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REVIEW 3 major objections 6 minor 65 references

Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case

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Pith's one-line read The paper proves that a single threshold, $s_p p=p-1$, decides whether weak solutions of the nonlocal (1,p)-Laplace equation gain fractional Sobolev regularity or a genuine gradient.

desk verdict First interior Sobolev/Hölder regularity for the nonlocal (1,p)-Laplace equation; the main theorems hold up, but Corollary 1.5 has a repairable gap in the choice of q. read the letter →

arxiv 2505.23288 v1 pith:EOCGOSUT submitted 2025-05-29 math.AP

Dingding Li , Chao Zhang This is my paper
classification math.AP MSC 35B6535D3035J6035R11
keywords Sobolevregularitynonlocal(1p)-LaplaceequationsuperquadraticcasefinitedifferencequotientstailestimatesMoseriterationHöldercontinuityfractionalspaces
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 studies locally bounded weak solutions of the nonlocal (1,p)-Laplace equation $(-\Delta_1)^{s_1}u+(-\Delta_p)^{s_p}u=0$ for $p\ge2$, where one fractional diffusion has linear growth and the other has $p$-growth. It tries to show that the fractional differentiability index $s_p p$ alone decides the interior Sobolev regularity: below the threshold $s_p p\le p-1$ solutions belong to $W^{\gamma,q}_{loc}$ for every $q\ge p$ and $\gamma

What carries the argument

The engine of the proof is the finite difference quotient $\tau_h u(x)=u(x+h)-u(x)$ combined with the second difference $\tau_h(\tau_h u)$, tested against the monotone function $J_\delta(a)=|a|^{\delta-2}a$ with a cut-off. Proposition 4.2 converts the equation into the energy estimate $\int_{B_r}|\tau_h(\tau_h u)|^q \le C |h|^{s_p p+\sigma(q-p+1)}(\cdots)$, where the factor $|h|^{s_p p}$ records the influence of the 1-growth through the sign quotient $Z$, and $\sigma(q-p+1)$ comes from the available $W^{\sigma,q}$ regularity of $u$. A tail estimate (Lemma 4.1) controls the long-range integrals. Lemmas 4.3 and 4.5 bootstrap the differentiability exponent through the update rule $\sigma_{i+1}=(1-(p-1)/q)\sigma_i+s_p p/q$, whose fixed point is $s_p p/(p-1)$; this is the ceiling in Theorem 1.3. In the high-$s_p$ regime, Lemmas 5.3 and 5.2 first push the differentiability past $p-s_p p$ so that the difference quotients are controlled in $L^p$ with a factor $|h|^p$, which via Lemma 2.13 yields $\nabla u\in(L^p_{loc})^N$; Lemmas 5.6–5.8 then iterate the same machinery to raise the integrability of $\nabla u$ to every $q\ge p$.

What would settle it

Run the induction in Section 4 for a concrete parameter set, for example $p=100$, $s_p=0.989$, $q=200$, and check whether $\gamma/\alpha>\mu_p^{s_p}$ holds for all $\gamma$ in $(s_p, s_p p/(p-1))$ at the first step of the iteration, where $\alpha$ is chosen as in the proof of Theorem 1.3. A negative gap for any admissible $\gamma$ would show the written bootstrap does not establish the claimed range; if the gap is always positive, the assertion is consistent.

Watch

Extended reading notes

Core claim

The central claim is that the nonlocal (1,p)-Laplace equation has an interior Sobolev regularity theory controlled by the single number $s_p p$. In the regime $s_p\in(0,(p-1)/p]$ (Theorem 1.3), any locally bounded weak solution in the sense of Definition 1.1 satisfies $u\in W^{\gamma,q}_{loc}(\Omega)$ for every $q\ge p$ and every $\gamma\in(0,s_p p/(p-1))$, together with the quantitative estimate $[u]^q_{W^{\gamma,q}(B_r)}\le C\,(T+[u]_{W^{s_p,p}(B_R)}+1)^q/(R-r)^\kappa$. In the regime $s_p\in((p-1)/p,1)$ (Theorem 1.4), the gradient exists: $\nabla u\in(L^q_{loc}(\Omega))^N$ for every $q\ge p$, with $\|\nabla u\|^q_{L^q(B_r)}\le C\,(T+[u]_{W^{s_p,p}(B_R)}+1)^q/(R-r)^\kappa$. Via fractional Morrey embeddings, these statements yield $C^{0,\gamma}_{loc}$ Hölder continuity with $\gamma<s_p p/(p-1)$ in the first case and any $\gamma<1$ in the second.

Load-bearing premise

The load-bearing premise is that, in the middle range $s_p\in(2/p,(p-1)/p]$, the induction step of Theorem 1.3 may be applied because the starting differentiability $\gamma/\alpha$ is strictly larger than the threshold $\mu_p^{s_p}=(s_p p-2)/(p-2)$; the paper asserts this gap without proof, and if it fails for some admissible $\gamma$, the induction does not cover the full claimed interval.

