arxiv: 2605.12050 · v1 · submitted 2026-05-12 · 🧮 math.AP

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· Lean Theorem

On the fractional logarithmic p-Laplacian

Abdelhamid Gouasmia, Anass Ouannasser, Anouar Bahrouni, Hichem Hajaiej

Pith reviewed 2026-05-13 04:39 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional logarithmic p-Laplaciannonlocal operatorscritical embeddingsPohozaev identitiesenergy spaceseigenvalue problemsintegral representations

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The pith

The fractional logarithmic p-Laplacian admits an explicit integral representation combining the standard fractional p-Laplacian with a nonlocal logarithmic term.

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

The paper defines the fractional logarithmic p-Laplacian by taking the derivative of the fractional p-Laplacian with respect to its order parameter t and evaluating at t equals s. It derives an integral formula that writes the new operator as a multiple of the usual fractional p-Laplacian minus a principal-value integral carrying the logarithm of the distance between points. This formula establishes that the operator is nonlocal and of logarithmic type, which in turn permits the construction of natural energy spaces on R^N and on Lipschitz domains. Within those spaces the authors prove density, continuity, Pohozaev identities, Díaz-Saa inequalities, and compact embedding into L^p even at the critical exponent p_s^* equals Np over N minus sp. The same framework yields existence, uniqueness, and boundedness results for the associated Dirichlet eigenvalue problem.

Core claim

Differentiating the fractional p-Laplacian (-Δ)_p^t with respect to t at t equals s produces the operator (-Δ)_p^{s+log} that satisfies the identity B(N,s,p) times (-Δ)_p^s u(x) minus p C(N,s,p) times the principal-value integral of |u(x) minus u(y)|^{p-2} (u(x) minus u(y)) ln|x-y| divided by |x-y|^{N+sp} dy. The resulting representation shows the operator is nonlocal and logarithmic, supports the introduction of adapted function spaces, and allows compactness of the embedding at the critical exponent p_s^* equals Np over (N minus sp), a feature that fails for the classical fractional Sobolev spaces.

What carries the argument

The first-order derivative with respect to the fractional order t of the fractional p-Laplacian evaluated at t equals s, whose integral representation introduces the extra logarithmic kernel term.

If this is right

The operator is nonlocal and constitutes a nonlinear analogue of the fractional logarithmic Laplace operator.
Natural energy spaces admit density of smooth functions, continuity properties, and compact embeddings into L^p at the critical exponent p_s^* equals Np over (N minus sp).
Pohozaev-type identities and Díaz-Saa inequalities hold for functionals associated with the operator.
The Dirichlet eigenvalue problem driven by the operator possesses existence, uniqueness, and boundedness of solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit logarithmic term may serve as a perturbation that connects standard fractional problems to their logarithmic counterparts or yields new comparison principles.
Compactness at the critical exponent suggests that direct variational methods can treat nonlinear equations involving this operator without extra concentration-compactness arguments.
Numerical evaluation of the representation on radial test functions could independently confirm the constants B and C appearing in the formula.

Load-bearing premise

The map that sends the order parameter t to the fractional p-Laplacian of a fixed function u is differentiable at t equals s, allowing differentiation under the integral sign.

What would settle it

A direct differentiation of the integral kernel that defines the fractional p-Laplacian with respect to s, performed on a smooth compactly supported test function, that fails to recover the stated combination of the fractional p-Laplacian term and the logarithmic principal-value integral.

read the original abstract

In this paper, we introduce and investigate the fractional logarithmic $p$-Laplacian $(-\Delta)_{p}^{s+\log}$, defined as the first-order derivative with respect to the parameter $t$ of the fractional $p$-Laplacian $(-\Delta)_{p}^{t}$ evaluated at $t=s$. We establish that this operator admits the following integral representation \[ \begin{aligned} (-\Delta)_{p}^{s+\log} u(x) &= B(N,s,p)(-\Delta)_{p}^{s}u(x)\\ &\quad -pC(N,s,p)\mathrm{P.V.}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ln |x-y|}{|x-y|^{N+sp}}dy, \end{aligned} \] where $C(N,s,p)$ denotes the standard normalization constant associated with the fractional $p$-Laplacian, and $B(N,s,p)=\frac{d}{ds}\left(\ln C(N,s,p)\right)$. As a consequence of this representation, it follows that the operator is nonlocal and of logarithmic type, and may be viewed as a nonlinear analogue of the fractional logarithmic Laplace operator recently introduced by Chen et al. \cite{Chen-Chen-Hauer}. We further develop the associated functional framework in both $\mathbb{R}^{N}$ and bounded Lipschitz domains by introducing the natural energy spaces adapted to problems driven by $(-\Delta)_{p}^{s+\log}$. Within this framework, fundamental functional inequalities are established, in particular Pohozaev-type identities and D\'{\i}az-Saa inequalities, which are of independent interest and applicable to a broader class of problems. Moreover, we derive results concerning density, continuity, and compact embedding properties. We emphasize that the compactness of the embedding is proved at the critical exponent $p^{*}_{s}=\frac{Np}{N-sp}$, which distinguishes the present setting from the classical Sobolev and fractional Sobolev frameworks. Finally, as an application, we investigate the associated Dirichlet eigenvalue problem and derive existence, uniqueness, and boundedness results for the corresponding solutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines the fractional logarithmic p-Laplacian by differentiating the standard fractional p-Laplacian at order s, derives its integral representation, and proves some embeddings and eigenvalue results, but the interchange of derivative and integral needs explicit justification.

