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Sharp remainder of the $L^{p}$-Poincar\'e inequality for Baouendi-Grushin vector fields

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Pith's one-line read An exact remainder formula for the Lp-Poincaré inequality on Grushin vector fields.

desk verdict The paper's central complex-valued identity is false; a real-valued restriction could be viable. read the letter →

arxiv 2507.01681 v2 pith:42RNOKN5 submitted 2025-07-02 math.AP

Kuralay Apseit , Nurgissa Yessirkegenov , Amir Zhangirbayev This is my paper · ORCID
classification math.AP MSC 39B6235B4435A01
keywords PoincaréinequalityBaouendi-Grushinvectorfieldsp-Grushinoperatorsharpremainderformulacomplex-valuedfunctionsporousmediumequationoptimalconstant
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 establishes an exact remainder formula for the Lp-Poincaré inequality on domains in $\mathbb{R}^{m+k}$ carrying the Baouendi-Grushin vector fields $X_i=\partial_{x_i}$, $Y_j=|x|^\gamma\partial_{y_j}$. For every complex-valued $u$ and every non-zero complex-valued test function $\varphi$, the gap between $\int_\Omega |\nabla_\gamma u|^p\,dz$ and the eigenvalue term is written as the integral of a nonnegative functional $C_p$ that vanishes precisely when $u/\varphi$ is constant. Taking $\varphi$ to be an eigenfunction of the p-Grushin operator turns this identity into a sharp remainder formula, and from it the authors recover the Lp-Poincaré inequality with the optimal constant $1/\lambda_1$ under the assumption that the first eigenfunction is strictly positive. The same inequality is then used to prove finite-time blow-up and global existence criteria for a doubly nonlinear porous medium equation with the degenerate p-Grushin operator. The results recover the known $p=2$, $\gamma=0$ cases and make explicit a remainder constant that had previously been left implicit.

What carries the argument

The load-bearing object is the nonnegative functional $C_p(\xi,\eta)$ defined in (1.4), a p-analogue of the squared-distance term that appears in the classical $p=2$ remainder. Choosing $\xi=\nabla_\gamma u$ and $\eta=\nabla_\gamma u-(\nabla_\gamma\varphi/\varphi)u$, the functional is rewritten as $R_p(\xi,\eta)=|\nabla_\gamma u|^p-|\nabla_\gamma\varphi|^{p-2}\nabla_\gamma(|u|^p/|\varphi|^{p-2}\varphi)\cdot\nabla_\gamma\varphi$, and integration by parts converts $\int_\Omega R_p\,dz$ into the right-hand side of (3.1). The proof is a divergence-form calculation; sharpness comes from the equivalence $C_p(\xi,\eta)=0$ iff $\nabla_\gamma(u/\varphi)=0$, which characterizes equality in the derived Poincaré inequality as $u/\varphi_1=\mathrm{const}$.

What would settle it

Take a concrete bounded product domain such as a rectangle in $\mathbb{R}^{m+k}$ with $\gamma>0$ and $p\neq 2$, and compute the first eigenfunction of $-\Delta_{\gamma,p}$ numerically. If it has a nodal set and is not strictly positive, then the assumption in Remark 3.3 and Corollary 3.4(2) fails for that domain. As a separate check, one can verify the identity (3.1) directly on a simple pair $(u,\varphi)$ by evaluating both sides; any mismatch would disprove the claimed remainder formula.

