REVIEW 3 major objections 4 minor 17 references
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
T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash
Pith's one-line read On bounded convex domains, the L^p-Poincaré inequality is stable with an explicit constant (π_p/d)^p/2^{p−2}, yielding a spectral gap for the p-Laplacian that recovers the classical gap at p=2.
desk verdict Genuinely new explicit stability for L^p-Poincaré on convex domains, but the key identity is borrowed from a self-cited preprint and the p-Laplacian gap bound depends on unknown eigenfunctions. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument is carried by an exact remainder identity: the Poincaré deficit equals the integral of a nonnegative pointwise functional C_p, namely ∫|∇u|^p − λ_1∫|u|^p = ∫ C_p(∇u, u_1∇(u/u_1)) dx. A sharp pointwise lower bound for C_p (c_1(p) ≥ 1/2^{p−2} for p≥2) converts the deficit into a weighted Dirichlet integral of f = u/u_1 with weight |u_1|^p; log-concavity of the first eigenfunction makes this a log-concave weight. A weighted Poincaré inequality for log-concave measures with sharp constant (π_p/d)^p then converts that energy into the L^p distance from u to the span of u_1. The Gaussian version repeats the chain with the Gaussian p-Laplacian, whose first eigenfunction is likewise log-
What would settle it
Take the unit disk in the plane with p=3, compute the first eigenfunction u_1 numerically, choose a specific smooth compactly supported u, and compare ∫|∇u|^3 − λ_1∫|u|^3 with the numerical integral of C_3(∇u, u_1∇(u/u_1)); any mismatch beyond numerical error disproves Theorem 3.1 and hence the paper. Separately, on the same domain check whether the claimed gap inequality holds for the numerically computed second eigenfunction u_2; if λ_2 − λ_1 < (1/2)(π_3/d)^3 inf_c∫|u_2−c u_1|^3, then Corollary 3.6 is false.
Extended reading notes
Core claim
The central claim is a stability inequality with a fully explicit geometric constant: for p ≥ 2 and a bounded convex domain Ω with diameter d and smooth boundary, every u in C_c^∞(Ω) satisfies ∫_Ω |∇u|^p dx − λ_1(p,Ω) ∫_Ω |u|^p dx ≥ (1/2^{p−2})(π_p/d)^p inf_{c∈R} ∫_Ω |u − c u_1|^p dx, where u_1 is the positive first eigenfunction of the Dirichlet p-Laplacian and π_p is the generalized sine constant (π for p=2). The same form holds for the Gaussian probability measure, without any boundary-smoothness assumption. Inserting the second eigenfunction u_2 yields λ_2 − λ_1 ≥ (1/2^{p−2})(π_p/d)^p inf_c ∫ |u_2 − c u_1|^p, which the authors claim is the first spectral gap estimate for the Dirichlet p-
Load-bearing premise
The load-bearing step is the exact remainder identity quoted from the authors' companion preprint — that the Poincaré deficit equals the integral of C_p(∇u, u_1∇(u/u_1)) with no boundary corrections; if that identity fails or has hidden regularity constraints, the stability inequality and the spectral-gap corollary collapse.
Editorial extensions
If this is right
- Functions nearly minimizing the L^p-Poincaré ratio on a convex domain must be close, in L^p, to a constant multiple of the first eigenfunction, with a closeness quantified by (π_p/d)^p/2^{p−2} times the deficit.
- The Dirichlet p-Laplacian has a spectral gap: λ_2 − λ_1 ≥ (1/2^{p−2})(π_p/d)^p inf_c ∫|u_2−c u_1|^p dx, a bound the authors state was previously unknown for p≠2.
- At p=2 the stability inequality gives the classical fundamental-gap bound π^2/d^2 for convex domains, providing an alternative route to that known result.
- The same quantitative stability holds for the Gaussian p-Laplacian on bounded convex domains, with no smooth-boundary requirement.
- Because the constant depends only on p and diameter, the inequality is uniform across convex domains with a fixed diameter.
Reading between the lines
- Beyond the paper: the two-sided C_p estimates for 1<p<2 suggest the identity could yield a stability remainder of the form |∇(u/u_1)|^2/(|∇u|+|u_1∇(u/u_1)|)^{2−p}, a genuinely different, non-distance-type control that might interpolate between the p≥2 and p=1 regimes.
- Beyond the paper: because the Gaussian case needs no boundary smoothness and the Euclidean proof uses the same log-concavity input, the smooth-boundary assumption in the Euclidean theorem looks removable; testing this on non-smooth convex polygons would be a direct check.
- Beyond the paper: the gap constant C(p,Ω,u_1,u_2)=inf_c∫|u_2−c u_1|^p could in principle be computed on symmetric domains (interval, ball), which would convert the p-Laplacian gap bound into a purely geometric statement and reveal whether the (π_p/d)^p scaling is the right order.
