{"id":"16111de6-3cc0-4718-8661-46ce46f1e037","arxiv_id":"2608.01079","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The logarithmic p-Laplacian on hyperbolic space is introduced, with a full pointwise integral representation and an extension problem characterization that extends to Euclidean space.","lead":"The paper defines the logarithmic p-Laplacian on hyperbolic space as the slope of the fractional p-Laplacian as its exponent tends to zero. It gives explicit integral formulas for this operator and shows an extension problem that also fills a gap for the Euclidean case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The weakest point is the unproved pointwise limit (1.3), K_{n,s,p}(ρ)/s → σ_{n,p}P_n(ρ,0), on which the dominated-convergence steps in Theorem 1.2(b) rest; (1.1)-(1.2) are cited from [11], but (1.3) is only asserted.","rationale":"The reader identified the kernel estimates (1.1)-(1.2) and the limit (1.3) as the weakest assumption, which matches my reading. However, I sharpen the concern: (1.1)-(1.2) are genuinely cited from [11, Proposition 1.2], while (1.3) appears in the text only as an unproved 'Note'. Since (1.3) is used directly in the dominated-convergence limits that produce the integral representation, it is the single most load-bearing unsecured premise. The rest of the paper is detailed and self-consistent, with explicit constants and no internal contradiction found; the extension theorems and the Euclidean specialization are plausible corollaries of the stated identities. The concern is therefore not a demonstrated error but a missing verification of a central externally supported limit, so acceptance should be conditional on confirming or proving (1.3).","tokens_in":31515,"tokens_out":24829,"duration_ms":208713,"concrete_test":"Check whether [11, Proposition 1.2] actually states (1.3); if it does not, derive (1.3) from the explicit kernel formulas in Section 1 using Bessel-function asymptotics as s→0, for both odd and even n. Independently, evaluate K_{n,s,p}(ρ)/s for small s at fixed ρ (e.g., 0.1, 1, 10) for n=2,3 and p=3/2,2,3, and compare with σ_{n,p}P_n(ρ,0); also verify the s-uniform bounds (1.1)-(1.2) on these examples for s∈(0,1/2). If the limit or the uniform bounds fail, Theorem 1.2(b) is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.2(b) is proved by splitting (-Δ_{H^n})_p^s f into H1+H2+H3 and passing to the limit in H1(s)/s and H2(s)/s via dominated convergence; see (4.2) and (4.7). These passages use two ingredients: the s-uniform bounds (1.1)-(1.2) and the pointwise limit (1.3). The bounds are explicitly cited from [11, Proposition 1.2], but (1.3) is merely 'noted' in the text and is not derived or cited. If [11] does not already contain this limit, or if the normalization σ_{n,p} differs from the one in (1.3), then the conclusions (4.2) and (4.7) are unproved steps rather than consequences of the cited proposition. The kernel definitions in Section 1 are explicit enough that (1.3) can and should be checked directly; the final integral representation in Theorem 1.2(b) collapses if this limit fails. This is a concern about external support and a missing internal proof, not an observed contradiction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the logarithmic p-Laplacian on the hyperbolic space H^n as the derivative at s=0 of the fractional p-Laplacian, after subtracting the limit A_{n,p}Φ_p(f(x)). The main results are: (i) Theorem 1.1 computes the limit of (−Δ_{H^n})_p^s f as s→0+ with explicit constants A_{n,p} in (3.14) and (3.25); (ii) Theorem 1.2 establishes a pointwise integral representation of the logarithmic operator, with a local version depending on log R and a global version with a compensated integral; (iii) Theorem 1.3 characterizes the operator as the limit of a Poisson-extension expression; (iv) Theorem 1.4 gives an analogous extension formula for the Euclidean logarithmic p-Laplacian, completing a result from the literature. The proofs rely on detailed asymptotic estimates for the kernel K_{n,s,p}, a Leibniz-type formula for (sinhρ)^{-1}∂_ρ, and dominated convergence arguments.","tokens_in":31791,"tokens_out":15091,"duration_ms":118216,"significance":"If the results are correct, the paper provides the first extension-problem realization of the logarithmic p-Laplacian in the hyperbolic setting and also supplies an extension formula for the Euclidean logarithmic p-Laplacian, which had been missing from prior work. The explicit constants A_{n,p} in (3.14) and (3.25) are a useful feature. The main theorems are quantitative and the proofs are largely self-contained, building on external results [11], [15], and [7]. The principal caveats are the unproved pointwise limit (1.3) and the use of real-variable inequalities for complex-valued functions; both