{"id":"added6f7-8347-47fb-9af2-79ff6ca99b6f","arxiv_id":"2506.20121","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims existence and classification of fundamental solutions of the logarithmic Laplacian in all dimensions via a modified division problem, but the central distribution is ill-defined as stated.","lead":"This mathematical note tries to construct fundamental solutions of the logarithmic Laplacian in all dimensions by solving a Fourier division problem in Lizorkin distributions, and it adds a Liouville theorem. The proposed main object, the renormalized inverse of log(|xi|^2), is not a well-defined distribution as written, so the central existence proof does not go through.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 4's exterior integral diverges for generic Schwartz (and even Lizorkin) test functions, so the central object Ê_log is not a distribution and the Section 2 computation acts on an undefined object.","rationale":"The reader's rejection is correct, and the weakest assumption identified is exactly the one that fails. I checked Definition 4 against concrete test functions; the exterior integral diverges in a way that is not a minor technicality. This is the load-bearing flaw because all of the paper's main claims use Ê_log: Theorem 2(2) classifies fundamental solutions as Ê_log plus a single layer, and Theorem 3 constructs E from Ê_log via comparison with the Helmholtz fundamental solution. Without a well-defined Ê_log, those statements are not merely unproven; they are statements about a nonexistent distribution. The Liouville part, Theorem 2(1), is argued separately and may be salvageable, but it does not support the fundamental-solution claims. The paper honestly states in Remark 8 that local integrability near zero is unresolved, and Remark 4 hints at the missing regularization; these cautions do not repair the invalid definition. The concrete check with ψ_0 is decisive: one Schwartz function in Ψ makes the divergence explicit. Therefore no amount of subsequent estimation in Section 3 can validate the conclusion until Definition 4 is replaced by a genuine finite-part definition and the Section 2 computation is redone. Since this confirms the reader's REJECT verdict, the stressed verdict is unchanged.","tokens_in":12023,"tokens_out":7421,"duration_ms":78654,"concrete_test":"Take ψ_0(ξ) = |ξ|^2 e^{−|ξ|^2} ∈ Ψ(R^d) and evaluate the Definition 4 outer integral. The integrand is (1/2)(r^2 e^{−r^2} − e^{−1}) r^{d−1}/log r dr dσ; the first term is integrable on (2,∞), but the second term gives −(e^{−1}/2)|S^{d−1}| ∫_2^∞ r^{d−1}/log r dr = −∞. This single computation settles that ⟨Ê^2_log, ψ_0⟩ is not finite, so Definition 4 does not define a distribution even on Ψ(R^d). If a finite part is supplied, Section 2's identity must be re-derived with the extra single-layer term included.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction in Definition 4 is not a well-defined distribution. For φ ∈ S(R^d), the outer integral ⟨Ê^2_log, φ⟩ = (1/2) ∫_{|ξ|>2} [φ(ξ) − φ(ξ/|ξ|)] / log|ξ| dξ does not converge: the term φ(ξ/|ξ|) is constant along rays while the volume element gives r^{d−1} dr, so ∫_2^∞ r^{d−1}/log r dr diverges. This is not rescued by restricting to Lizorkin test functions: ψ_0(ξ) = |ξ|^2 e^{-−|ξ|^2} belongs to Ψ(R^d), yet ψ_0(ξ/|ξ|) = e^{−1} ≠ 0, so the same divergence occurs. Consequently Ê_log is neither in S'(R^d) nor in Ψ'(R^d), and the proof of existence in Section 2 applies log(|·|^2)Ê_log to an object that has not been defined. The cancellation of the subtraction term in that computation is formally correct only after multiplication by log(|ξ|^2), since log(|ξ/|ξ||^2) = 0, but the transpose identity ⟨log(|·|^2)Ê, ψ⟩ = ⟨Ê, log(|·|^2)ψ⟩ requires Ê to be a distribution on all of S(R^d). Remark 4 acknowledges that a genuine Hadamard finite part would add single-layer terms; Definition 4 omits that regularization. Thus Theorem 2(2) and Theorem 3, and the claimed resolution of the Chen–Véron bound in dimensions 1 and 2, rest on an undefined object.