The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay

Gabrielle Nornberg; Sebasti\'an Flores Sep\'ulveda

REVIEW 3 major objections 5 minor 23 references

The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Polynomial decay forces triviality for nonlocal elliptic equations.

desk verdict 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. read the letter →

arxiv 2501.00969 v1 pith:PO3QY7Z7 submitted 2025-01-01 math.AP

Sebastián Flores Sepúlveda , Gabrielle Nornberg This is my paper · ORCID

classification math.AP MSC 47G2045K0535B4035D40

keywords LandisconjecturefractionalLaplacianuniquecontinuationatinfinitypolynomialdecayfullynonlinearintegro-differentialoperatorsviscositysolutionsweakHarnackinequality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $u\equiv 0$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

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.

The 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.

I 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.

Minor: 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Proposition 11] 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.
[Theorem 1, first case] 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.
[Theorem 8, after (9)] 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).

minor comments (5)

[Proposition 6, statement] 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.
[Proposition 11] 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.
[Theorem 1, comparison step] 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.
[Proposition 12] There is a typo in the statement: 'for very t \in [0,1]' should read 'for every t \in [0,1]'.
[Throughout] The abstract and acknowledgments contain a typo: 'Aknowlegments' should be 'Acknowledgments'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorem reduces to explicit spectral hypotheses and standard external tools; the only self-citation is background.

full rationale

Theorem 1 is derived from the explicit assumption λ±1(I+V,G)>0 for every bounded subdomain G, together with standard nonlocal tools: the weak Harnack inequality (Proposition 6, adapted from Ros-Oton–Serra), Hölder regularity (Proposition 7, from Quaas–Salort–Xia), the solvability result for bounded-domain Dirichlet problems (Proposition 5, from Mou), and the Leray–Schauder alternative. Proposition 11 constructs the positive entire supersolution ψ by an exhaustion argument; this construction uses the assumed maximum/minimum principle in an essential way, but that principle is a hypothesis of the theorem, not a disguised version of the conclusion. The comparison step then uses ψ and the same maximum principle to force u≤0, and the Harnack lower bound (11) contradicts the assumed decay (4). No fitted parameters are renamed as predictions, no empirical subset is used to calibrate the result, and no 'uniqueness theorem' from the authors' prior work is invoked to forbid alternatives. The only self-citation is reference [22], by Moreira dos Santos, Nornberg, Schiera, and Tavares, cited for 'analogous definitions in the local case'; this is not load-bearing for the nonlocal argument. The skeptical concern about Proposition 11, that the constants m_j = 1/u_j(x0) may fail to be uniform in j and hence the L1_ωs convergence by dominated convergence is not justified, is a possible correctness gap in the proof, not a circularity: it does not reduce the theorem's claim to its own assumptions by definition. Therefore the circularity score is 1, reflecting only a minor, non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no fitted parameters or new entities. It relies on the explicit ellipticity class for kernels, the sup/inf structure of I, the spectral positivity condition λ±_1(I+V,G)>0, and standard results from the literature (weak Harnack, pointwise evaluation, Dirichlet solvability, Hölder regularity, Leray-Schauder alternative).

assumptions (9)

domain assumption Kernels satisfy 0 < λ/|y|^{N+2s} ≤ K(y) ≤ Λ/|y|^{N+2s}
Equation (1) defines the class L; this is the standard ellipticity assumption for integro-differential operators of order 2s.
domain assumption I is either sup_{a∈A} L_a or inf_{a∈A} L_a with L_a ∈ L
Equation (3) defines the fully nonlinear operators studied; the proof uses the Pucci-type inequalities M^- ≤ I ≤ M^+.
domain assumption λ±_1(I+V,G) > 0 for every bounded subdomain G
Definition 2; this is the pivotal hypothesis that both maximum and minimum principles hold, used to construct ψ (Proposition 11) and to run the comparison argument in Theorem 1.
domain assumption V ∈ C(R^N) ∩ L∞(R^N)
Assumed in Theorem 1; ensures the Dirichlet problem and the scaling arguments apply.
standard math Weak Harnack inequality (adapted from Theorem 2.2 in [19])
Proposition 6; the proof in the paper reproduces an adaptation, and the result is used to derive the large-radius version Theorem 8.
standard math Pointwise evaluation for viscosity solutions (Lemma 3.3 in [5])
Used in Proposition 6 to evaluate M^-u at a touching point.
standard math Dirichlet solvability for fully nonlinear nonlocal equations (Corollary 5.7 in [17])
Proposition 5; used in Proposition 11 for the Leray-Schauder construction.
standard math Hölder regularity for viscosity solutions (Theorem 3.6 in [18])
Proposition 7; used for compactness of the solution operator S.
standard math Leray-Schauder alternative (Corollary 1.19 in [1])
Used in Proposition 11 to obtain a fixed point of S.