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Extended reading notes

Core claim

The central claim, Theorem 3.1, is the identity $$\int_\$\Omega$ C_p\left(\nabla_\gamma u,\ \nabla_\gamma u-\frac{\nabla_\gamma\varphi}{\varphi}u\right)dz = \int_\$\Omega$ |\nabla_\gamma u|^p\,dz + \int_\$\Omega$ \frac{|u|^p}{|\varphi|^{p-2}\varphi}\,\Delta_{\gamma,p}\varphi\,dz,$$ valid for all complex-valued $u\in W^{1,p}_\gamma(\Omega)$ and all non-zero, twice-differentiable complex-valued $\varphi$, where $C_p(\xi,\eta)=|\xi|^p-|\xi-\eta|^p-p|\xi-\eta|^{p-2}\operatorname{Re}((\xi-\eta)\cdot\eta)\ge 0$. Because $C_p$ vanishes exactly when $\nabla_\gamma(u/\varphi)=0$, the identity is sharp: when $\varphi$ is a positive eigenfunction with $-\Delta_{\gamma,p}\varphi=\lambda|\varphi|^{p-2}\varphi$, the right-hand side becomes $\int_\Omega|\nabla_\gamma u|^p\,dz-\lambda\int_\Omega|u|^p\,dz$, and equality holds only for $u=c\varphi$. With $\lambda=\lambda_1$ and $\varphi=\varphi_1$ this gives the optimal constant $1/\lambda_1$ in the Lp-Poincaré inequality, together with quantitative remainder estimates for $p\ge 2$ and $1<p<2$, recovering and making explicit previously known or implicit results. The paper flags that strict positivity of $\varphi_1$ is an assumption, since the cited spectral theory leaves the sign unknown.

Load-bearing premise

The argument needs the first eigenfunction $\varphi_1$ of the p-Grushin operator to be strictly positive on the domain; the paper itself notes that the cited spectral theory leaves this sign unknown, so if $\varphi_1$ changes sign, the sharp constant and equality case in Corollary 3.4(2) do not follow.

Editorial extensions

If this is right

  • For every admissible eigenpair $(\lambda,\varphi)$, the Poincaré gap $\int_\Omega|\nabla_\gamma u|^p\,dz-\lambda\int_\Omega|u|^p\,dz$ is nonnegative and exactly measures how far $u$ is from being a constant multiple of $\varphi$.
  • With $\varphi=\varphi_1$ and $\lambda=\lambda_1$, the constant $1/\lambda_1$ in the Lp-Poincaré inequality is optimal, and equality holds only for $u=c\varphi_1$.
  • The explicit lower bounds on the remainder for $p\ge 2$ and $1<p<2$ give quantitative improvements of the Poincaré inequality, recovering a known improved inequality with an explicit constant.
  • For the doubly nonlinear porous medium equation $u_t-\Delta_{\gamma,p}(u^\ell)=f(u)$, the sharp Poincaré inequality yields finite-time blow-up under one growth condition on $f$, with an upper bound on the blow-up time, and global existence under a reversed condition.
  • When $p=2$ or $\gamma=0$, the identities reproduce previously known sharp remainder formulas, and the complex-valued formulation covers functions that prior real-valued results did not.

Reading between the lines

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

  • Inference beyond the paper: because the proof of (3.1) is purely a divergence-form computation, the same remainder identity is likely to hold on any sub-Riemannian structure with a horizontal gradient and a divergence theorem; the Grushin vector fields are a working example rather than the whole scope.
  • Inference beyond the paper: the explicit $C_p$ functional measures a distance from $u$ to the one-dimensional subspace spanned by $\varphi$, so the identity should yield quantitative stability estimates for the Poincaré inequality with an explicit spectral-gap dependence, not just equality cases.
  • Inference beyond the paper: the strict positivity of $\varphi_1$ could be tested computationally on product domains; if sign-changing first eigenfunctions occur, the optimal-constant statement would require either replacing $\varphi_1$ by its absolute value or a separate minimization argument.
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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 / 3 minor

Summary. The paper claims a sharp remainder formula for the Lp-Poincar\'e inequality for Baouendi-Grushin vector fields, stated as identity (3.1) for all complex-valued u in W^{1,p}_γ(Ω) and all nonzero twice-differentiable complex-valued φ. From this identity the authors derive the Lp-Poincar\'e inequality with optimal constant 1/λ1 (Corollary 3.4), remainder estimates (Corollaries 3.9 and 3.11), and applications to finite-time blow-up and global existence for a doubly nonlinear porous medium equation (Theorems 4.1 and 4.3). The central step is Theorem 3.12, which asserts that the functional Cp(ξ,η) equals an integrated-by-parts expression Rp(ξ,η).

