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Sharp Quantitative Stability for the Affine \(p\)-Sobolev Inequality, Part I: The Case \(2\le p<n\)

T0 review · 1 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For 2 ≤ p < n, the affine p-Sobolev inequality is quantitatively stable: the normalized deficit controls the distance to the affine extremal manifold to the power p, and this exponent cannot be improved.

desk verdict Sharp affine p-Sobolev stability, likely true and locally solid, but Lemma 4.1's indexing and the uniform decoupling (4.6) need repair before the global step is airtight. read the letter →

arxiv 2606.09555 v2 pith:3IFLZ6H4 submitted 2026-06-08 math.AP

Song Fan , Gui-Dong Li , Jianjun Zhang This is my paper · ORCID
classification math.AP MSC 46E3526D1035A23
keywords affinep-SobolevinequalityquantitativestabilitysharpexponentextremalmanifoldSL(n)invarianceHessiansecondvariationprofiledecompositiondirectionalenergydecoupling
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

Quantitative stability for inequalities asks how close a near-optimizer must be to the set of exact optimizers. This paper answers that question for the affine L^p-Sobolev inequality, a sharp inequality that refines the classical Sobolev inequality and is invariant under all volume-preserving linear changes of coordinates. The authors prove that for 2 ≤ p < n, if a function u has affine energy close to the optimal value, then u is close, in a gradient L^p sense and after an optimal affine renormalization, to some affine image of the standard bubble — with distance raised to the p-th power controlling the deficit. They also show that p is the smallest exponent that can work. The proof identifies the second variation of the affine energy as the classical Sobolev Hessian minus a variance term, and shows that this term enlarges the zero-modes precisely to the tangent space of the affine extremal manifold.

What carries the argument

The engine is the second variation of the affine energy E at the normalized bubble U. The affine energy is a negative mean over directions ξ of directional L^p energies, so its Hessian splits as the classical Sobolev Hessian minus a variance term R_p that measures how much the directional linear responses L_ξ(ϕ) fluctuate over the sphere. A spherical harmonic decomposition and a standard angle-averaging identity show that R_p affects only the even angular sectors; in the degree-two sector it cancels the positive classical contribution exactly, creating the new zero directions x·B∇U for trace-free symmetric matrices B. The kernel of the affine Hessian is thereby identified as the tangent spac

What would settle it

Take two copies of the bubble U placed at distance R apart, u_R(x) = U(x - R e_1) + U(x + R e_1), and compute the directional energies A_ξ(u_R) for ξ nearly perpendicular to e_1 as R grows. If the coupling between the two bubbles in such directions does not vanish uniformly in ξ, the key decoupling step (4.6) fails and the global argument would need a different proof; if it does vanish uniformly, one can test the exponent-p inequality directly on these sequences.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for 2 ≤ p < n there exists a constant c_{n,p} > 0 such that every nonzero u ∈ W^{1,p}(R^n) satisfies E(u)/(S_{n,p}‖u‖_{L^{p*}}) − 1 ≥ c_{n,p} · [inf_{a∈R, A∈SL(n), λ>0, x₀∈R^n} ‖∇(T_{λA,x₀}u − aU)‖_{L^p}/‖∇(T_{λA,x₀}u)‖_{L^p}]^p. The infimum measures, after optimal affine normalization, the gradient distance from u to the manifold M_aff of affine images of the bubble U. Thus near-extremizers of the affine Sobolev inequality are quantitatively close to M_aff at order p, and the exponent p is optimal: no α < p can replace it. The proof also establishes that the kernel of the affine Hessian at U is exactly the tangent space of M_aff, which is strictly larger th

Load-bearing premise

The proof depends on the claim that when a near-extremizing sequence is split into separate bumps, the total energy in every direction is essentially the sum of the energies of the bumps, with an error uniformly small over all directions; this uniformity is asserted by a compactness argument but not proved in detail near the most degenerate directions.

Editorial extensions

If this is right

  • Near-extremizers with vanishing affine deficit converge, after affine renormalization and scaling, to an affine image of the bubble U.
  • The distance-to-extremals deficit controls the gradient distance to M_aff with power p, and no smaller power works.
  • The zero-directions of the affine second variation are exactly the infinitesimal affine reparametrizations of the bubble — the full tangent space of M_aff.
  • The local spectral-gap estimate is the quantitative content behind the stability inequality, so the proof reduces the global problem to a compactness statement.