are fixable.","major_comments":[{"comment":"The pointwise limit lim_{s→0+} K_{n,s,p}(ρ)/s = σ_{n,p}P_n(ρ,0) is asserted with the phrase 'Note that' and is neither proved nor cited. This limit is load-bearing: it is used in the dominated convergence arguments leading to (4.2) and (4.7) in the proof of Theorem 1.2(b), and it is needed for the proof of Theorem 1.3 as well. Although the limit is plausible from the explicit kernel formulas and the asymptotics of the modified Bessel functions, the paper should supply a direct verification or a precise reference in [11] with the same normalization. Without this, the integral representation in Theorem 1.2(b) is not fully established.","section":"Section 1, Eq. (1.3)"},{"comment":"The main theorems are stated for complex-valued functions in C_c(H^n)∩Lip^α_loc, but the pointwise inequality (4.6) is quoted from [15, Lemmas 2 and 3], which are real-variable results. The same inequality is applied to complex arguments in the proofs of Theorem 1.2(b) and Theorem 1.3. The authors should either restrict the main statements to real-valued functions or add a short argument showing that the estimates extend to complex-valued Φ_p; the extension is standard via a path-integral estimate, but it is not automatic and should be addressed.","section":"Section 4, Eq. (4.6)"}],"minor_comments":[{"comment":"The title header contains a typo: 'SP ACES' should be 'SPACES'.","section":"Title page"},{"comment":"There are several typographical errors: 'choosen' should be 'chosen', 'ocurrence' should be 'occurrence', 'symplest' should be 'simplest', and 'Faà di Brunos's formula' should be 'Faà di Bruno's formula'.","section":"Throughout"},{"comment":"The lower bound in (1.1) is written as 'sρ^{-n-sp}/C'; writing it as (s/C)ρ^{-n-sp} would be clearer and would avoid possible ambiguity.","section":"Section 1, Eqs. (1.1)-(1.2)"},{"comment":"The definition of (−Δ_{H^n})_p^s includes 'P.V.', but in the proofs the integrals are shown to be absolutely convergent for the considered functions; this could be noted explicitly to avoid confusion.","section":"Section 1, definitions"},{"comment":"The statement says 'for a certain α_{n,p}∈R', but the proof at the end of Section 6 computes α_{n,p} explicitly; including this value in the statement would make the result more informative.","section":"Theorem 1.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution and the central results appear defensible. The main concern is the unproved limit (1.3), which is essential for the dominated convergence steps; this should be fixed before publication. The complex-valued extension of (4.6) also needs a short justification. The authors should also clarify the relation of their hyperbolic logarithmic operator to the results in [5, Section 4] on general manifolds, since the present paper cites that section without discussing the overlap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good paper, worth refereeing. It introduces log(-Δ_H^n)_p, proves that the fractional p-Laplacian rescaled by s converges to A_{n,p}Φ_p(f(x)) with an explicit constant, gives a pointwise integral representation, solves an extension problem, and adds a Euclidean extension theorem for log(-Δ)_p that was missing from the literature. The proofs are detailed and self-consistent. The hyperbolic results are genuinely new, and Theorem 1.4 is a useful completion of [7]. Credit is earned here.\n\nThe stress-test flags (1.3), the pointwise limit K_{n,s,p}(ρ)/s → σ_{n,p}P_n(ρ,0), which the text merely \"notes\" and then uses in the dominated-convergence steps of Theorem 1.2(b). The concern is fair as a matter of exposition: a load-bearing limit should be proved or cited, and a referee should ask for a short derivation. But from the explicit kernel formulas and the standard continuity of the Bessel expression in s for fixed ρ>0, the limit is immediate; I do not think the theorems are at risk. It is a minor gap, not a structural flaw.\n\nTwo smaller issues. First, several constants in Theorems 1.2 and 1.3 are only shown to exist as limits, never computed; acceptable, but it limits immediate applications. Second, the positioning against [5] is thinner than it should be. Since [5] already treats logarithmic operators on manifolds, the paper should state plainly what is new for p>1 on hyperbolic space and why the earlier work does not cover it. This does not affect validity.\n\nThe citation pattern looks honest: no self-citations, the asymptotic kernel bounds (1.1)-(1.2) come from [11], and the pointwise bound on Φ_p differences is from [15]. I found no circularity.