{"id":"e3e8d931-bb9f-4eee-bf89-13d1ef0e0a0b","arxiv_id":"2506.19311","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Bochner integral formula defines the logarithmic Laplacian on complete Riemannian manifolds, with pointwise kernel formulas under Ricci lower bounds and sharp estimates on hyperbolic space.","lead":"This paper defines the logarithmic Laplacian, a nonlocal operator, on curved spaces using a heat-semigroup integral formula, and derives explicit kernel formulas on hyperbolic space. It gives a unified definition on compact and noncompact manifolds and connects two different definitions via the concept of diffusion mass loss.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The zero eigenvalue is silently excluded: on compact or finite-volume complete manifolds p_t(x,y) does not decay, so the Bochner integral diverges for nonzero-mean f and the pointwise formula's K2 is infinite.","rationale":"The reader's weakest_assumption is correct and, in fact, the defect is more basic: the same zero-mode issue invalidates the abstract Bochner formula in Theorem 1.1, not just the pointwise kernel formula in Theorem 1.10/3.4. This is an internal consistency problem, not a disagreement with community consensus: the paper explicitly claims to cover both compact and noncompact settings, and its definition of Hlog on compact manifolds includes constants, but the Frullani representation of log λ has no meaning at λ=0. The compact-manifold section defines log(-Δ) by an eigen-sum over positive eigenvalues, so the operator itself is well-defined; the missing step is that the Bochner and pointwise representations only hold on the orthogonal complement of the kernel, or under an assumption excluding L^2-harmonic functions. The hyperbolic-space results are not affected because σ(-Δ_Hn) starts at (n-1)^2/4 > 0, and the Euclidean recovery has no zero mode either. The fix is straightforward: state Theorem 1.1, Theorem 1.10, and Theorem 3.4 for f in the mean-zero subspace on compact or finite-volume manifolds, or add a hypothesis such as 'no nontrivial L^2-harmonic functions' for the noncompact statement. With that repair, the constructive content—the spectral/Bochner framework and the Euclidean and hyperbolic kernel computations—would be valuable. I therefore keep the reader's CONDITIONAL verdict; no further adjustment is needed.","tokens_in":39142,"tokens_out":8291,"duration_ms":92840,"concrete_test":"Take M=S^n with the round metric and f≡1. f∈Hlog(M) since the constant is in the domain of the compact spectral definition, but the Bochner integrand at λ=0 is (e^{-t}-1)/t, and ∫_1^∞ (e^{-t}-1)/t dt = -∞, so Theorem 1.1/2.12 fails. In the same example, use the eigen-expansion p_t(x,y)=1/Vol(M)+Σ_{j≥1} e^{-λ_j t} φ_j(x)φ_j(y) to compute K2(x,y)=∫_1^∞ p_t(x,y)/t dt; the first term gives (1/Vol(M)) log T + O(1), so K2 diverges and the Theorem 1.10/3.4 formula is not defined. The flat torus T^n gives the same conclusion with an even simpler eigenbasis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 1.1 and 1.10/3.4 are not valid as stated on 'any complete Riemannian manifold.' Their proofs put the spectral measure at λ=0 into the Frullani formula, but log 0 is not defined: for the constant function, e^{-t}-e^{-tλ} = e^{-t}-1 and ∫_1^∞ (e^{-t}-1)/t dt diverges (integrand ~ -1/t). The proof of Theorem 2.12's long-time estimate uses E1(λ) ≤ C(1+|log λ|), which is false at λ=0 since E1(0)=∞. The same zero-mode failure appears in Theorem 3.4: under Ric_g ≥ -(n-1)k, stochastic completeness only gives ∫ p_t = 1, not decay. On any compact manifold, and also on finite-volume complete manifolds such as finite-area hyperbolic surfaces, p_t(x,y) → 1/Vol(M) > 0, so K2(x,y)=∫_1^∞ p_t(x,y)/t dt = ∞; the line in the proof saying the long-time interchange is 'justified by rapid decay of p_t(x,y) as t→∞' is therefore incorrect. The statement should restrict to the spectral subspace orthogonal to L^2-harmonic functions (mean-zero on compact/finite-volume manifolds) or assume there are no nontrivial L^2-harmonic functions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a Bochner integral formula