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Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs

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Pith's one-line read Using the graph's heat kernel as the weight for a fractional gradient, this paper constructs fractional Sobolev spaces on locally finite graphs and proves existence of positive and ground-state solutions for a fractional p-Laplace…

desk verdict New fractional graph Sobolev framework with plausible existence theorems; the compact embedding is the load-bearing proof debt. read the letter →

arxiv 2506.07694 v1 pith:QLR6IVZV submitted 2025-06-09 math.AP

Mengjie Zhang , Yong Lin , Yunyan Yang This is my paper
classification math.AP MSC 35A1535R0235R1146E35
keywords fractionalSobolevspacep-LaplaceoperatorvariationalmethodheatkernellocallyfinitegraphstochasticcompletenessgroundstatesolutionnonlinearSchrödingerequation
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

This paper builds fractional Sobolev spaces and a fractional $p$-Laplace operator on connected, locally finite, stochastically complete graphs, where previous fractional results were mostly confined to lattice graphs. The fractional gradient is defined by weighting vertex differences with a kernel made from the graph's heat kernel, and the paper shows the resulting spaces are reflexive Banach spaces that embed into $L^q$ for every $q\in[p,\infty]$. With this toolkit it proves that the nonlinear equation $(-\Delta)_p^s u + h|u|^{p-2}u = f(x,u)$ has a strictly positive solution, and, under the extra monotonicity assumption that $f(x,y)/y^{p-1}$ is increasing in $y$, a strictly positive ground state solution. The significance is a variational existence theory for nonlocal $p$-Laplace equations on discrete graphs.

What carries the argument

The load-bearing object is the heat-kernel-weighted kernel $W_s(x,y)=\frac{s}{\Gamma(1-s)}\mu(x)\mu(y)\int_0^\infty p(t,x,y)t^{-1-s}\,dt$, whose finiteness is guaranteed by stochastic completeness and the on-diagonal heat-kernel bound. It defines the fractional gradient $\nabla^s u(x)$ whose components are $\sqrt{\widetilde W_s(x,y)/(2\mu(x))}\,(u(x)-u(y))$, and the fractional $p$-Laplacian $(-\Delta)_p^s u = -\operatorname{div}_s(|\nabla^s u|^{p-2}\nabla^s u)$. The identity that carries the variational argument is the integration-by-parts formula $\int_V \varphi(-\Delta)_p^s u\,d\mu = \int_V |\nabla^s u|^{p-2}\nabla^s u\cdot\nabla^s\varphi\,d\mu$, which identifies weak solutions of the equation with critical points of the energy functional $E_{s,p}(u)=\frac{1}{p}\int_V(|\nabla^s u|^p + h|u|^p)\,d\mu - \int_V F(x,u^+)\,d\mu$.

What would settle it

A concrete check is to take a stochastically incomplete graph, where $\sum_{y\in V}p(t_0,x_0,y)\mu(y)<1$ for some $t_0,x_0$, and evaluate $\sum_{y\neq x}W_s(x,y)$ from (4): if the sum diverges, the kernel estimate fails and the fractional objects in the paper are not defined.

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Extended reading notes

Core claim

The central claim is that the heat kernel of a graph can define a nonlocal gradient with enough structure to support the full variational machinery. With $W_s(x,y)=\frac{s}{\Gamma(1-s)}\mu(x)\mu(y)\int_0^\infty p(t,x,y)t^{-1-s}\,dt$, the fractional gradient $\nabla^s$ and the fractional $p$-Laplace operator $(-\Delta)_p^s$ are well defined, and the paper proves that $W^{s,p}(V)$ and $W_0^{s,p}(V)$ are reflexive Banach spaces with $W^{s,p}(V)\hookrightarrow L^q(V)$ for $q\in[p,\infty]$. The two main existence theorems state that, under positivity and coercivity of the potential $h$ and the structural conditions (A1)-(A4) on $f$, the equation $(-\Delta)_p^s u + h|u|^{p-2}u = f(x,u)$ admits a strictly positive solution, and with the additional monotonicity (A5) it admits a strictly positive ground state solution. The proof route is the mountain-pass theorem and the Nehari manifold, both made to work by the integration-by-parts identity and the compact embedding of $H^{s,p}$ into $L^q$.

