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

Dachun Yang; Rupert L. Frank; Yinqin Li

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

T0 review · 0 major / 7 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Sharp stability for affine Sobolev inequality proved for p≥2

desk verdict Sharp stability for the affine Sobolev inequality, p>=2, with a new affine spectral gap inequality as the key ingredient read the letter →

arxiv 2607.06415 v2 pith:KWPGVB7Q submitted 2026-07-07 math.AP math.CAmath.FA

Rupert L. Frank , Yinqin Li , Dachun Yang This is my paper · ORCID

classification math.APmath.CAmath.FA MSC 46E3526D10

keywords affineSobolevinequalityquantitativestabilitysharpexponentspectralgapp-LaplaciansphericalharmonicsGegenbauerpolynomialscriticalpoints

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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 proves that the affine Sobolev inequality admits sharp quantitative stability for exponents p at least 2, meaning that any function whose affine Sobolev ratio is close to the best constant must be close, in a precise and optimal sense, to the family of extremal functions. The affine Sobolev inequality is a strengthening of the classical Sobolev inequality that replaces the L^p norm of the gradient with the affine Sobolev energy E_p, a quantity obtained by averaging directional gradient norms over the unit sphere with a negative power. This energy is invariant under all volume-preserving affine transformations, a much larger symmetry group than the classical inequality enjoys. The authors show that the deficit E_p(f)^p - S_aff^p ||f||_{p*}^p controls two terms simultaneously: the p-th power of the distance from f to the nearest extremal function, and a weighted quadratic term involving |∇v|^{p-2} |∇(f-v)|^2. Both exponents p and 2 are proved to be sharp. A parallel result establishes stability for near-critical points of the affine Sobolev functional in the single-bubble regime, with the optimal exponent p-1 on the distance. The central new ingredient is an affine spectral gap inequality that controls a variance-type term on the sphere arising from the nonlinear outer average in the affine energy, a term with no classical counterpart.

What carries the argument

The proof combines a compactness reduction via profile decomposition, a nonlinear Taylor expansion of the affine energy E_p that produces a variance term Var(f,g) on the sphere, and an affine spectral gap inequality proved by decomposing perturbations into spherical harmonics and verifying strict monotonicity of a sequence of eigenvalue ratios involving Gegenbauer polynomial coefficients.

What would settle it

Construct a sequence of functions f_epsilon = U + epsilon * phi for perturbations phi in the orthocomplement of the tangent space, showing that the deficit is of order epsilon^2 while the p-th power distance is of order epsilon^p with p > 2, confirming that the quadratic term is the leading contribution and cannot be replaced by any higher power.

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Extended reading notes

Core claim

The key discovery is that the affine Sobolev energy, despite being a nonlinear nonlocal functional involving a negative-power average of directional gradient norms over the sphere, admits a local second-order expansion whose leading correction includes a variance term on S^{n-1} that has no analogue in the classical Sobolev inequality. This variance term detects the extra trace-free affine directions in the tangent space of the extremal manifold. The authors prove an affine spectral gap inequality showing that this variance term, combined with the weighted quadratic form from the directional derivatives, is bounded below by a positive multiple of the weighted L^2 norm of the perturbation, as

Load-bearing premise

The load-bearing new ingredient is an affine spectral gap inequality that requires, for even spherical harmonic degrees at least 4, the strict monotonicity of a sequence of eigenvalue ratios. This monotonicity is verified by an explicit computation showing the ratio lambda_{2k+2,p}/lambda_{2k,p} is strictly less than 1, with lambda_{2,p} = 1, but the verification is intricate and specific to the structure of Gegenbauer polynomial coefficients.

Editorial extensions

If this is right

The stability result with the sharp quadratic term may enable quantitative convergence rates for numerical approximations of the affine Sobolev constant, paralleling applications of classical Sobolev stability to finite element methods.
The affine spectral gap inequality established here could serve as a foundation for studying multi-bubble stability of the affine Sobolev inequality, where the interaction between multiple extremal functions must be controlled.
The techniques for handling the nonlinear outer average and variance term may extend to other affine-invariant inequalities, such as the affine Moser-Trudinger or affine Morrey-Sobolev inequalities, where similar nonlocal averaging structures appear.
The restriction to p at least 2 is technical; the spectral gap argument for odd spherical harmonics is immediate but the even case requires an intricate monotonicity verification, suggesting that new ideas may be needed for the range 1 < p < 2.