for the logarithmic Laplacian on a complete Riemannian manifold, log(-Δ) = ∫_0^∞ (e^{-t} I - e^{tΔ})/t dt, derived from the scalar Frullani identity via the spectral theorem. It shows that on Euclidean space this formula recovers the pointwise kernel representation of Chen–Weth, and under a Ricci lower bound it derives a pointwise representation involving heat-kernel integrals K1 and K2. The paper also compares spectral and heat-kernel definitions of fractional and logarithmic Laplacians, relates their discrepancy to stochastic completeness, and obtains sharp kernel asymptotics and L^p-continuity results on real hyperbolic space. The Euclidean recovery and the hyperbolic-space kernel estimates are the strongest parts of the manuscript.","tokens_in":39399,"tokens_out":3386,"duration_ms":37384,"significance":"If the main results were valid in the stated generality, the Bochner formula would give a unified functional-calculus definition of the logarithmic Laplacian on manifolds and a practical route to pointwise kernel formulas, genuinely useful for PDE and geometric analysis. The paper is self-contained, uses no fitted parameters, and verifies the new definition against the known Euclidean formula. The hyperbolic-space kernel estimates (Propositions 4.6 and 4.8) and the resulting pointwise/L^p theory are substantial contributions. However, the central claims as stated extend to manifolds with an L^2 kernel for Δ, where the construction breaks down; this is a load-bearing gap that must be repaired before the main theorems can be accepted in their current form.","major_comments":[{"comment":"The proof of Theorem 2.12 uses the bound E1(λ) ≤ C(1+|log λ|) for all λ > 0, but this fails at λ = 0, where E1(0) = ∞. For any f with nonzero projection onto ker Δ, e.g. a nonzero constant on a compact manifold, the scalar integrand at λ = 0 is (e^{-t} - 1)/t, whose integral from 1 to ∞ diverges like -∫ t^{-1} dt. Consequently the Bochner integral does not converge for such f, and Theorem 1.1 as stated 'for every f ∈ Hlog(M)' is false. The theorem should be restricted to the spectral subspace orthogonal to ker Δ, and the definition of Hlog in Definition 1.3 must explicitly exclude or separately handle the point λ = 0.","section":"§2.2, Theorem 2.12 (Theorem 1.1)"},{"comment":"The pointwise formula of Theorem 3.4 requires K2(x,y) = ∫_1^∞ p_t(x,y)/t dt to be finite, but the stated hypothesis Ric_g ≥ -(n-1)k does not imply any decay of p_t as t → ∞. It only gives stochastic completeness, i.e. ∫_M p_t dvol = 1. On a compact manifold, or a finite-volume complete manifold with such a curvature bound, p_t(x,y) → 1/Vol(M) > 0, so K2(x,y) = ∞ for all x,y and the formula fails for functions with nonzero mean. The line in the proof saying the long-time interchange is 'justified by rapid decay of p_t(x,y) as t → ∞' is therefore incorrect. The theorem needs an additional assumption, such as the absence of nontrivial L^2-harmonic functions, a positive bottom of the L^2-spectrum, or restriction of f to the mean-zero subspace.","section":"§3.2, Theorem 3.4 (Theorem 1.10)"},{"comment":"The Li–Yau estimate displayed in (3.4) contains the factor exp(-μ1(M)t) with μ1(M) = inf σ(-Δ) ≥ 0. When μ1(M) = 0, which is the case for Euclidean space and for compact manifolds, this factor provides no decay, so the estimate cannot justify the Fubini interchange needed for the K2 term. The proof of Theorem 3.4 implicitly assumes a spectral gap or heat-kernel decay that is not part of the hypotheses; this is the technical source of the failure described in the previous comment.","section":"§3.2, Eq. (3.4)"}],"minor_comments":[{"comment":"The sentence 'On a compact manifold, such as colsed manifold' contains a typo: 'colsed' should be 'closed'.","section":"§1, p. 3"},{"comment":"The word 'Silimarly' appears in the proof of Lemma 2.13; it should be 'Similarly'.","section":"§2.3, proof of Lemma 2.13 and Theorem 2.15"},{"comment":"In the display for E({0})f, the integral is written as ∫_M f(x) dvol_Hn(x); since the manifold is a general compact M, this should be dvol(y) (or dvol(x)) and the subscript Hn should be