\n\nBottom line: send this to a serious referee. With a proof of (1.3) and a better comparison with [5], it will be a clean, citable contribution to nonlocal operators on manifolds.","headline":"A solid, useful paper that defines the logarithmic p-Laplacian on hyperbolic spaces with a pointwise representation and an extension problem; the main caveat is a missing proof of a key pointwise kernel limit, but that limit is readily checked and does not threaten the theorems.","tokens_in":32338,"tokens_out":4905,"would_cite":true,"duration_ms":43312,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","47G20","42B37","47A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines the logarithmic p-Laplacian on hyperbolic space as the derivative at s=0 of the fractional p-Laplacian and proves an explicit pointwise integral representation.","keywords":["fractional p-Laplacian","logarithmic p-Laplacian","hyperbolic space","extension problem","integral representation","Poisson kernel","Bessel kernel","nonlocal operator"],"falsifier":"Take a compactly supported radial function $f$ on $\\mathbb H^2$ or $\\mathbb H^3$, compute $(-\\Delta_{\\mathbb H^n})_p^s f(x)$ numerically as $s\\to 0^+$, subtract $A_{n,p}\\Phi_p(f(x))$, divide by $s$, and compare with the right-hand side of the formula; a mismatch growing like $\\log R$ as the support radius $R$ changes would show the logarithmic term is wrong.","tokens_in":31325,"feed_emoji":"📐","tokens_out":8573,"duration_ms":69708,"temperature":0.7,"pith_summary":"The paper introduces the logarithmic p-Laplacian on hyperbolic space $\\mathbb H^n$ for $n\\ge 2$ as the derivative at $s=0$ of the fractional p-Laplacian. For compactly supported functions that are locally $\\alpha$-Hölder for some $\\alpha\\in(0,1)$, it proves that the rescaled fractional operator tends to a constant multiple of $|f|^{p-2}f$, and that the next term in $s$ is given by an explicit integral involving a Poisson-type kernel. It also shows that this logarithmic operator solves an extension problem, and adapts the same argument to give an extension characterization of the Euclidean logarithmic p-Laplacian, a property that had been missing. A reader should care because the result turns a formal derivative of a nonlocal operator into a usable formula and opens a route to boundary-value problems for this operator on hyperbolic space.","feed_headline":"Hyperbolic log p-Laplacian gets an explicit integral formula","feed_subtitle":"The derivative at s=0 of the fractional p-Laplacian is computed pointwise and via an extension problem.","key_machinery":"The central object is the hyperbolic fractional $p$-Laplacian kernel $K_{n,s,p}(\\rho)$, built from Bessel functions and the operator $(\\frac{1}{\\sinh\\rho}\\partial_\\rho)$, together with its companion Poisson-type kernel $P_n(\\rho,t)$. The argument is carried by three asymptotic facts: near $\\rho=0$, $K_{n,s,p}(\\rho)$ behaves like $s\\rho^{-n-sp}$; at infinity it behaves like $s\\rho^{-(1+sp)/2}e^{-(n-1)\\rho}$; and $K_{n,s,p}(\\rho)/s$ converges pointwise to $\\sigma_{n,p}P_n(\\rho,0)$. These bounds turn every limit into a dominated-convergence calculation, and Lemma 2.1's expansion of the operator $(\\frac{1}{\\sinh\\rho}\\partial_\\rho)^m$ into a finite sum lets the authors peel off logarithmic singular terms and identify the constants $A_{n,p}$, $\\alpha_{n,p}$, and $\\beta_{n,p}$.","core_discovery":"On $\\mathbb H^n$, for $f\\in C_c(\\mathbb H^n)\\cap \\mathrm{Lip}^\\alpha_{\\mathrm{loc}}(\\mathbb H^n)$, the paper establishes that $\\lim_{s\\to 0^+} (-\\Delta_{\\mathbb H^n})_p^s f(x)=A_{n,p}\\Phi_p(f(x))$, with $\\Phi_p(z)=|z|^{p-2}z$, and defines the logarithmic operator by subtracting this term and dividing by $s$. The principal formula states that $$(\\log(-\\Delta_{\\mathbb H^n})_p f)(x)=\\alpha_{n,p}\\Phi_p(f(x))+\\sigma_{n,p}\\left[\\int_{B(x,1)}\\Phi_p(f(x)-f(y))P_n(d_{\\mathbb H^n}(x,y),0)\\,dy+\\int_{\\mathbb H^n\\setminus B(x,1)}(\\Phi_p(f(x)-f(y))-\\Phi_p(f(x)))P_n(d_{\\mathbb H^n}(x,y),0)\\,dy\\right],$$ where $P_n$ is the hyperbolic Poisson-type kernel and $\\sigma_{n,p}$ is the $s$-normalized limit of the fractional kernel. The proof also shows the operator appears as a $t\\to 0^+$ limit of an extension involving $P_n$, and the same argument yields an extension characterization of the Euclidean logarithmic $p$-Laplacian.","pith_inferences":["A natural next step, implicit in the representation, is to extend the operator to functions with weaker smoothness by reading the two integrals as a principal value; the paper's local Hölder condition is sufficient but not necessary for the formulas.","For $p=2$ the new representation should coincide with the logarithmic Laplace-Beltrami operator on hyperbolic space; checking this explicitly would test the normalization of the Poisson kernel and the constants.","The method is not tied to the hyperboloid model: any noncompact symmetric space whose fractional kernel obeys the same two-scale bounds and pointwise limit should admit an