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The logarithmic $p$-Laplacian on hyperbolic spaces

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.01079 v1 pith:A7BJRE7X submitted 2026-08-02 math.AP

Jorge J. Betancor , Lourdes Rodr\'{i}guez-Mesa This is my paper · ORCID
classification math.AP MSC 35R1147G2042B3747A60
keywords fractionalp-LaplacianlogarithmichyperbolicspaceextensionproblemintegralrepresentationPoissonkernelBesselnonlocaloperator
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 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.

What carries the argument

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 $\beta_{n,p}$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 1, Eq. (1.3)] 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.
  2. [Section 4, Eq. (4.6)] 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.
minor comments (5)
  1. [Title page] The title header contains a typo: 'SP ACES' should be 'SPACES'.
  2. [Throughout] 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'.
  3. [Section 1, Eqs. (1.1)-(1.2)] 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.
  4. [Section 1, definitions] 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.
  5. [Theorem 1.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the hyperbolic logarithmic p-Laplacian representation is derived from explicit kernel definitions and prior external results, not from the target conclusion.

full rationale

The paper contains no self-citations, and no claimed prediction reduces by construction to a fitted input or to a self-referential uniqueness statement. The operator log(-Delta_H^n)_p is defined as the s-derivative of the already-defined fractional p-Laplacian, and Theorem 1.2 derives its integral representation directly from that definition together with dominated convergence and explicit kernel asymptotics; the representation is not assumed at the outset. The kernel estimates (1.1)-(1.2) are cited from the independent prior work [11, Proposition 1.2], and the pointwise limit in (1.3) is asserted as a 'Note that' without proof or citation. This is a genuine support gap: the dominated-convergence passages in (4.2) and (4.7) rely on (1.3), and if that limit fails those proofs would be incomplete. However, this is a missing analytic verification about an explicit kernel, not a circular step: the target integral representation is not being used as its own input, and the constants A_{n,p}, alpha_{n,p}, beta_{n,p}, gamma_n, delta_{n,p} are produced by limits rather than fitted to the desired formula. Theorem 1.4 similarly uses the known Euclidean representation from [7, Theorem 1.1] as an external benchmark to derive the new extension characterization, which is a legitimate use of independent prior work. Overall, the derivation chain is not circular; it is dependent on external results whose completeness is a correctness concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the constants A_{n,p}, sigma_{n,p}, and the limit constants alpha, beta, gamma, delta are derived from the kernels, not chosen by hand. The paper relies on prior external results for kernel bounds, Bessel asymptotics, and the Euclidean pointwise representation, all of which are cited.

assumptions (4)
  • domain assumption The kernel estimates (1.1)-(1.2) and the limit (1.3) for K_{n,s,p} hold as stated in [11, Proposition 1.2].
    These bounds are used in every dominated convergence argument in Sections 3-5 to compute limits as s tends to 0 and as t tends to 0.
  • standard math The Bessel function identities and asymptotics (2.11)-(2.13) from [13] hold.
    The proofs of Theorem 1.1 and Theorem 1.2 replace K_{nu} by its asymptotic form to extract the leading s-dependence.
  • domain assumption The pointwise representation (6.1) of the Euclidean logarithmic p-Laplacian from [7, Theorem 1.1] is valid for f in C_alpha^c(R^n).
    Theorem 1.4 derives the Euclidean extension formula using this representation and the constants eta_{n,p} and delta_{n,p}.
  • domain assumption The estimate (4.6) for differences of Phi_p(z)=|z|^{p-2}z, cited from [15, Lemmas 2 and 3], holds for 1<p<infinity.
    This Lipschitz-type bound is used to dominate the singular integrals involving Phi_p(f(x)-f(y))-Phi_p(f(x)) in Sections 4 and 5.

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Pith. "Pith review of The logarithmic $p$-Laplacian on hyperbolic spaces." pith.science (2026). https://pith.science/paper/A7BJRE7X

@misc{pith2026260801079,
  author       = {Pith},
  title        = {Pith review of: The logarithmic $p$-Laplacian on hyperbolic spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7BJRE7X}},
  note         = {Machine review of arXiv:2608.01079}
}
abstract

In this paper, the logarithmic $p$-Laplacian operator $\log (-\Delta_{\mathbb H ^n})_p$ on the hyperbolic space $\mathbb H^n$, with $n\geq 2$, is introduced. We prove that if $f$ is a locally Lipschitz function of exponent $\alpha \in (0,1)$ with compact support in $\mathbb H^n$, then, for a suitable constant $A_{n,p}>0$, $$ \lim_{s\rightarrow 0^+}(-\Delta_{\mathbb H ^n})_p^sf(x)=A_{n,p}|f(x)|^{p-2}f(x),\quad x\in \mathbb H^n, $$ where $(-\Delta_{\mathbb H ^n})_p^s$ denotes the $s$-fractional $p$-Laplacian on $\mathbb H^n$. We establish a pointwise integral representation for the operator $\log (-\Delta_{\mathbb H ^n})_p=\frac{d}{ds}(-\Delta_{\mathbb H^n})_p^s\,_{|s=0}$. Furthermore, we show that $\log (-\Delta_{\mathbb H ^n})_p$ can be realized as the solution of a suitable extension problem and provide an extension theorem that yields the operator $\log (-\Delta)_p$ in $\mathbb R^n$. To the best of our knowledge, this property has not been established for the Euclidean logarithmic $p$-Laplacian $\log (-\Delta)_p$.

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