read the letter

The core new piece is the definition of (–Δ)_p^{s+log} u as the t-derivative of (–Δ)_p^t u at t=s, together with the explicit integral formula that splits into a multiple of the usual fractional p-Laplacian plus a principal-value integral carrying the extra ln|x–y| factor. This is a direct nonlinear extension of the linear logarithmic Laplacian from Chen–Chen–Hauer, and the authors use it to set up energy spaces on R^N and on Lipschitz domains. They also record Pohozaev identities and Díaz–Saa inequalities, which are standard but still useful once the operator is in place, and they obtain existence, uniqueness, and boundedness for the associated Dirichlet eigenvalue problem. The compact embedding into L^{p_s^*} at the critical exponent is presented as a distinguishing feature of this logarithmic setting. That claim, if it holds, is the part most likely to be cited by others working on nonlocal p-problems. The soft spot is the passage from the difference quotient to the differentiated kernel. Differentiating the kernel |x–y|^{-N–tp} with respect to t produces the log term, but to move the derivative inside the principal-value integral one needs a domination argument that works uniformly for u in the energy space W^{s,p} and for t near s. The abstract and the weakest-assumption note both flag this step; if the paper only verifies it on smooth compactly supported functions and then appeals to density without controlling the logarithmic singularity, the representation does not automatically carry over to the functions used for the embedding and eigenvalue theorems. The rest of the functional analysis looks routine once the representation is granted. This is a specialized note in nonlocal analysis. Readers already working on fractional p-Laplacians or on parameter-dependent nonlocal operators will find the setup and the eigenvalue application worth seeing. I would send it to peer review so that referees can check the domination argument and the critical embedding in detail.

Referee Report

2 major / 2 minor

Summary. The paper defines the fractional logarithmic p-Laplacian as the t-derivative at t=s of the fractional p-Laplacian (−Δ)_p^t. It derives the integral representation (−Δ)_p^{s+log} u(x) = B(N,s,p)(−Δ)_p^s u(x) − p C(N,s,p) P.V. ∫ |u(x)−u(y)|^{p−2}(u(x)−u(y)) ln|x−y| / |x−y|^{N+sp} dy, establishes the associated energy spaces on R^N and bounded domains, proves Pohozaev-type identities and Díaz-Saa inequalities, obtains density/continuity/compact embedding results (including compactness at the critical exponent p_s^* = Np/(N−sp)), and applies the framework to the Dirichlet eigenvalue problem to obtain existence, uniqueness, and boundedness of solutions.

Significance. If the differentiation-under-the-integral step is rigorously justified for the full energy space and the embedding claims hold, the work supplies a nonlinear logarithmic nonlocal operator together with a functional setting and inequalities that could support further variational analysis. The compact embedding statement at the critical exponent, if correct, would be a notable departure from standard fractional Sobolev theory.

major comments (2)

[Definition of (−Δ)_p^{s+log} and derivation of the integral representation] The central representation formula is obtained by differentiating the kernel |x−y|^{-N−tp} with respect to t and passing the derivative inside the principal-value integral. A complete justification (via dominated convergence or an equivalent argument) must be supplied that controls the difference quotient uniformly for t near s and for all u in the energy space W^{s,p}; verification only on C_c^∞ followed by density does not automatically extend because of the additional logarithmic singularity.
[Compact embedding results] The compactness of the embedding at the critical exponent p_s^* is asserted to hold in bounded Lipschitz domains and is presented as distinguishing the setting from classical fractional Sobolev theory. The proof must be examined in detail, as the standard Rellich–Kondrachov theorem yields compactness only for subcritical exponents; any argument that reaches the critical exponent requires explicit control of the logarithmic perturbation and cannot rely on the usual concentration-compactness or profile decomposition without additional structure.

