equations, used to construct the fixed-point solutions in Proposition 11."},{"cited_title":"Regularity theory for ful ly nonlinear integro-differential equations","cited_arxiv_id":null,"evidence_quote":"Provides the pointwise evaluation lemma (Lemma 3) that lets viscosity supersolutions be tested at a touching point."},{"cited_title":"Fern´ andez-Real and X","cited_arxiv_id":null,"evidence_quote":"Supplies the viscosity framework, stability under limits, and the Pucci-type inequality between $Iu-Iv$, $M^-$, and $M^+$, used throughout."},{"cited_title":"Principal eigenvalues of fully nonlinear integro- differential elliptic equations with a drift term","cited_arxiv_id":null,"evidence_quote":"Yields the regularity estimate for viscosity solutions and the equivalence between principal eigenvalues and maximum/minimum principles used in Definition 2."},{"cited_title":"Solutions of Quasilinear Seco nd-Order Elliptic Boundary Value Problems via Degree Theory","cited_arxiv_id":null,"evidence_quote":"Supplies the Leray-Schauder alternative used to obtain the positive global supersolution $\\psi$."},{"cited_title":"Positive solutions of th e nonlinear Schr¨ odinger equa- tion with the fractional Laplacian","cited_arxiv_id":null,"evidence_quote":"Provides explicit positive solutions of the fractional nonlinear Schr\\\"odinger equation with decay $|x|^{-(N+2s)}$, used to indicate sharpness."},{"cited_title":"Quantitative local and g lobal a priori estimates for fractional nonlinear diffusion equations","cited_arxiv_id":null,"evidence_quote":"Gives the lemma that radial functions with $|x|^{-(N+2s)}$ decay are mapped by the fractional Laplacian to comparable functions, supporting the sharpness discussion."}],"review_version":1}