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an alternative construction of a fundamental solution for the logarithmic Laplacian in all dimensions by solving the division problem log(|·|^2)Ê = 1 in the space of Lizorkin distributions. It defines a candidate distribution Ê_log via two subtracted integrals, claims that this object yields a fundamental solution, proves a Liouville theorem identifying harmonic distributions with single layers on the unit sphere, and uses these tools to give a partial resolution of the Chen–Véron conjecture in dimensions 1 and 2. The approach is inspired by the Malgrange–Ehrenpreis theory and is developed through a comparison with the classical Helmholtz fundamental solution.","tokens_in":12257,"tokens_out":14893,"duration_ms":138798,"significance":"If correct, the paper would extend the existence of fundamental solutions of the logarithmic Laplacian to all dimensions, provide a new Fourier-side proof of a Liouville theorem, and make progress on the Chen–Véron conjecture in dimensions 1 and 2. The strategy of renormalizing the inverse logarithmic symbol and comparing with the Helmholtz solution is a natural and potentially valuable idea. However, the central object of the paper, Ê_log, is not a well-defined distribution, so the main theorems are not established as written. The paper does not include machine-checked proofs or reproducible code; its contribution is purely analytic and, in its current form, the core construction fails.","major_comments":[{"comment":"The pairing ⟨Ê^2_log, φ⟩ = (1/2)∫_{|ξ|>2} [φ(ξ) − φ(ξ/|ξ|)]/log|ξ| dξ is divergent for any φ ∈ S(R^d) whose restriction to the unit sphere is not identically zero, because φ(ξ/|ξ|) is independent of |ξ| and ∫_2^∞ r^{d−1}/log r dr diverges for every d ≥ 1. Such φ also exist in the Lizorkin space Ψ(R^d), for instance a smooth function supported in an annulus containing S^{d−1} with nonvanishing values on the sphere. Hence Ê_log is not a distribution on S′(R^d) or Ψ′(R^d), and the existence step in Section 2 applies log(|·|^2) to an undefined object.","section":"Definition 4"},{"comment":"The computation of ⟨log(|·|^2)Ê_log, ψ⟩ discards the subtraction terms ψ(ξ/|ξ|). After multiplication by log(|ξ|^2), these terms become 2ψ(ξ/|ξ|), and the exterior contribution ∫_{|ξ|>2} 2ψ(ξ/|ξ|) dξ is not zero and, as written, diverges for generic ψ. The claimed identity ⟨log(|·|^2)Ê_log, ψ⟩ = ∫ψ dξ is therefore not a consequence of Definition 4 and is not justified by the displayed calculation.","section":"Section 2, proof of existence"},{"comment":"The object called E^2_log in the proof of Theorem 3 is the inverse Fourier transform of the locally integrable function 1_{|ξ|>2}/log(|ξ|^2), which is different from the Ê^2_log of Definition 4, where the subtraction φ(ξ/|ξ|) appears. The bounds established for this different object do not control the fundamental solution constructed from Ê_log, and the decomposition E = Φ + E^1_rem + E^2_rem does not follow from Theorem 2(2) without the ill-defined Ê_log.","section":"Section 3, proof of Theorem 3"},{"comment":"The identity ψ − Eτ = (1−Λ)ψ + ψ̃ and the assertion ⟨û, (1−Λ)ψ⟩ = 0 from supp(û) ⊂ S^{d−1} require that (1−Λ)ψ vanish in a neighborhood of S^{d−1}. The definition of Λ in Definition 5 only ensures Λ = 1 on S^{d−1}, not in a neighborhood; a distribution supported on S^{d−1} can pair nontrivially with a function that merely vanishes on the surface, as the radial derivative example in Remark 6 illustrates. This gap is repairable by requiring Λ ≡ 1 near S^{d−1}, but the proof as written is incomplete.","section":"Section 2, proof of the Liouville theorem, Step 2"}],"minor_comments":[{"comment":"The notation Ê_Flog appears where Ê_log is meant; please correct the typo.","section":"Section 2"},{"comment":"The outline refers to 'Appendix 3', but the appendix is unnumbered; renumber the cross-reference.","section":"Section 1.3"},{"comment":"The region ||ξ|−1| < 1 is the annulus 0 < |ξ| < 2; this should be stated explicitly, since the origin is not included but lies in the closure.","section":"Definition 4"},{"comment":"The phrase 'the unique extension from Z′(R^d) to S′(R^d)' in the paragraph after (12) needs a precise statement; the convention in Definition 3 and Remark 3 does not define a canonical extension for all elements of Z′(R^d).","section":"Section 3"},{"comment":"The citation '[1, Chapter 6]' does not match the bibliography format; assign Strichartz a proper reference key and use it