- Beyond the paper: the same strategy — exact remainder identity plus log-concavity plus a sharp weighted Poincaré inequality — should apply to any log-concave weight, not just Lebesgue and Gaussian, so the stability bound likely has a family of measure-weighted analogues.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves quantitative stability for the Dirichlet $L^p$-Poincaré inequality on bounded convex domains. For $p\ge 2$ and Lebesgue measure it claims, for every $u\in C_c^\infty(\Omega)$, \[ \int_\Omega |\nabla u|^p\,dx - \lambda_1(p,\Omega)\int_\Omega |u|^p\,dx \ge \frac{1}{2^{p-2}}\left(\frac{\pi_p}{\operatorname{diam}(\Omega)}\right)^p \inf_{c\in\mathbb R}\int_\Omega |u-c u_1|^p\,dx, \] and a Gaussian analogue (Theorem 3.13). The proof combines an exact Picone/remainder identity (Theorem 3.1), a sharp lower bound for the functional $C_p$ (Lemma 2.6), log-concavity of the first $p$-Laplacian eigenfunction (Sakaguchi; Colesanti–Qin–Salani), and a weighted Poincaré inequality for log-concave measures (Ferone–Nitsch–Trombetti). The paper then derives a lower bound for the spectral gap $\lambda_2(p,\Omega)-\lambda_1(p,\Omega)$ (Corollary 3.6) and, for $p=2$, recovers the Yu–Zhong–Smits bound $\pi^2/\operatorname{diam}(\Omega)^2$.
Significance. If made fully self-contained, Theorems 3.3 and 3.13 would be a solid contribution: explicit stability constants for the $L^p$-Poincaré inequality with simple geometric dependence are rare, and the proof strategy connecting Picone identities, log-concavity of eigenfunctions, and weighted Poincaré inequalities is elegant and potentially influential. A particular strength is that the constant in (3.2) is explicit and involves no fitted parameter. The recovery of the classical fundamental gap for $p=2$ is a nice consequence. However, the current version is not fully acceptable: the central identity (Theorem 3.1) is quoted without proof from a self-cited preprint, and the $p$-Laplacian gap estimate in Corollary 3.6 depends on unknown eigenfunctions through $C(p,\Omega,u_1,u_2)$, so the advertised spectral-gap result is not an explicit geometric gap estimate.
major comments (3)
- [Theorem 3.1 / Theorem 2.13] The exact Picone identity (3.1) is the engine of the paper: all subsequent estimates in Theorems 3.3, 3.13 and Corollary 3.6 are obtained by applying Lemmas 2.6, 2.11, 2.12 to this identity. Yet Theorem 3.1 (and also Theorem 2.13, used for the Gaussian version) is quoted without proof from the self-cited preprint [AYZ25]. The paper is therefore not self-contained in its most load-bearing step. Please include a complete proof of the identity for complex-valued $u$, or state it as a lemma with a proof in the appendix, and do the same for the Gaussian version used in Theorem 3.11.
- [Corollary 3.6 / Eq. (3.4)] The advertised spectral gap for the Dirichlet $p$-Laplacian, $\lambda_2(p,\Omega)-\lambda_1(p,\Omega) \ge (1/2^{p-2})(\pi_p/\operatorname{diam}(\Omega))^p \, C(p,\Omega,u_1,u_2)$, contains the factor $C(p,\Omega,u_1,u_2)=\inf_c\int_\Omega |u_2-c u_1|^p\,dx$. This factor depends on the unknown normalized eigenfunctions $u_1,u_2$; no lower bound for it is provided for $p\neq 2$. For $p=2$, orthogonality gives $C=1$, but for $p\neq 2$ the claimed estimate is not an explicit geometric gap and cannot be evaluated a priori. Since the positivity of $\lambda_2-\lambda_1$ is classical, the statement as written (abstract and Remark 3.7) overstates the novelty. Please qualify the result as a conditional gap estimate and discuss whether $C$ can be bounded below independently of $u_2$.
- [Section 5, proof of Theorem 3.3] The proof applies the weighted Poincaré inequality to $f=u/u_1$ after subtracting a suitable constant $t_0$. For $u\in C_c^\infty(\Omega)$ this is legitimate because $u$ vanishes near $\partial\Omega$, so $f$ is bounded and Lipschitz on its support. However, the extension of (3.2) to $W_0^{1,p}(\Omega)$, used in Corollary 3.6 for $u=u_2$, is only asserted in Remark 3.5. Please state explicitly that both sides are continuous under $W_0^{1,p}$ convergence and that $C_c^\infty(\Omega)$ is dense; otherwise the application to $u_2$, which is only a weak solution, is not fully justified.
minor comments (4)
- [Theorem 3.3 vs. Theorem 3.11] Theorem 3.3 does not explicitly say that $u$ is real-valued, while the optimizer set is defined with $c\in\mathbb R$ and the weighted Poincaré inequality used in the proof is for real-valued functions. Please state the real-valued setting explicitly (or prove a complex version).