removed.","section":"§3.1, p. 30"},{"comment":"There are several small typographical errors: 'sufficies' should be 'suffices' in the proof of Proposition 4.19, and 'remind' should be 'remainder' in the proof of Proposition 4.17.","section":"§4.4"},{"comment":"The notation Klog(x,y) = ∫_0^∞ (e^{-t}δ_x(y) - p_t(x,y))/t dt uses δ_x as a distribution; it would be clearer to state that the formula holds in the sense of distributions or to specify the integrability condition (3.3) before using it.","section":"§3.2"}],"recommendation":"major_revision","confidential_remarks":"The zero-eigenvalue issue is substantive but clearly fixable by restating the theorems on the spectral subspace orthogonal to ker Δ or under a positive spectral gap. The hyperbolic-space section and the Euclidean recovery are convincing and should be preserved. I do not see a novelty concern, and the acknowledgment of Pramanik's independent work is appropriate. The paper's scope fits the journal, provided the main theorems are corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea here is good and new: writing log(−Δ) as the Frullani-type Bochner integral ∫₀^∞ (e^{-t} − e^{tΔ})/t dt and using heat-kernel estimates to get pointwise formulas is a real step forward for noncompact curved spaces. The Euclidean recovery is a convincing sanity check, and the hyperbolic-space computations look careful and correct. I also like the mass-loss comparison between spectral and heat-kernel definitions; that is a clean, parameter-free observation. The paper is self-contained, honestly written, and does not dress up fitted results as predictions. It deserves a serious referee.\n\nThe soft spot is not minor, though. Theorem 1.1 as stated is false for functions in the kernel of Δ, for example constants on a compact manifold. The Frullani identity is for λ > 0; at λ = 0 the integrand e^{-t} − 1 behaves like −1/t and the t-integral diverges. The proof even uses E₁(λ) ≤ C(1 + |log λ|), which is false at λ = 0. The fix is straightforward: state the Bochner formula on the spectral subspace orthogonal to the L²-harmonic functions, or assume the manifold has no nonzero L²-harmonic functions. The same issue invalidates the pointwise formula in Theorem 3.4 on compact and finite-volume complete manifolds, where p_t(x,y) → 1/Vol(M) > 0 and K₂(x,y) = ∫₁^∞ p_t(x,y)/t dt = ∞. The line in the proof that the interchange is justified by 'rapid decay of p_t(x,y) as t → ∞' is simply wrong under the stated Ricci lower bound alone. Again a spectral-subspace restriction or a no-harmonic-functions assumption fixes it.\n\nSo: the hyperbolic-space results stand, the Euclidean check stands, and the Bochner formula is a useful tool once restricted correctly. The paper overcovers itself in Theorems 1.1 and 1.10/3.4. I would send it to review, with the clear expectation that the author restricts the domain of the theorems. This is the kind of paper that a knowledgeable referee can fix without changing the substance.","headline":"A genuinely useful Bochner-integral framework for log(-Delta) on manifolds, but the headline theorems overclaim: they silently fail on compact and finite-volume manifolds because the zero mode makes the time integrals diverge.","tokens_in":742,"tokens_out":747,"would_cite":true,"duration_ms":23645,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","58J35","47A60","58J50","53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces a Bochner integral formula that defines log(−Δ) on any complete Riemannian manifold, unifying the Euclidean, compact, and noncompact cases and yielding explicit pointwise kernels under Ricci lower bounds.","keywords":["logarithmic Laplacian","Bochner integral","heat kernel","fractional Laplacian","Riemannian manifolds","functional calculus","hyperbolic space","stochastic completeness"],"falsifier":"Take a closed manifold, for instance a round sphere, which satisfies $\\mathrm{Ric}_g \\ge -(n-1)k$, and apply Theorem 1.10 to the constant function $f \\equiv 1$. The spectral logarithmic Laplacian is $0$, but the claimed formula contains $-\\int_M K_2(x,y)\\,d\\mathrm{vol}(y)$, and because $\\int_M