consistently.","section":"Definition 5"}],"recommendation":"reject","confidential_remarks":"The central construction in Definition 4 is not a distribution, and the main theorems rest on this undefined object. This is a load-bearing error that cannot be fixed by minor editing; a proper Hadamard regularization would introduce additional single-layer terms, as the author notes in Remark 4, and the proofs would need substantial reworking. The Liouville step with Λ also needs repair. If the author can produce a coherent definition and rework the proofs, the underlying idea may still be of interest, but the current manuscript does not support its claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The stress-test note is right, and the reader's verdict lands in the right place. Definition 4's exterior integral diverges for generic Schwartz test functions because φ(ξ/|ξ|) is constant along rays, and ∫_2^∞ r^{d−1}/log r dr diverges. It also fails for Lizorkin test functions: ψ(ξ)=|ξ|^2 e^{−|ξ|^2} is in Ψ(R^d) but ψ(ξ/|ξ|)=e^{−1}, so the same divergence appears. Thus Ê_log is neither in S'(R^d) nor Ψ'(R^d), and Theorem 2(2) and Theorem 3 do not follow. This is not a minor typo; it is the main object of the paper, and the Section 2 computation is a formal manipulation of something that has not been defined.\n\nThat said, the paper is not a waste. The Liouville theorem section is a largely independent argument and reads plausibly: the support of û is confined to the unit sphere (up to a possible point at 0 that the author's convention should explicitly exclude), and the single-layer classification via the Taylor subtraction is a clean idea. That part might survive essentially unchanged once the quotient by polynomials is made precise. The comparison with the Helmholtz fundamental solution in Section 3 is also a good strategy, and it would be a nice way to obtain the d=1,2 bound once a correct regularized inverse of log(|ξ|^2) is available. The author is honest about the unresolved local integrability near zero (Remark 8) and acknowledges an earlier error in the acknowledgments; that is the behavior of someone actually engaging with the mathematics.\n\nSoft spots, in proportion: the divergent exterior integral is load-bearing, and the Section 3 bound estimates an E2_log that differs from the object defined earlier. The support-at-zero omission in the Liouville proof is a smaller gap, fixable by working in the quotient S'/polynomials. The untracked Fourier constants in the bound are a minor inconvenience, unlikely to change the result.\n\nWho is this for? Someone working on the logarithmic Laplacian or on division problems for non-smooth symbols might find the strategy and the Liouville part useful as a starting point, but the paper in its current form should not be the cited reference for the main existence theorem. I would not cite it yet. I would, however, send it to a serious referee: the flaw is technical and possibly repairable, and the Liouville argument deserves scrutiny and probably publication after revision. My recommendation is reject in current form, but with a clear and constructive path toward a fix.","headline":"The central distribution in Definition 4 is undefined, so the main existence theorems collapse, but the Liouville argument and the overall strategy are salvageable.","tokens_in":12877,"tokens_out":7707,"would_cite":false,"duration_ms":78506,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A08","35JXX","35R11","42B37","42B10","42B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs a fundamental solution of the logarithmic Laplacian in every dimension $d\\ge 1$ within Lizorkin distributions, and classifies all such solutions as the explicit one plus a surface-layer distribution on the unit sphere.","keywords":["logarithmic Laplacian","fundamental solution","Lizorkin distributions","division problem","single layer distributions","Helmholtz equation","Liouville theorem","nonlocal operators"],"falsifier":"Compute the Definition 4 pairing on $\\varphi(\\xi)=e^{-|\\xi|^2}\\tau(\\xi/|\\xi|)$ with $\\tau\\in C^\\infty(S^{d-1})$ having nonzero spherical mean. The exterior term equals $\\frac12\\left(\\int_{S^{d-1}}\\tau\\,d\\sigma\\right)\\int_2^\\infty \\frac{e^{-r^2}-e^{-1}}{\\log r}r^{d-1}\\,dr$, which