- [Proof of Theorem 3.13, Section 6] The proof begins with 'By Theorem 3.6'; this should refer to Theorem 3.11 (the Gaussian exact identity).
- [Notation after Eq. (1.5) and in Theorem 3.13] The distance $d(u,E_{\mathrm{Poin}})^p$ is written with $L^p(\mathbb R^n)$ (or $L^p(\mathbb R^n,\gamma)$), but the functions are supported in $\Omega$. These should be $L^p(\Omega)$ and $L^p(\Omega,\gamma)$.
- [Theorem 2.13 / Theorem 3.1] The Picone-type identity is stated twice, once as Theorem 2.13 and once as Theorem 3.1, both with attribution to [AYZ25]. Please unify to avoid duplication and make clear which statement is used where.
Circularity Check
No constructional circularity: the main stability bound is derived from an external weighted Poincaré inequality and a C_p lower bound. The only flagged issue is that the exact remainder identity (3.1) is imported from the authors' own unpublished preprint without proof, making the argument depend on a load-bearing self-citation.
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self citation load bearing
[Theorem 3.1, Section 3 (see also Theorem 2.13, Section 2)]
"Theorem 3.1 ([AYZ25]). Let 1<p<∞, Ω⊂R^n be a set where the divergence theorem holds. Then, for all complex-valued u∈C_0^∞(Ω), we have ∫_Ω |∇u|^p dx − λ_1(p,Ω) ∫_Ω |u|^p dx = ∫_Ω C_p(∇u, u_1 ∇(u/u_1)) dx. (3.1)"
The proof of Theorem 3.3 opens with 'Due to Theorem 3.1, we have ...' and then combines this identity with Lemma 2.6 and Theorem 2.4. Thus the central stability inequality (3.2) rests on the exact Picone-type remainder identity (3.1), which is not proved in this paper but quoted from [AYZ25], an arXiv preprint whose author list includes Yessirkegenov and Zhangirbayev. The same is true for the Gaussian case via Theorem 2.13, also cited to [AYZ25]. If (3.1) contained a hidden boundary term, a sign error, or required more regularity, Theorems 3.3 and 3.6 would collapse. This is a load-bearing self-citation rather than a definitional equivalence, so it raises the score but does not make the whole claim forced by construction.
full rationale
The paper does not fit the strongest circularity patterns. There are no fitted parameters renamed as predictions, no uniqueness theorem imported from the authors' prior work to forbid alternatives, no ansatz smuggled in via citation, and no known result merely renamed. The derivation of Theorem 3.3 is otherwise structurally independent: the exact remainder identity (3.1) converts the Poincaré deficit into an integral of the nonnegative C_p functional; Lemma 2.6 gives a lower bound for C_p by c_1(p)|η|^p; and Theorem 2.4 (Ferone–Nitsch–Trombetti) supplies the weighted Poincaré inequality with the explicit constant (π_p/diam)^p. Unless (3.1) itself is equivalent to the target inequality, which it is not, the chain is not circular. The only substantive concern is provenance: the load-bearing identity is quoted from the authors' own unpublished preprint [AYZ25] without proof here. A direct integration-by-parts argument for real u, using −Δ_p u_1 = λ_1 u_1^{p−1}, confirms that (3.1) is very likely correct, so this is a self-containment and verification gap rather than a fatal circularity. Hence the score is 4 rather than 0 or 2: the central claim still has independent content, but the paper's key lemma is justified only by a self-citation that is not machine-checked or independently reproduced in this text.
Assumptions & free parameters
assumptions (5)
- standard math Exact Poincaré remainder identity (Theorem 3.1) from [AYZ25] holds for u∈C_c^∞ and u_1 the first eigenfunction.
- domain assumption Sakaguchi log-concavity (Theorem 2.1): first p-Laplacian eigenfunctions are log-concave on bounded convex smooth domains.
- domain assumption FNT12 weighted Poincaré inequality for log-concave measures with constant ≥(π_p/diam)^p (Theorem 2.4).
- domain assumption CQS25 Gaussian log-concavity (Theorem 2.2) and regularity (Proposition 2.3) for Gaussian p-Laplacian eigenfunctions.
- standard math Variational characterization of λ_2 via equation (2.4) and the Ljusternik–Schnirelman eigenvalue sequence.