Reading between the lines

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

  • If the uniform directional decoupling in (4.6) is made fully rigorous — the paper only sketches it — the same global argument would likely extend the stability theorem to endpoint cases or to related affine-invariant inequalities where the negative-mean structure appears.
  • The explicit ratio identity for the angle-averaging coefficients suggests a closed formula for the constants in the spectral gap; computing c_{n,p} explicitly would make the stability bound quantitative rather than existential.
  • The restriction p ≥ 2 is used in the convexity inequality and the sign of the variance correction; a natural test is whether the same stability statement, with perhaps a different exponent, holds for 1 < p < 2.
  • The construction testing optimality with distant translated bumps indicates that the sharp exponent is driven by 'splitting' a bubble off into the far field; a similar phenomenon might occur for fractional affine Sobolev stability, where the Hilbertian case p = 2 is already understood.
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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

1 major / 5 minor

Summary. The paper proves a sharp quantitative stability estimate for the affine L^p-Sobolev inequality of Lutwak--Yang--Zhang for 2≤p<n. For every nonzero u∈Ẇ^{1,p}(R^n), the affine-Sobolev deficit controls the p-th power of the Ẇ^{1,p}-distance to the full affine extremal manifold M_aff, and the exponent p is shown to be optimal. The proof combines a local second-variation analysis of the affine energy, a spectral-gap result for the affine Hessian whose kernel is enlarged by trace-free degree-two modes, and a global compactness argument based on profile decomposition after a John-ellipsoid normalization. The sharpness is tested on translated bumps.

Significance. If correct, this is the first sharp gradient-type stability theorem for the affine Sobolev inequality, with the optimal exponent p. It extends the classical Figalli--Zhang theory to the SL(n)-invariant affine setting and identifies explicitly the enlarged extremal manifold. The paper is substantial: it contains a detailed second-variation computation, a Funk--Hecke/Saalschütz evaluation of the spectral coefficients, a calibration identity for the degree-two sector, and a self-contained sharpness construction. The overall architecture is sound, and the main unresolved point is a uniformity issue in the directional profile decomposition used in the global compactness step.

major comments (1)
  1. [§4, Eq. (4.6)] The directional decoupling A_ξ(u_k)=Σ_j A_ξ(W_j)+A_ξ(r_k^F)+o_k(1) uniformly in ξ is load-bearing for the no-splitting step and is not proved. The stated derivation applies the usual gradient decoupling to M_{ξ,η}=P_ξ+η(I-P_ξ) and lets η↓0, claiming uniformity 'by compactness of S^{n-1}'. This is insufficient: for each fixed invertible M the decoupling error depends on the condition number of M, which degenerates as η→0 at the equator, and no uniform-in-(ξ,η) estimate is supplied. If (4.6) fails, the chain (4.7) and the exclusion of multi-bubble splitting collapse. A repair appears plausible: prove pointwise-in-ξ decoupling using standard directional profile decomposition, then use the uniform Lipschitz bound |A_ξ(v)-A_η(v)|≤C∥∇v∥_p^p |ξ-η| to pass to uniformity in ξ. The manuscript should either provide this argument or a different honest proof of (4.6).
minor comments (5)
  1. [§4, proof of Lemma 4.2] Typo: 'hlod' should be 'hold'.
  2. [§3.2] The notation eε_j = ε_j λ_j is introduced but never used after (3.24); consider simplifying the normalization step.
  3. [§4, proof of Proposition 4.1] The sentence 'Then, by (2.8) yields' is ungrammatical; the reference to (2.8) in (4.12) should be clarified.
  4. [§4.1, sharpness proof] The expression 'cbδaff,p(uε,R)' in (4.21) mixes notations; clarify the role of c and the subscript p.
  5. [References] Reference [17] is given only as 'arXiv preprint'; provide a full arXiv number or journal data if available.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central derivation is self-contained; the only self-citation is a non-load-bearing analogy, and the flagged (4.6) uniformity issue is an analytic gap, not a circular step.

full rationale

The proof chain is independent of the desired conclusion. The local stability estimate rests on the external one-directional expansion of Figalli–Zhang [20], the classical Sobolev Hessian [19,20,37], and explicit Funk–Hecke/Saalschütz computations; the affine kernel is verified by invariance (5.26) and sector-by-sector estimates, not assumed. The global compactness step uses the external equality classification of Lutwak–Yang–Zhang [33] and standard profile decomposition; no fitted parameter is renamed as a prediction, and the optimal-exponent test uses an explicit translated-bump family. The only self-citation, [17], is used as an analogy ('parallel to the affine fractional Hilbertian case') and as background, never as a load-bearing input, so it does not generate circularity. The one substantive concern is the proof of (4.6), where uniform directional decoupling is asserted after 'letting η↓0' and 'by compactness of S^{n-1}'; this is a possible omitted uniformity argument near the singular projector, but it is an analytic gap about profile decomposition, not a reduction of the theorem to its inputs. Accordingly, no circular step can be exhibited, and the manuscript is best assessed as essentially self-contained with a minor non-load-bearing self-citation.