Load-bearing premise

The argument assumes the graph is stochastically complete, meaning its heat kernel conserves total mass $\sum_{y\in V}p(t,x,y)\mu(y)=1$ and is smooth enough at $t=0$; on a graph where this fails, the kernel $W_s$ can cease to be finite and the fractional gradient, Sobolev spaces, and $p$-Laplacian are not defined.

Editorial extensions

If this is right

  • If Theorems 4 and 5 hold, the fractional $p$-Laplace equation (12) has at least one strictly positive solution whenever the heat kernel and nonlinearity meet the stated conditions, and a strictly positive ground state when $f(x,y)/y^{p-1}$ is increasing.
  • The embedding $W^{s,p}(V)\hookrightarrow L^q(V)$ for every $q\in[p,\infty]$ becomes a standard tool, so other variational problems on graphs can be treated with the same spaces.
  • Because the construction works on every connected, locally finite, stochastically complete graph, lattice results are a special case rather than the whole theory.
  • The framework applies only for $p\geq 2$; the inclusion $C_c(V)\subseteq W^{s,p}(V)$ for $1<p<2$ is not proved, so the existence theorems do not yet cover that range.

Reading between the lines

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

  • Editorial extension: the same heat kernel could be used to define fractional Laplacian eigenvalue problems on general graphs, extending the lattice eigenvalue estimates that motivated the kernel's form.
  • Editorial extension: because the proof relies mainly on finiteness and symmetry of $W_s$, graphs that are only locally stochastically complete, or weighted graphs with a different time scale, might support a modified kernel with the same variational structure.
  • Editorial extension: a concrete test of the $p\in[2,\infty)$ restriction is whether a two-point test function on a graph with a single non-zero edge still has finite $W^{s,p}$ energy for $1<p<2$; if it does, the missing inclusion may be provable by a finer estimate.
  • Editorial extension: if the compact embedding of $H^{s,p}$ into $L^q$ survives under graph perturbations, the mountain-pass and Nehari arguments could adapt to sequences of graphs, yielding discrete-to-continuum convergence for fractional $p$-Laplace problems.
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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

4 major / 5 minor

Summary. The paper develops a theory of fractional Sobolev spaces W^{s,p}(V) on connected, locally finite, stochastically complete graphs using a heat-kernel-defined interaction kernel W_s(x,y), and introduces a fractional p-Laplace operator (-Δ)_p^s. It proves that W^{s,p}(V) is a reflexive Banach space, embeds into L^q for q∈[p,∞], and that positive/negative parts preserve membership. The main results, Theorems 4 and 5, assert existence of a strictly positive solution and, under an additional monotonicity condition, a strictly positive ground state solution to the nonlinear Schrödinger-type equation (-Δ)_p^s u + h|u|^{p-2}u = f(x,u), under assumptions (13)-(14) and (A1)-(A4) or (A5). The proofs use the mountain-pass theorem and a Nehari manifold argument.

Significance. If the main theorems are correct, the paper provides a general variational framework for fractional p-Laplace problems on locally finite graphs, going beyond the lattice-graph settings of earlier work. The construction via stochastic completeness and the heat kernel is natural, and the paper explicitly identifies the p∈[2,∞) restriction in Remark 6 rather than hiding it. The paper also includes several explicit estimates, such as the kernel finiteness estimate (4) and the vector inequality (35), which are useful. However, the central existence theorems depend on three lemmas (10, 13, and 16) whose proofs are delegated to earlier papers with only a sentence of adaptation; in particular, the compact embedding Lemma 13 is the engine of the Palais-Smale condition and the attainment of the Nehari minimum. The current manuscript does not give the reader enough detail to verify that the fractional kernel W_s satisfies the hypotheses needed for those arguments, so the central claim is not fully established within the paper.