Editorial extensions

If this is right

  • If Theorem 1.3 holds, every locally bounded weak solution with $s_p p\le p-1$ has fractional differentiability up to $s_p p/(p-1)$ in every $L^q$ with $q\ge p$, so the solution is smoother than the a priori $W^{s_p,p}$ class.
  • If Theorem 1.4 holds, the same equation with $s_p p>p-1$ produces a genuine gradient in $(L^q_{loc})^N$ for all $q\ge p$, making $s_p=(p-1)/p$ the transition point at which gradients appear.
  • Corollaries 1.5 and 1.6 convert these Sobolev statements into explicit Hölder exponents: $C^{0,\gamma}$ for $\gamma<s_p p/(p-1)$ in the low range and $C^{0,\gamma}$ for every $\gamma<1$ in the high range.
  • The quantitative estimates on nested balls make the regularity statements local and stable, with constants depending only on $N,p,q,s_1,s_p$ and on the distance to the boundary through $R-r$.

Reading between the lines

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

  • Beyond the paper, the threshold $s_p=(p-1)/p$ is likely a structural mark of the 1-versus-$p$ growth competition, suggesting the same dichotomy for parabolic (1,p)-Laplace systems once a tail estimate is available.
  • The update rule $\sigma_{i+1}=(1-(p-1)/q)\sigma_i+s_p p/q$ is a linear self-improvement law; analogous recursions should control other mixed-growth fractional operators, so the method may transfer to $(1,\ell)$-Laplace equations with $1<\ell<p$.
  • One testable extension is to check whether the ceiling $s_p p/(p-1)$ is optimal by constructing radial or layered solutions of the pure nonlocal 1-Laplacian whose fractional differentiability stops exactly at that exponent.
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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

3 major / 6 minor

Summary. The paper proves interior Sobolev regularity for weak solutions of the nonlocal (1,p)-Laplace equation (−Δ1)^{s1}u+(−Δp)^{sp}u=0 in the superquadratic case p≥2. For sp≤(p−1)/p it establishes u∈W^{γ,q}_{loc} for every q≥p and γ∈(0,spp/(p−1)); for sp>(p−1)/p it proves that ∇u exists and ∇u∈(L^q_{loc})^N for every q≥p, with explicit quantitative estimates on balls. The proof combines finite difference quotients, Caccioppoli-type inequalities, tail estimates, a Moser-type iteration, and an induction that raises the integrability exponent while preserving or slightly increasing the fractional differentiability. Hölder continuity is derived as a corollary via fractional Sobolev embedding.

Significance. If the results are correct, this is a valuable contribution to the regularity theory of nonlocal problems with (1,p)-growth, a setting where even basic Sobolev regularity was open. The paper offers a clean threshold at sp=(p−1)/p for the existence of the gradient, and the quantitative nature of the estimates makes the arguments adaptable to other nonlocal problems with mixed growth. The main novelty lies in coping with the non-smooth Z-term of the 1-Laplacian by letting the p-growth be controlled by the 1-growth, which is a genuinely new mechanism. The manuscript is self-contained in its main bootstrap, and the technical lemmas from B"ogelein et al. and Brasco-Lindgren are used appropriately. However, as detailed below, several proof steps are incomplete as written, although they appear repairable by standard arguments.

major comments (3)
  1. [Section 4, proof of Theorem 1.3] The induction in the case sp∈(2/p,(p−1)/p] only proves (4.19) for a fixed γ with γ>sp, because the base case applies Lemma 4.5 with σ=sp, which yields W^{γ,p} only for γ>sp. The theorem asserts the result for every γ∈(0,spp/(p−1)). The specific assertion γ/α>μ^sp_p used in the induction is justified for γ>sp, since α<sp(p−2)/(spp−2) and hence γ/α>sp/α>μ^sp_p; the reader's concern about that inequality does not apply to the induction as written. However, the small-γ range is not covered by the written proof. This gap is repairable by noting that for any γ≤sp one can choose γ′∈(sp,spp/(p−1)) and use the already proved W^{γ′,q}_{loc} together with the elementary embedding W^{γ′,q}(BR)⊂W^{γ,q}(BR) for γ<γ′. Please add this reduction explicitly.
  2. [Corollary 1.5] The proof selects q=2N/(spp/(p−1)−γ) and invokes Theorem 1.3, which requires q≥p. This condition can fail; for example, N=2, p=100, sp=0.99, γ=0.1 gives q≈4.4<p. The Hölder conclusion is nevertheless true, but the written argument is incomplete. A repair is to choose q′≥max(p, 2N/(spp/(p−1)−γ)) and any γ̃∈(γ,spp/(p−1)) with γ̃−N/q′≥γ, and then apply Lemma 2.7. The proof should be revised accordingly.
  3. [Lemma 5.3] The iteration applies Lemma 5.1 with σ=σ_{i−1} as long as σ_i<1 and β_i≤1, but Lemma 5.1 also requires the starting differentiability σ≤p−spp. Since α>p−spp and σ_i→α, there is an index after which σ_i>p−spp, and Lemma 5.1 is no longer applicable. The proof should stop at the first i with σ_i>p−spp and then invoke Lemma 5.2 to obtain the gradient. This is a local fix, but it must be written in the proof.
minor comments (6)
  1. [Equation (4.10)] The term ∥u∥_{L∞(B_{\tilde R})} in (4.10) should be raised to the power q (or replaced by the L^q norm) to match Lemma 2.9; the prefactor 7R^N/(σ(R−r)^q) also does not match the constants in Lemma 2.9 for general d.
  2. [Equation (4.13)] The same power issue as in (4.10) appears in (4.13): the L∞-norm inside the brackets should carry the exponent q, and the exponents in the last display should be rechecked.
  3. [Proof of Lemma 4.3] The parameter ε defined in Proposition 4.2 is introduced but is not used explicitly in the final estimate of the proof; please clarify its role or remove it if it is only needed for the Hölder estimates.
  4. [Remark 1.7] The displayed sequence γ_{i+1}=spp/q+γ_i(q−p+1)/q is not the same as the iteration used in Lemmas 4.3 and 5.3; please align the notation or explain the connection.
  5. [Theorem 1.4] The statement says the constants depend on γ, but Theorem 1.4 itself contains no γ; the dependence on γ belongs only to Corollary 1.6. Please correct the statement.
  6. [Remark 3.4] The phrase 'estimate the weak solution u from blow' should read 'from below'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Sobolev-regularity proof is a self-contained bootstrap from the weak formulation, with self-citations only contextual and external lemmas load-bearing.