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

The proof depends on (i) the LYZ affine Sobolev theory and its equality classification [33]; (ii) the Figalli–Neumayer/Zhang machinery for classical p-Sobolev stability (nonlinear expansion, L^{p*} expansion, compact embedding, nondegeneracy) [19,20,37]; (iii) standard harmonic analysis (Funk–Hecke, Saalschütz) [1,13,16]; (iv) John's ellipsoid theorem [38]; and (v) implicit Hardy-type weighted inequalities for the p-bubble in Lemma 3.1. No free parameters and no invented entities; [17] (self-citation) is used only as an analogy. The constant c_{n,p} is existential, not fitted; the exponent p is the theorem's content.

assumptions (6)
  • domain assumption Equality classification for the affine L^p-Sobolev inequality: E(u)^p = S^p_{n,p}‖u‖^p_{L^{p*}} forces u ∈ M_aff (affine image of a Talenti bubble) [33].
    Used in Prop. 4.1 to identify the limiting extremal profile W after the profile decomposition; cited, not re-derived. Load-bearing for the global step.
  • domain assumption One-directional pointwise nonlinear expansion (2.18): A_ξ(U+εφ) ≥ A_0 + εL_ξ(φ) + ε²(1−κ*)B_{ξ,ε}(φ) + c_{κ*}ε^p A_ξ(φ) [20, Lemma 2.1(ii)].
    Input to Lemma 2.1; an external quantitative-convexity lemma for the p-Dirichlet functional. Structural for the local estimate.
  • standard math L^{p*} expansion of the Sobolev norm (2.23) [19, Lemma 3.2] and compact weighted embedding (3.1) H_U ↪↪ L²(U^{p*−2}dx) [19, Cor. 6.2] / [20, Prop. 3.2].
    Used in §2.1 and §3 (Prop. 3.1, Lemma 3.3); standard in the Sobolev-stability literature.
  • standard math Classical nondegeneracy of the critical p-Laplace bubble: ker Q_Sob,p = span{U, Z_0, ∂_{x1}U, ..., ∂_{xn}U} [19,20,37].
    Used in §5.2 and §5.8 to identify radial/translation sectors of the affine kernel.
  • standard math John's ellipsoid theorem (existence of E with E ⊂ K_u ⊂ √n E) [38, Thm 10.12.2].
    Core of Lemma 4.1's affine normalization; combined with the currently mis-indexed change-of-variables identity.
  • domain assumption Hardy-type weighted estimates for the bubble U used in Lemma 3.1: ∫|U'|^{p−2}(|f'|² + λ_ℓ f²/r²)r^{n−1} and ∫|U'|^{p−2}(|(ah)'|² + ...) are controlled by ∫U^{p*−2}f² plus the form value (eqs. (3.3), (3.7)).
    Asserted via sectorial forms; the underlying Hardy inequalities for the p-bubble are not proved in the paper but are standard (cf. [19]). If these failed, the Gårding bound in Prop. 3.1 would fail.

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Pith. "Pith review of Sharp Quantitative Stability for the Affine \(p\)-Sobolev Inequality, Part I: The Case \(2\le p<n\)." pith.science (2026). https://pith.science/paper/3IFLZ6H4

@misc{pith2026260609555,
  author       = {Pith},
  title        = {Pith review of: Sharp Quantitative Stability for the Affine \(p\)-Sobolev Inequality, Part I: The Case \(2\le p<n\)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3IFLZ6H4}},
  note         = {Machine review of arXiv:2606.09555}
}
abstract

We prove a sharp quantitative stability result for the affine \(L^p\)-Sobolev inequality, for \(p\ge2\), introduced by Lutwak--Yang--Zhang (\emph{J. Differential Geom.}, \textbf{62} (2002), 17--38). Moreover, the stability exponent is shown to be optimal, and equal to \(p\).

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Forward citations

Cited by 2 Pith papers

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

  1. Sharp Gradient Stability for the Sobolev Trace Inequality

    math.AP 2026-07 conditional novelty 7.0 of 10

    The Sobolev trace deficit controls the max{2,p}-th power of the gradient distance to the trace-bubble manifold for all 1<p<n.

  2. Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$

    math.AP 2026-07 accept novelty 7.0 of 10

    Sharp stability estimates with optimal exponents are established for the affine Sobolev inequality and its critical points for p≥2, including a new affine spectral gap inequality.

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