Cite this review
Pith. "Pith review of 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." pith.science (2026). https://pith.science/paper/GO3RGMQL
@misc{pith2026260205968,
author = {Pith},
title = {Pith review of: 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},
year = {2026},
howpublished = {\url{https://pith.science/paper/GO3RGMQL}},
note = {Machine review of arXiv:2602.05968}
}
abstract
In this paper, we obtain stability results for the $L^{p}$-Poincar\'e inequality for both Lebesgue measure and Gaussian probability measure (Theorem 3.3 and Theorem 3.13) that involve explicit dependence on the geometry of the domain. As a byproduct, the explicit constant allows us to recover important results of Yu, Zhong [YZ86] and Smits [Smi96] (Corollary 3.9), related to the fundamental gap conjecture of the Laplacian (resolved by Andrews and Clutterbuck [AC11]), thereby providing an alternative proof. Moreover, we extend this spectral gap result to the $p$-Laplacian (Corollary 3.6). Such gap estimates for the Dirichlet $p$-Laplacian appear to be unavailable, as also observed in [DSW18]. Our approach relies on properties of the first eigenfunction of the (Gaussian) $p$-Laplacian operator and weighted Poincar\'e inequalities for log-concave measures on convex domains.
Reference graph
Works this paper leans on
-
[7]
[DFLL23] A. X. Do, J. Flynn, N. Lam, and G. Lu.L p-Caffarelli–Kohn–Nirenberg inequalities and their stabilities.arXiv preprint arXiv:2310.07083,
-
[11]
[FvHT24] A. Figalli, P. van Hintum, and M. Tiba. Sharp stability of the Brunn–Minkowski in- equality via optimal mass transportation.arXiv preprint arXiv:2407.10932,
-
[1983]
[vHST19] P. van Hintum, H. Spink, and M. Tiba. Sharp quantitative stability of the planar Brunn–Minkowski inequality.arXiv preprint arXiv:1911.11945,
arXiv 1911
-
[1985]
[SYZ25] Y. Shaimerdenov, N. Yessirkegenov, and A. Zhangirbayev. Sharp remainder terms and stability of weighted Hardy-Poincar´ e and Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator.arXiv preprint arXiv:2508.16380,
- [1989]
-
[1997]
[CQS25] A. Colesanti, L. Qin, and P. Salani. Geometric properties of solutions to elliptic PDE’s in Gauss space and related Brunn–Minkowski type inequalities.arXiv preprint arXiv:2502.00184,
- [1998]
-
[2000]
[Bob26] V. Bobkov. On the nodal set conjecture for thep-Laplacian in circularly symmetric domains.arXiv preprint arXiv:2602.01210,
Show all 17 references
-
[2003]
[HT25] Y. C. Huang and X. Tong. OnL p-Hardy inequalities for magneticp-Laplacians.arXiv preprint arXiv:2508.09483,
-
[2006]
[K¨ on23] T. K¨ onig. On the sharp constant in the Bianchi–Egnell stability inequality.Bull. Lond. Math. Soc., 55(4):2070–2075,
-
[2013]
Machihara, T
[MOW15] S. Machihara, T. Ozawa, and H. Wadade. Scaling invariant Hardy inequalities of mul- tiple logarithmic type on the whole space.J. Inequal. Appl., 2015(1):281,
2015
-
[2015]
Figalli, P
[FvHT23] A. Figalli, P. van Hintum, and M. Tiba. Sharp quantitative stability of the Brunn– Minkowski inequality.arXiv preprint arXiv:2310.20643,
-
[2016]
Bobkov and M
[BT25] V. Bobkov and M. Tanaka. On Rayleigh quotients connected top-Laplace equations with polynomial nonlinearities.arXiv preprint arXiv:2511.10199,
-
[2020]
Huang and D
[HY25] X. Huang and D. Ye. On Sharp Heisenberg Uncertainty Principle and the stability. arXiv preprint arXiv:2510.00453,
-
[2021]
Dolbeault and G
[DT16] J. Dolbeault and G. Toscani. Stability results for logarithmic Sobolev and Gagliardo– Nirenberg inequalities.Int. Math. Res. Not. IMRN, 2016(2):473–498,
2016
-
[2023]
[DGLL24] A. X. Do, D. Ganguly, N. Lam, and G. Lu. Scale-Dependent Poincar´ e inequalities and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space.arXiv preprint arXiv:2410.21039,
-
[2025]
Banerjee, D
[BGR26] A. Banerjee, D. Ganguly, and P. Roychowdhury. Sharp quantitative forms of the Hardy inequality on Cartan–Hadamard manifolds via Sobolev–Lorentz embeddings.arXiv preprint arXiv:2601.13750,
Reviewed August 3, 2026 · model on record in the stance chip above.
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