p_t(x,y)\\,d\\mathrm{vol}(y)=1$ on a closed manifold, this term equals $\\int_1^\\infty t^{-1}\\,dt = \\infty$, so the identity would force $0 = -\\infty$.","tokens_in":38919,"feed_emoji":"🧮","tokens_out":10168,"duration_ms":89117,"temperature":0.7,"pith_summary":"This paper proposes a single Bochner integral formula, $\\log(-\\Delta)f = \\int_0^\\infty (e^{-t}f - e^{t\\Delta}f)\\,dt/t$, as the definition of the logarithmic Laplacian on every complete Riemannian manifold. The same formula recovers the classical pointwise expression on Euclidean space, and under a Ricci lower bound it yields an explicit pointwise kernel representation split into short- and long-time heat-kernel parts. The author further shows that the difference between spectral and heat-kernel definitions of fractional and logarithmic Laplacians is a multiplication operator built from the mass-loss function, so the two definitions coincide precisely when the manifold is stochastically complete. On real hyperbolic space, sharp heat-kernel asymptotics give kernel estimates that support a pointwise theory for weighted integrable functions with Dini continuity.","feed_headline":"One integral formula defines the logarithmic Laplacian on manifolds","feed_subtitle":"It recovers the Euclidean operator and gives pointwise heat-kernel formulas under Ricci lower bounds.","key_machinery":"The load-bearing object is the scalar identity $\\log\\lambda = \\int_0^\\infty (e^{-t}-e^{-\\lambda t})\\,dt/t$, fed through the spectral theorem so that $e^{-\\lambda t}$ becomes the heat semigroup $e^{t\\Delta}$. This yields the Bochner formula for $\\log(-\\Delta)$. The argument splits the time integral at $t=1$: the short-time piece pairs the heat kernel with $f(x)-f(y)$ and becomes $K_1$; the long-time piece pairs with $f(y)$ and becomes $K_2$. Convergence is controlled by Gaussian heat-kernel upper bounds and volume comparison estimates, while the spectral-versus-heat-kernel comparison is carried by the mass-loss function; its large-time limit is the non-explosion probability and encodes stochastic completeness.","core_discovery":"The paper's central claim is that the logarithmic Laplacian can be defined on any complete Riemannian manifold by the Bochner integral $\\log(-\\Delta)f = \\int_0^\\infty (e^{-t}f - e^{t\\Delta}f)\\,dt/t$, converging in $L^2$ for $f$ in the logarithmic Sobolev space $H_{\\log}(M)$. Applied inside the spectral calculus, the scalar identity $\\log\\lambda = \\int_0^\\infty (e^{-t}-e^{-\\lambda t})\\,dt/t$ turns this abstract operator into a concrete object controlled by heat-kernel integrals. Under $\\mathrm{Ric}_g \\ge -(n-1)k$, the paper derives the pointwise representation $\\log(-\\Delta)_{\\mathrm{spec}} f(x) = \\int_M K_1(x,y)(f(x)-f(y))\\,d\\mathrm{vol}(y) - \\int_M K_2(x,y)f(y)\\,d\\mathrm{vol}(y) + \\Gamma'(1)f(x)$ for H\\\"older compactly supported $f$, with $K_1 = \\int_0^1 p_t(x,y)\\,dt/t$ and $K_2 = \\int_1^\\infty p_t(x,y)\\,dt/t$. The same framework shows that the discrepancy between spectral and heat-kernel definitions is a multiplication operator involving the mass-loss function $r(t,x)=1-\\int_M p_t(x,y)\\,d\\mathrm{vol}(y)$, which vanishes exactly when the manifold is stochastically complete.","pith_inferences":[],"forward_implications":["The logarithmic Laplacian now has a definition on any complete Riemannian manifold, with $H_{\\log}(M)$ as its natural domain, so Dirichlet and spectral problems can be posed on curved spaces.","On manifolds with Ricci curvature bounded below, the operator has an explicit pointwise integral formula, making it as accessible as the fractional Laplacian for PDE analysis.","Spectral and heat-kernel definitions of both fractional and logarithmic Laplacians coincide exactly on stochastically complete manifolds; on non-stochastically complete ones their difference is a multiplication operator determined by the mass-loss function.","On hyperbolic space $\\mathbb{H}^n$, the pointwise formula extends beyond compactly supported smooth functions to weighted $L^1$ functions that are locally Dini continuous, and $\\log(-\\Delta_{\\mathbb{H}^n})f$ lies in $L^p$ for $1