analogous logarithmic p-Laplacian and extension problem."],"forward_implications":["The logarithmic p-Laplacian on $\\mathbb H^n$ is well defined on $C_c(\\mathbb H^n)\\cap\\mathrm{Lip}^\\alpha_{\\mathrm{loc}}(\\mathbb H^n)$ and acts through a locally integrable kernel representation, so Dirichlet-type problems for this operator can be formulated on hyperbolic domains.","The same derivative-at-zero procedure yields an extension problem whose solution recovers the operator, providing a tool for studying boundary regularity and maximum principles in the hyperbolic setting.","The Euclidean logarithmic p-Laplacian also admits an extension characterization, filling a gap noted for the Euclidean operator.","The constants $A_{n,p}$, $\\alpha_{n,p}$, and $\\beta_{n,p}$ are determined by explicit recursive formulas in the proof, so concrete computations on $\\mathbb H^2$ and $\\mathbb H^3$ are possible."],"supporting_citations":[{"why":"Supplies the kernel estimates (1.1)-(1.2) and the pointwise limit (1.3) on which every dominated-convergence step rests.","marker":"[11]"},{"why":"Defines the Euclidean logarithmic p-Laplacian and its integral representation, the result that Theorem 1.4 complements.","marker":"[7]"},{"why":"Introduces the fractional Laplacian on noncompact manifolds and provides the hyperbolic-space kernel framework used here.","marker":"[1]"},{"why":"Gives the extension-problem method that Theorems 1.3 and 1.4 adapt.","marker":"[2]"},{"why":"Provides the extension characterization of the logarithmic Laplacian that serves as the template for the hyperbolic argument.","marker":"[3]"},{"why":"Supplies inequality (4.6) controlling $\\Phi_p$ differences, needed to pass the $s$-limit inside the singular integrals.","marker":"[15]"},{"why":"Three representations of the fractional p-Laplacian that motivate defining the logarithmic operator as the derivative at $s=0$.","marker":"[6]"}],"fun_headline_variants":["Hyperbolic log p-Laplacian: pointwise formula and extension","Derivative of fractional p-Laplacian on H^n yields log operator","Log p-Laplacian on hyperbolic space gets explicit integral form","New limit: log p-Laplacian from fractional kernel on H^n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a pair of asymptotic bounds and a pointwise limit for the fractional p-Laplacian kernel; if those fail in some dimension or for some $p$, the constants $A_{n,p}$ and the integral representation are not established.","fun_headline_variants_meta":{"raw":{"variants":["Hyperbolic log p-Laplacian: pointwise formula and extension","Derivative of fractional p-Laplacian on H^n yields log operator","Log p-Laplacian on hyperbolic space gets explicit integral form","New limit: log p-Laplacian from fractional kernel on H^n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000477,"raw_usage":{"total_tokens":2447,"prompt_tokens":1111,"completion_tokens":1336,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":1260}},"tokens_in":727,"tokens_out":1336,"duration_ms":9265,"temperature":1.0,"reasoning_tokens":1260,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:13:17.375248+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a compactly supported radial function $f$ on $\\mathbb H^2$ or $\\mathbb H^3$, compute $(-\\Delta_{\\mathbb H^n})_p^s f(x)$ numerically as $s\\to 0^+$, subtract $A_{n,p}\\Phi_p(f(x))$, divide by $s$, and compare with the right-hand side of the formula; a mismatch growing like $\\log R$ as the support radius $R$ changes would show the logarithmic term is wrong.","supporting_citations":[{"cited_title":"The fractional p - L aplacian on hyperbolic spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the kernel estimates (1.1)-(1.2) and the pointwise limit (1.3) on which every dominated-convergence step rests."},{"cited_title":"o., Jarohs, S., and Sk, F","cited_arxiv_id":null,"evidence_quote":"Defines the Euclidean logarithmic p-Laplacian and its integral representation, the result that Theorem 1.4 complements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the fractional Laplacian on noncompact manifolds and provides the hyperbolic-space kernel framework used here."},{"cited_title":"An extension problem related to the fractional L aplacian","cited_arxiv_id":null,"evidence_quote":"Gives the extension-problem method that Theorems 1.3 and 1.4 adapt."},{"cited_title":"H\\\"older estimates for viscosity solutions of equations of fractional p - L aplace type","cited_arxiv_id":null,"evidence_quote":"Supplies inequality (4.6) controlling $\\Phi_p$ differences, needed to pass the $s$-limit inside the singular integrals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Three representations of the fractional p-Laplacian that motivate defining the logarithmic operator as the derivative at $s=0$."}],"review_version":1}