minor comments (2)

[Functional framework] Clarify the precise definition of the energy space adapted to (−Δ)_p^{s+log} (norm, seminorm, or full space) and its relation to the standard fractional Sobolev space W^{s,p}.
[Representation formula] The constant B(N,s,p) is defined as d/ds (ln C(N,s,p)); verify that this derivative exists and is continuous for the admissible range of s and p.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comments. We address each point below and indicate the revisions we will make.

read point-by-point responses

Referee: [Definition of (−Δ)_p^{s+log} and derivation of the integral representation] The central representation formula is obtained by differentiating the kernel |x−y|^{-N−tp} with respect to t and passing the derivative inside the principal-value integral. A complete justification (via dominated convergence or an equivalent argument) must be supplied that controls the difference quotient uniformly for t near s and for all u in the energy space W^{s,p}; verification only on C_c^∞ followed by density does not automatically extend because of the additional logarithmic singularity.

Authors: We agree that the current derivation, which proceeds from C_c^∞ via density, requires a direct justification for the full space to control the difference quotients uniformly near t = s. In the revised version we will insert a dedicated lemma that applies the dominated convergence theorem directly on W^{s,p}. The key estimate bounds the difference quotient of the kernel by an integrable majorant of the form C |u(x)−u(y)|^{p−1} (1 + |ln |x−y||) / |x−y|^{N+sp} whose integrability follows from the definition of the energy space; this majorant is independent of t in a small interval around s. The revised proof will therefore not rely solely on density. revision: yes
Referee: [Compact embedding results] The compactness of the embedding at the critical exponent p_s^* is asserted to hold in bounded Lipschitz domains and is presented as distinguishing the setting from classical fractional Sobolev theory. The proof must be examined in detail, as the standard Rellich–Kondrachov theorem yields compactness only for subcritical exponents; any argument that reaches the critical exponent requires explicit control of the logarithmic perturbation and cannot rely on the usual concentration-compactness or profile decomposition without additional structure.

Authors: The compactness at the critical exponent is obtained precisely because the logarithmic perturbation supplies an extra integrability factor that rules out concentration. In the revised manuscript we will expand the proof of Theorem 4.3 (or its equivalent) by inserting a profile-decomposition argument adapted to the log-weighted energy. After extracting a possible bubble, the logarithmic term produces a strictly positive remainder that contradicts the critical-energy assumption unless the bubble vanishes; the argument therefore does not reduce to the standard fractional Sobolev case. We will also add a remark clarifying why the usual Lions-type concentration-compactness lemma must be modified by the log weight. If the referee wishes to see an intermediate version of these estimates, we are happy to provide them. revision: partial

Circularity Check

0 steps flagged

Operator defined as derivative; integral representation derived via direct differentiation without circularity

full rationale

The paper explicitly defines the fractional logarithmic p-Laplacian as the first-order derivative with respect to the parameter t of the fractional p-Laplacian evaluated at t = s. The integral representation is then obtained by differentiating the standard integral expression for (-Δ)_p^t u, which involves differentiating both the normalization constant C(N,t,p) and the kernel with respect to t, producing the logarithmic factor. This is a straightforward calculus step rather than a circular redefinition or a fitted prediction. The subsequent development of energy spaces, inequalities, and embedding results relies on this derived representation but does not feed back into the definition of the operator itself. No load-bearing self-citations or ansatzes smuggled via prior work are evident in the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work assumes standard properties of the fractional p-Laplacian and Sobolev spaces from the literature, with the new operator constructed directly from differentiation.

axioms (1)

domain assumption The fractional p-Laplacian is differentiable in the parameter s
Invoked to define the new operator as the first-order derivative at t=s.

invented entities (1)

fractional logarithmic p-Laplacian no independent evidence
purpose: Nonlocal nonlinear operator with logarithmic kernel
Defined in the paper via differentiation of the standard operator.

pith-pipeline@v0.9.0 · 5714 in / 1288 out tokens · 118093 ms · 2026-05-13T04:39:11.779237+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
We establish that this operator admits the following integral representation: (−Δ)_p^{s+log} u(x) = B(N,s,p)(−Δ)_p^s u(x) − p C(N,s,p) P.V. ∫ |u(x)−u(y)|^{p−2}(u(x)−u(y)) ln|x−y| / |x−y|^{N+sp} dy
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_fourth_deriv_at_zero echoes
d/dt |_{t=s} (−Δ)_p^t u and differentiation under the integral

Reference graph

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