diverges to $-\\infty$; unless a finite-part prescription is supplied, $\\hat E_{\\log}$ is not a tempered distribution and the identity $\\log(|\\xi|^2)\\hat E_{\\log}=1$ in $\\Psi'$ is not established.","tokens_in":11697,"feed_emoji":"📐","tokens_out":11961,"duration_ms":111083,"temperature":0.7,"pith_summary":"The paper aims to prove that the logarithmic Laplacian — the operator obtained by differentiating the fractional Laplacian with respect to its exponent at zero — has a fundamental solution in every dimension $d\\ge 1$, not just $d\\ge 3$ as previously known. The proof recasts the problem as a Fourier division problem $\\log(|\\xi|^2)\\Psi = 1$ and solves it with a renormalized kernel that tames the singularity on the unit sphere. If the argument is right, the set of all fundamental solutions is completely classified: they differ by harmonic functions, and harmonic functions for this operator are exactly generalized eigenfunctions of the ordinary Laplacian with eigenvalue 1. In dimensions 1 and 2 the constructed solution satisfies the decay bound conjectured by Chen and Véron away from the origin, leaving only local integrability near zero open.","feed_headline":"Logarithmic Laplacian gets fundamental solutions in every dimension","feed_subtitle":"A frequency-space division trick plus sphere-layer corrections delivers the conjectured decay in dimensions 1 and 2","key_machinery":"The load-bearing object is the renormalized Fourier kernel $\\hat E_{\\log}=\\hat E_{\\log}^1+\\hat E_{\\log}^2$, defined for Schwartz $\\varphi$ by\n$$\\langle \\hat E_{\\log},\\varphi\\rangle = \\frac12\\int_{||\\xi|-1|<1}\\frac{\\varphi(\\xi)-\\varphi(\\xi/|\\xi|)}{\\log|\\xi|}\\,d\\xi + \\frac12\\int_{|\\xi|>2}\\frac{\\varphi(\\xi)-\\varphi(\\xi/|\\xi|)}{\\log|\\xi|}\\,d\\xi.$$\nSubtracting the radial projection cancels the pole of $1/\\log|\\xi|$ on the unit sphere and makes the identity $\\log(|\\xi|^2)\\hat E_{\\log}=1$ hold on the Lizorkin test space $\\Psi(\\mathbb{R}^d)$. The second mechanism is the comparison with the Helmholtz fundamental solution $\\Phi$: the difference between the logarithmic and Helmholtz kernels is a single-layer distribution on $S^{d-1}$, i.e. a distribution that acts through a functional on the sphere, and single layers are annihilated by $\\log(|\\xi|^2)$ on the sphere. That comparison converts the Fourier remainders into Bessel integrals whose asymptotics give the decay in dimensions 1 and 2.","core_discovery":"The central claim is that in the frequency domain the existence of a fundamental solution is a division problem. The paper defines a tempered distribution $\\hat E_{\\log}$ by pairing a Schwartz function $\\varphi$ with $\\frac12(\\varphi(\\xi)-\\varphi(\\xi/|\\xi|))/\\log|\\xi|$ near the unit sphere and for $|\\xi|>2$; subtracting the radial projection removes the singularity at $|\\xi|=1$ and makes the Fourier multiplier $\\log(|\\xi|^2)$ invertible on Lizorkin test functions. It then shows $\\log(|\\xi|^2)\\hat E_{\\log}=1$ in $\\Psi'(\\mathbb{R}^d)$, so $E=\\mathcal F^{-1}\\hat E_{\\log}$ solves $\\log(-\\Delta)E=\\delta_0$ in $Z'(\\mathbb{R}^d)$. The Liouville theorem identifies every solution of the homogeneous equation with a single-layer distribution supported on the unit sphere, giving the full classification. For dimensions 1 and 2, subtracting a classical Helmholtz fundamental solution leaves remainder terms whose Fourier inverses are controlled by Bessel estimates, yielding $|E(x)|\\le C|x|^{-(d-1)/2}$ for $|x|\\ge 2$.","pith_inferences":["Beyond the paper: if the classification is correct, selecting a distinguished fundamental solution is the same as selecting a single-layer correction on the unit sphere; a natural next step is to identify the correction that enforces a Sommerfeld-type radiation condition.","Beyond the paper: the same division-problem template — regularize the symbol by subtracting a projection onto the singular set — should apply to other operators whose symbol is non-smooth on a submanifold rather than at a point.","Beyond the paper: the full Chen–Véron conjecture is now pinned to one object, the remainder $G_2$ from Section 3; proving or disproving that $G_2$ is a regular distribution near 0 would settle it."],"forward_implications":["In every dimension $d\\ge 1$, the logarithmic Laplacian has a fundamental solution in the Lizorkin distribution space, extending the previously known range $d\\ge 3$.","The full solution set is an affine space over the harmonic functions, and being harmonic for the logarithmic Laplacian is equivalent to being a generalized eigenfunction of $-\\Delta$ with eigenvalue 1.","In dimensions 1 and 2, there is a fundamental solution with the conjectured decay $|E(x)|\\lesssim |x|^{-(d-1)/2}$ away from the origin, so the Chen–Véron conjecture is reduced to local integrability near zero.","The logarithmic Laplacian is well-defined on $Z'(\\mathbb{R}^d)$, even though it cannot act on all tempered distributions because polynomials are not controlled.","The Liouville theorem for the logarithmic Laplacian holds without any boundedness assumption on the solution."],"supporting_citations":[{"why":"Establishes existence for d≥3 via Riesz potentials and poses the conjecture about decay in d=1,2 that this paper partially resolves.","marker":"[CV24]"},{"why":"Introduces the logarithmic Laplacian and its Fourier symbol 2 log(|ξ|), the operator whose fundamental solution is sought.","marker":"[CW19]"},{"why":"Supplies the division-problem viewpoint that fundamental solutions for constant-coefficient operators can be obtained by dividing by the symbol.","marker":"[Mal56]"},{"why":"Grounds the Malgrange–Ehrenpreis approach to division problems that this paper modifies.","marker":"[Ehr54]"},{"why":"Continues the division-problem theory and supports the renormalization strategy for singular symbols.","marker":"[Ehr55]"},{"why":"Provides the classification of Helmholtz fundamental solutions as the classical symbol divided by |ξ|^2−1 plus a single-layer distribution, which the paper compares against.","marker":"[Sch66]"},{"why":"Gives the representation of generalized eigenfunctions of the Helmholtz equation as oscillatory surface integrals, used for the Liouville statement.","marker":"[Agm99]"},{"why":"Supplies the Bessel-function estimates that convert the Fourier remainders into the decay bounds in dimensions 1 and 2.","marker":"[Gra14]"},{"why":"Defines the Lizorkin spaces and distributions that provide the solution space Z'(R^d).","marker":"[Sam02]"}],"fun_headline_variants":["Division problem approach yields fundamental solutions for log Laplacian","Logarithmic Laplacian fundamental solutions via frequency-space division","All dimensions: fundamental solutions for the logarithmic Laplacian","New proof: fundamental solutions for logarithmic Laplacian in every dimension","Inverting the logarithmic Laplacian: a division problem solution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the pairing written in Definition 4 is a well-defined tempered distribution; for a generic Schwartz test function the exterior integral diverges, so the proof depends on a regularization that the paper does not spell out.","fun_headline_variants_meta":{"raw":{"variants":["Division problem approach yields fundamental solutions for log Laplacian","Logarithmic Laplacian fundamental solutions via frequency-space division","All dimensions: fundamental solutions for the logarithmic Laplacian","New proof: fundamental solutions for logarithmic Laplacian in every dimension","Inverting the logarithmic Laplacian: a division problem solution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000484,"raw_usage":{"total_tokens":2365,"prompt_tokens":897,"completion_tokens":1468,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":1383}},"tokens_in":513,"tokens_out":1468,"duration_ms":10232,"temperature":1.0,"reasoning_tokens":1383,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:23:21.221167+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Definition 4 pairing on $\\varphi(\\xi)=e^{-|\\xi|^2}\\tau(\\xi/|\\xi|)$ with $\\tau\\in C^\\infty(S^{d-1})$ having nonzero spherical mean. The exterior term equals $\\frac12\\left(\\int_{S^{d-1}}\\tau\\,d\\sigma\\right)\\int_2^\\infty \\frac{e^{-r^2}-e^{-1}}{\\log r}r^{d-1}\\,dr$, which diverges to $-\\infty$; unless a finite-part prescription is supplied, $\\hat E_{\\log}$ is not a tempered distribution and the identity $\\log(|\\xi|^2)\\hat E_{\\log}=1$ in $\\Psi'$ is not established.","supporting_citations":[],"review_version":2}