major comments (4)
  1. [§4, Lemma 13] Lemma 13 is the load-bearing step for both existence theorems: it supplies the compact embedding H^{s,p} ↪ L^q for every q∈[p,∞] that is used in Lemma 17 to prove the Palais-Smale condition and in Lemma 19 to pass to the limit in the Nehari minimization. The proof is omitted with only 'adapting a modified argument presented in the proof of ([16], Lemma 2.1)'. This is not a routine detail: the adaptation must produce both uniform vanishing at infinity via h→∞ and local precompactness on finite balls using only the weighted kernel W_s, and it is not obvious that the estimates in [16] survive the replacement of the discrete gradient by the infinite-dimensional fractional gradient of (5). The authors should either provide a full proof or state precisely which theorem in [16] applies verbatim after the indicated modifications.
  2. [§3, Lemma 10] Lemma 10 asserts W^{s,p}(V) embeds into L^q(V) for all q∈[p,∞] and is proved by 'adapting the argument presented in the proof of ([15], Theorem 7)', with no details. Since this embedding is used in the definition of H^{s,p} and in the argument that H^{s,p} is a closed subspace of W^{s,p}, and since the fractional kernel W_s does not appear in [15], the adaptation is not a formality. Please provide the proof or a precise reference to a statement that covers this specific kernel.
  3. [§4.1, Lemma 16] Lemma 16, which supplies the mountain-pass geometry (existence of u_0 with E_{s,p}(t u_0)→-∞ and a positive lower bound on the sphere ∥u∥_{H^{s,p}}=r), is stated without proof and attributed to ([16], Lemmas 3.2 and 3.3). These properties depend on the specific form of the fractional energy and on assumptions (A2)-(A4), so they should be proved or at least sketched in the paper. In particular, the verification of the local minimum condition uses the spectral quantity λ_p from (A4), which is new here.
  4. [Remark 6 and §2.2] The paper restricts to p∈[2,∞) and states in Remark 6 that C_c(V)⊆W^{s,p}(V) is unverified for 1<p<2. This limitation is acknowledged, and I do not treat it as an error. However, because Theorem 1(i) is used in the proof of Lemma 9 and in the definition of H^{s,p}, the reader should be told whether the p∈[2,∞) restriction also affects the validity of Lemma 13 or the mountain-pass construction beyond the C_c inclusion. If the restriction is only needed for the truncation argument in Lemma 12, that should be stated explicitly.
minor comments (5)
  1. [Title] The title reads 'Fractional Sobolev spaces and fractionalp-Laplace equations'; a space is missing between 'fractional' and 'p-Laplace'.
  2. [Equation (4)] The interchange of the sum over y and the integral over t in (4) is justified by positivity of the integrand, but this should be stated explicitly, as the reader may otherwise worry about the validity of the equality.
  3. [Lemma 8] The reflexivity of L^p is delegated to [32, Proposition 3.2] without proof. Since this is a standard consequence of uniform convexity of the vector-valued L^p norm, a brief indication of the Clarkson-type inequality would improve self-containedness.
  4. [Lemma 15] In the proof of inequality (35), the function φ(t,m) is minimized over m∈[-1,1], but the case analysis for critical points and endpoints is compressed. The lower bound for critical points should be written out fully, since the constant 1/(2^{p-2}p) is used in Lemma 17.
  5. [Proof of Theorem 5] The proof of Lemma 20 follows [2, Lemma 3.5] and is sketched. The argument is plausible, but the choice of v with ⟨E'_{s,p}(u_0),v⟩<0 and the construction of t_m should be given a few more details, as the continuity of φ(t,m) in both variables is essential.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the main theorems follow from explicit hypotheses via standard variational arguments; the only concern is heavy reliance on omitted proofs adapted from the authors' prior papers, which is verification debt rather than circularity.