full rationale

Walking the derivation chain: Definition 1.1 supplies u in W^{s1,1} ∩ W^{sp,p} and the Z-formalism; Proposition 4.2 derives the second-difference estimate from the weak form (1.3) using Lemma 2.3 and Lemma 4.1, both attributed to [6]; Lemma 4.3 converts that into the differentiability improvement beta = (1 - (p-1)/q) sigma + spp/q; Lemma 4.5 iterates to the fixed point spp/(p-1); the integrability lift in Theorem 1.3 uses only local boundedness and Lemma 4.5. Section 5 runs the same bootstrap with map beta = sigma/p + sp to reach W^{1,p}, then Lemmas 5.6-5.8 lift q. Every step starts from an assumption u in W^{sigma,q} and concludes a strictly stronger W^{alpha,q} regularity, so the conclusion is never used as an input. The only self-citations are [24] and [31], both in the introduction; the load-bearing technical lemmas come from external sources [6,7,17,22,26,28,36,39,41,43]. There are no fitted parameters, no data, and no prediction of a fitted quantity. Two review comments identify proof gaps—the unproved inequality gamma/alpha > mu in the induction for sp in (2/p,(p-1)/p], and Corollary 1.5's choice q = 2N/(spp/(p-1)-gamma) possibly violating q >= p—but these are correctness or completeness concerns, not circularity, because the argument does not assume the theorem it is proving.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the weak-solution framework for singular 1-growth (existence of Z), the decay/integrability condition for the p-tail, and standard fractional Sobolev and iteration tools. No free parameters are fitted; the constants C and exponents κ are universal. No new entities are introduced.

assumptions (5)
  • domain assumption There exists Z∈L∞(R^N×R^N) with Z∈sgn(u(x)−u(y)) representing the 1-Laplace quotient (Definition 1.1).
    This is the definition of weak solution for the singular nonlocal 1-Laplace term; existence is asserted via the direct method cited in [9] but not proved in this paper.
  • domain assumption u∈L^{p−1}_{spp}(R^N), a decay assumption for the p-growth term (Remark 1.2).
    Needed to make the tail functional finite; the paper notes no decay is needed for the 1-growth term.
  • standard math Algebraic inequalities in Lemmas 2.2 and 2.3, quoted from [6], hold for all real arguments.
    These are used to convert difference-quotient expressions into energy terms; they are stated without proof and cited.
  • standard math Fractional Sobolev embedding and interpolation lemmas (Lemmas 2.5-2.8) are valid as stated.
    Standard results from [27] used to derive Hölder and higher-integrability conclusions.
  • standard math The iteration Lemma 2.4 (from [41,65]) applies to the constructed sequences Y_j.
    Used in local boundedness and bootstrap arguments.

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Pith. "Pith review of Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case." pith.science (2026). https://pith.science/paper/EOCGOSUT

@misc{pith2026250523288,
  author       = {Pith},
  title        = {Pith review of: Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOCGOSUT}},
  note         = {Machine review of arXiv:2505.23288}
}
abstract

We investigate the interior Sobolev regularity of weak solutions to the nonlocal $(1, p)$-Laplace equations in the superquadratic case $p\ge 2$. As a product, the explicit H\"{o}lder continuity estimates of weak solutions are derived. The proof relies on a detailed analysis of the structural characteristics of $(1, p)$-growth in the nonlocal setting, combined with the finite difference quotient method, tail estimates, refined energy estimates, and a Moser-type iteration scheme.

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