Significance. If the main identity were correct, the paper would provide a useful extension of the p=2 result of Suragan and Yessirkegenov to 1<p<∞, give an explicit remainder estimate recovering Bobkov-Kolonitskii, and support a parabolic application. The paper also identifies a genuine gap in the literature, namely the unknown sign of the first p-Grushin eigenfunction, and it is honest about that gap in Remark 3.3. However, the central identity is false as stated for complex-valued functions, so the main theorem and all subsequent corollaries and applications rest on an invalid derivation. The significance of the intended results does not rescue the manuscript because the load-bearing argument is not merely missing a technical hypothesis: a concrete counterexample refutes (3.1).

major comments (3)
  1. [Theorem 3.12, proof of Theorem 3.1] The expansion of Rp(ξ,η) is invalid for complex-valued functions. The proof differentiates |u|^p and |φ|^{2-p}φ^{-1} as if u and φ were holomorphic, but the correct formulas are ∇|u|^p = p|u|^{p-2} Re(\bar u ∇u) and ∇(|φ|^{2-p}φ^{-1}) = (2-p)|φ|^{-p}φ^{-1} Re(\bar φ ∇φ) - |φ|^{2-p}φ^{-2} ∇φ. These conjugate terms are dropped in the displayed computation after (3.8), so the claimed identity Cp(ξ,η) = Rp(ξ,η) does not hold for complex-valued inputs. A concrete counterexample to (3.1) is p=2, γ=0, Ω=(0,1), u=x(1-x), and φ=e^{ix}: the left-hand side of (3.1) equals 11/30 while the right-hand side equals 3/10. Thus the central identity is false, not merely unproved.
  2. [Corollaries 3.4, 3.5, 3.9, 3.11 and Theorem 4.1] Every subsequent result that uses (3.1) is unsupported. In particular, the sharp remainder formula (3.3), the optimal Poincar\'e inequality (3.4), the remainder estimates (3.6) and Corollary 3.11, and the porous-medium blow-up and global-existence theorems all rely on the false identity. Since the counterexample in Theorem 3.12 concerns the exact setting advertised in the abstract (complex-valued functions), this is not a local fix: the derivation of (3.1) would need to be replaced by a genuinely different argument before any of the derived inequalities can be accepted.
  3. [Remark 3.3, Corollary 3.4(2)] The paper explicitly assumes that the first eigenfunction φ1 of the p-Grushin operator is strictly positive, while noting that the cited reference [MBS25] leaves the sign unknown. This assumption is load-bearing: it is needed to substitute the eigenvalue equation into (3.1), to form the quotient u/φ1, and to characterize extremizers by u/φ1 = const. If φ1 changes sign or vanishes, the identities and the optimality statement in Corollary 3.4(2) do not follow. Even if the complex-valued issue in Theorem 3.12 were repaired, this positivity gap would still leave the Grushin optimal-constant result conditional.
minor comments (3)
  1. [Throughout] The notation Δ_{γ,p}φ for complex-valued φ is used without specifying whether the p-Grushin operator is defined via real or complex derivative structure; this ambiguity is directly related to the main error and should be clarified in any revision.
  2. [Remark 3.8] The comparison with the L^{2m} result of Ozawa and Suragan is described only in words; a short displayed comparison of the hypotheses would help the reader see why the two formulas differ.
  3. [Proof of Corollary 3.4] The relation λ1^{1/p} = ||∇γφ1||_{Lp}/||φ1||_{Lp} is derived using an integration by parts that assumes sufficient regularity and boundary behavior of φ1; the weak formulation used by [MBS25] would need to be invoked explicitly to justify this step.

Circularity Check

1 steps flagged · score 2.0 of 10

Central remainder identity is independently derived; only the optimal-constant Poincaré inequality step is definitional because λ1 is introduced as the minimum Rayleigh quotient.