full rationale

Section 2 defines W_s from the heat kernel under the explicit stochastic-completeness assumption (1); inequality (4) is a consequence of (1)-(3), not an input. The fractional p-Laplacian (8) and the integration-by-parts identity (11) are mutually consistent by construction through the divergence definition (10), so Proposition 3 is a definitional identity rather than a derived prediction. Theorems 4 and 5 are obtained by standard mountain-pass and Nehari-manifold arguments from hypotheses (13)-(14) and (A1)-(A5); no parameter is fitted and no target quantity is smuggled into the assumptions. The main concern is proof debt: Lemma 10, Lemma 13, and Lemma 16 are stated with proofs omitted and referred to the authors' own prior results ([15], [16], [33], [38]). In particular, Lemma 13 (compact embedding H^{s,p} into L^q for q in [p,∞]) is the engine for the Palais-Smale condition (Lemma 17) and the Nehari attainment (Lemma 19), but the paper only says 'adapting a modified argument presented in the proof of ([16], Lemma 2.1). We omit the details of the proof here.' That is a completeness/correctness risk, not circularity: the cited lemmas are independent published mathematical results in the standard non-fractional graph setting and do not assume the theorem being proved. There is no equation in the paper where a claimed output reduces to an input by construction, no fitted parameter renamed as a prediction, and no uniqueness or ansatz imported solely from the authors' prior work. Hence no circular step is identified; score 1 reflects the minor but real self-citation and omitted-proof debt. A fully self-contained treatment of the compact embedding would eliminate even that concern.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper has no fitted numerical parameters; the only constants are structural, such as A_x from heat kernel regularity and λ_p from a Rayleigh quotient. The assumptions listed here are domain assumptions stated in the theorems, none of which is derived inside the paper. The new objects W_s, ∇^s, and (−Δ)_p^s are definitions rather than explanatory entities invoked post hoc.

assumptions (6)
  • domain assumption G is connected, locally finite, and stochastically complete with heat kernel p satisfying (1), p(t,x,y)>0, p(0,x,x)μ(x)=1, and t ↦ p is C^1 on [0, ∞).
    Stated in Section 2; used to define the kernel W_s in (2) and to obtain the finiteness bound (4).
  • domain assumption inf_{x∈V} μ(x) > 0.
    Assumed in Theorems 1, 2, 4, and 5; used in Lemma 7, in the completeness proof of Lemma 8, and in the compact embedding Lemma 13.
  • domain assumption p ∈ [2, ∞).
    The Theorems restrict to p at least 2; Remark 6 states that C_c ⊂ W^{s,p} is unverified for 1 < p < 2, so the full p-Laplace range is not covered.
  • domain assumption h satisfies (13)-(14) and f satisfies (A1)-(A5).
    These are the structural assumptions on the potential and nonlinearity used for the existence theorems in Section 4.
  • standard math Mountain pass theorem and Nehari manifold method are valid in the reflexive Banach space H^{s,p}.
    Used in Section 4 for Theorems 4 and 5; the paper applies the Ambrosetti-Rabinowitz theorem and a direct Nehari argument.
  • standard math ℓ²-valued L^p spaces are uniformly convex and reflexive for p ∈ [2, ∞).
    Used in Lemma 8 to establish reflexivity of L^p; the proof is delegated to the same argument as in [32, Proposition 3.2].
invented entities (2)
  • Fractional p-Laplace operator (−Δ)_p^s on locally finite graphs (equation (8))
    purpose: Nonlocal analogue of the p-Laplacian and the main operator studied in equation (12).
    It is a new mathematical definition, not an empirical entity; its validity is judged by the theorems proved with it, not by external data.
  • Heat-kernel based interaction kernel W_s(x,y) (equation (2))
    purpose: Defines the fractional gradient and fractional Laplacian on arbitrary locally finite graphs.
    Adapted from the lattice kernel in [36]; its finiteness rests on stochastic completeness and the heat kernel regularity assumptions.

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Pith. "Pith review of Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs." pith.science (2026). https://pith.science/paper/QLR6IVZV

@misc{pith2026250607694,
  author       = {Pith},
  title        = {Pith review of: Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLR6IVZV}},
  note         = {Machine review of arXiv:2506.07694}
}
abstract

Graph-based analysis holds both theoretical and applied significance, attracting considerable attention from researchers and yielding abundant results in recent years. However, research on fractional problems remains limited, with most of established results restricted to lattice graphs. In this paper, fractional Sobolev spaces are constructed on general graphs that are connected, locally finite and stochastically complete. Under certain assumptions, these spaces exhibit completeness, reflexivity, and other properties. Moreover, we propose a fractional $p$-Laplace operator, and study the existence of solutions to some nonlinear Schr\"odinger type equations involving this nonlocal operator. The main contribution of this paper is to establish a relatively comprehensive set of analytical tools for studying fractional problems on graphs.

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Forward citations

Cited by 1 Pith paper

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  1. Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs

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