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Cite this review

Pith. "Pith review of The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay." pith.science (2026). https://pith.science/paper/PO3QY7Z7

@misc{pith2026250100969,
  author       = {Pith},
  title        = {Pith review of: The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PO3QY7Z7}},
  note         = {Machine review of arXiv:2501.00969}
}

abstract

We obtain a unique continuation result at infinity for fully nonlinear elliptic integro-differential operators of order 2s which satisfy the maximum and minimum principles in bounded subdomains, under the decay assumption $o(|x|^{-(N+2s)})$ at infinity. Our result is new even in the case of the fractional Laplacian, as it unveils the nonlocal nature of the decay in Landis conjecture, evolving from exponential to polynomial.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Solutions of Quasilinear Seco nd-Order Elliptic Boundary Value Problems via Degree Theory

C. Bandle and W. Reichel. “Solutions of Quasilinear Seco nd-Order Elliptic Boundary Value Problems via Degree Theory”. In: Handbook of Diﬀerential Equations: Station- ary Diﬀerential Equations . Ed. by M. Chipot and P. Quittner. First. Vol. 1. Amster- dam: Elsevier, 2004, pp. 1–70

work page 2004

[2] [2]

Quantitative local and g lobal a priori estimates for fractional nonlinear diﬀusion equations

M. Bonforte and J. L. V´ azquez. “Quantitative local and g lobal a priori estimates for fractional nonlinear diﬀusion equations”. In: Advances in Mathematics 250 (2014), pp. 242–284

work page 2014

[3] [3]

On localization in the continu ous Anderson-Bernoulli model in higher dimension

J. Bourgain and C. Kenig. “On localization in the continu ous Anderson-Bernoulli model in higher dimension”. In: Inventiones mathematicae 161.2 (2005), pp. 389–426

work page 2005

[4] [4]

Bucur, E

C. Bucur, E. Valdinoci, et al. Nonlocal diﬀusion and applications . Vol. 20. Springer, 2016

work page 2016

[5] [5]

Regularity theory for ful ly nonlinear integro-diﬀerential equations

L. Caﬀarelli and L. Silvestre. “Regularity theory for ful ly nonlinear integro-diﬀerential equations”. In: Communications on Pure and Applied Mathematics 62.5 (2009), pp. 597– 638

work page 2009

[6] [6]

The Landis conjecture via Liouv ille comparison principle and criticality theory

U. Das and Y. Pinchover. “The Landis conjecture via Liouv ille comparison principle and criticality theory”. In: arXiv preprint arXiv:2405.11695 (2024)

arXiv 2024

[7] [7]

The Landis conjectur e for variable coeﬃcient second-order elliptic PDEs

B. Davey, C. Kenig, and J.-N. Wang. “The Landis conjectur e for variable coeﬃcient second-order elliptic PDEs”. In: Transactions of the American Mathematical Society 369.11 (2017), pp. 8209–8237

work page 2017

[8] [8]

On Landis’ conjectur e in the plane when the po- tential has an exponentially decaying negative part

B. Davey, C. Kenig, and J.-N. Wang. “On Landis’ conjectur e in the plane when the po- tential has an exponentially decaying negative part”. In: St. Petersburg Mathematical Journal 31.2 (2020), pp. 337–353

work page 2020