  1. self definitional [Corollary 3.4(2) and its proof, displayed equations (3.16)-(3.17)]
    "Dropping the remainder term in (3.15) and dividing both sides by λ1 > 0, we get ∫_D |u|^p dz ≤ 1/λ1 ∫_D |∇γu|^p dz, (3.16) where λ1 = min_{u≠0} (∫_D |∇γu|^p dz)/(∫_D |u|^p dz) (3.17) by [MBS25, Proposition 4.2]."

    The inequality (3.16) is exactly the statement that the Rayleigh quotient in (3.17) is bounded below by λ1. Since λ1 is defined as the minimum of that quotient, the inequality is true by construction once (3.17) is granted; the sharp-remainder identity (3.3) adds nothing to the inequality itself. The information that does not reduce to the definition is the remainder formula (3.3), the equality case u/φ1 = const, and the subsequent remainder estimates. This is therefore a minor self-definitional step in a corollary, not a collapse of the paper's central derivation.

full rationale

The paper's main identity, Theorem 3.1, is obtained by expanding the algebraically specified functional Cp, proving Rp = Cp by direct differentiation, and applying the divergence formula; no parameter is fitted and no target result is inserted as an ansatz. Corollaries 3.9 and 3.11 and the PME applications rest on the same identity together with external remainder estimates from [CKLL24] and [CT24]. The cited [MBS25] result is not authored by the present authors and is used for eigenvalue existence and the variational characterization, not to smuggle in an assumption. The admitted positivity assumption in Remark 3.3 is a stated hypothesis, not a circular import. The only reduction to an input is Corollary 3.4(2), where the optimal Poincaré inequality is just the definition of λ1 as the minimum Rayleigh quotient; even there the remainder identity supplies the extremal characterization. Thus the circularity burden is low. A separate concern—whether the complex-valued differentiation in Theorem 3.12 is valid—would be a mathematical correctness issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim depends on the assumed spectral property of the p-Grushin operator, on the divergence formula, and on a false implicit calculus rule for complex functions. No free parameters are fitted and no new entities are introduced.

assumptions (3)
  • domain assumption The first eigenvalue λ1 of −Δγ,p admits a strictly positive eigenfunction φ1 on D.
    Assumed in Remark 3.3; used in Corollary 3.4 to derive the sharp remainder formula and the extremizer characterization. Not proven in the cited literature.
  • standard math The domain Ω supports the divergence theorem for the Baouendi-Grushin vector fields.
    Used in the proof of Theorem 3.1 to integrate by parts and obtain the integral identity.
  • ad hoc to paper For complex-valued u, the paper implicitly assumes ∇|u|^p = p|u|^{p−2} u ∇u without a conjugate term.
    This false shortcut appears in the proof of Theorem 3.12 and invalidates the complex-valued statement of the identity.

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Pith. "Pith review of Sharp remainder of the $L^{p}$-Poincar\'e inequality for Baouendi-Grushin vector fields." pith.science (2026). https://pith.science/paper/42RNOKN5

@misc{pith2026250701681,
  author       = {Pith},
  title        = {Pith review of: Sharp remainder of the $L^p$-Poincar\'e inequality for Baouendi-Grushin vector fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42RNOKN5}},
  note         = {Machine review of arXiv:2507.01681}
}
abstract

In this paper, we establish a sharp remainder formula for the Poincar\'e inequality for Baouendi-Grushin vector fields in the setting of $L^{p}$ for complex-valued functions. In special cases, we recover previously known results. Consequently, we also derive the $L^{p}$-Poincar\'e inequality with an explicit optimal constant under a certain assumption. Additionally, we provide estimates of the remainder term for $p\geq2$ and $1<p<2\leq n<\infty$. As an application, we obtain a blow-up in finite time and global existence of the positive solutions to the initial-boundary value problem of the doubly nonlinear porous medium equation involving a degenerate nonlinear operator $\Delta_{\gamma,p}$.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Stability of the $L^{p}$-Poincar\'e inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps

    math.AP 2026-02 conditional novelty 7.0 of 10

    An explicit geometric stability constant is derived for the L^p-Poincaré inequality on convex domains, yielding a new but partially non-explicit spectral-gap bound for the p-Laplacian.

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