Reading between the lines

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

The sharpness of the quadratic exponent 2 for p > 2 suggests that the affine Sobolev inequality has a fundamentally different local geometry than the classical Sobolev inequality near its extremals, where the exponent is p rather than 2 for the leading distance term.
The affine spectral gap inequality on spherical harmonics may have independent interest in spectral theory, as it relates the spectrum of the linearized p-Laplacian to the variance operator coming from the nonlinear sphere average, potentially connecting to representation-theoretic properties of Gegenbauer polynomials.
The parallel between the exponents (p, 2) in the stability theorem and (p-1) in the critical point theorem mirrors the classical Sobolev pattern, suggesting a universal relationship between the stability exponent and the critical point exponent for inequalities with similar variational structure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves sharp stability for the affine Sobolev inequality for p in [2,n), with optimal exponents on both the distance term (power p) and the weighted quadratic term (power 2), plus a corresponding critical-point stability result with exponent p-1. The main new ingredient is the affine spectral gap inequality (Theorem 4.2), which handles a variance term on the sphere that has no classical counterpart. This variance term arises because the affine energy involves a negative-power spherical average of directional derivatives, and controlling it is the central difficulty the paper resolves. The sharpness constructions in Section 6 are clean and match the exponents claimed in the main theorems. The overall strategy follows the Figalli-Zhang framework adapted to the affine setting, which is the natural approach, and the execution is careful throughout. The compactness argument in Section 2, the Taylor-type expansions in Section 3, and the nonlinear extension of the spectral gap (Theorem 4.11) via contradiction/compactness are all present and internally consistent. The concurrent work [19] obtained a weaker result without the quadratic term, which confirms the novelty here. The stress-test concern about Proposition 4.6(iv) checks out: the key computation showing the sequence {lambda_{2k,p}} is strictly decreasing with lambda_{2,p}=1 is verified by an explicit ratio calculation, and the bound is correct for k>=1, n>=2, p in [2,n). The nonlinear extension in Theorem 4.11 uses a standard contradiction argument with Newton-Leibniz decomposition and dominated convergence, which is handled correctly. The restriction to p>=2 is acknowledged as technical, and removing it is left as an open problem. This is a genuine limitation but not a flaw — the p<2 case likely requires different tools. I did not find errors in the proofs I checked in detail. The paper is written for specialists in sharp Sobolev inequalities and geometric analysis. It deserves a serious referee who can verify the spectral gap computations line by line, particularly the Sturm-Liouville analysis and the passage to the limit in Theorem 4.11.

Referee Report

0 major / 7 minor

Summary. The paper proves sharp quantitative stability for the affine Sobolev inequality for $p�[2,n)$, including both stability of the deficit (Theorem 1.1) and stability of critical points in the absence of bubbling (Theorem 1.3). The right-hand side of the main stability inequality (1.3) contains two terms: a $p$-th power distance to the optimizer manifold $M_{aff}$ and a weighted quadratic term $|∇v|^{p-2}|∇(f∘B−v)|^2$. Both exponents ($p$ and $2$) are shown to be sharp in Section 6. The proof follows the Figalli–Zhang framework for the classical Sobolev inequality, adapted to the affine setting. The central new ingredient is an affine spectral gap inequality (Theorem 4.2), which controls a variance term on $S^{n-1}$ arising from the nonlinear outer average in the affine energy. The spectral gap is proved via spherical harmonic decomposition, reducing to a classical gap for $ℓ≤2$ and to an explicit monotonicity computation for even $ℓ≥4$ (Proposition 4.6). A nonlinear extension (Theorem 4.11) is obtained by a contradiction/compactness argument.

Significance. The paper makes a substantial contribution to the stability theory of sharp functional inequalities. The affine Sobolev inequality is a strengthening of the classical Sobolev inequality with a larger symmetry group, and its stability analysis requires handling the nonlocal structure of the affine energy. The key technical achievement is the affine spectral gap inequality (Theorem 4.2/4.11), which is the load-bearing new ingredient and is verified by an explicit computation involving Gegenbauer polynomial coefficients (Proposition 4.6(iv)). The sharpness constructions in Section 6 are concrete and falsifiable. The result improves upon the concurrent work [19] by including the quadratic term, which is shown to be sharp and relevant for applications. The paper also provides a complete proof of stability for critical points (Theorem 1.3) with sharp exponent $p-1$. The overall structure is clear and the proofs are detailed.

minor comments (7)

Appendix A, line containing 'deinfed': typo, should be 'defined'.
Remark 1.4(b): 'be definition' should be 'by definition'.
The notation switches between $T_{λ,S,x}$ and $T^{(q)}_{λ,A,x}$; a brief remark in Section 1.3 or Section 2 clarifying the relationship once more would help the reader.
In the proof of Theorem 4.11, the passage from (4.27) to the liminf estimates for $I_j$ and $II_j$ involves several applications of dominated convergence and weak convergence. The logic is correct but dense; adding a sentence summarizing why the cross-terms vanish would improve readability.
The acknowledgment of AI tools is appropriate; no action needed.
Some cross-references to equations in Section 3 could be made more precise (e.g., the reference to (3.13) in the proof of Proposition 3.4). This is a minor presentation issue.
In Section 6.2, the condition (6.12) is introduced after the construction of $f_ε$; stating it earlier or more prominently would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment. The referee's report recommends minor revision but does not list specific major comments requiring changes. We address the substance of the report below.

read point-by-point responses

Referee: The referee provides a detailed and accurate summary of the paper's structure, main results (Theorems 1.1 and 1.3), the affine spectral gap inequality (Theorems 4.2/4.11), the sharpness constructions in Section 6, and the overall proof strategy following the Figalli–Zhang framework adapted to the affine setting.

Authors: We thank the referee for the careful and accurate summary of our paper. The description of the main results, the role of the affine spectral gap inequality as the central new ingredient, the spherical harmonic decomposition and Gegenbauer polynomial computation in Proposition 4.6, and the sharpness constructions in Section 6 are all faithfully represented. We confirm that the referee's characterization of the paper's contribution and structure is correct. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found. The derivation is self-contained with external benchmarks.

full rationale

The paper's central new ingredient—the affine spectral gap inequality (Theorem 4.2)—is proved from scratch via spherical harmonic decomposition (Proposition 4.6). Parts (i)–(iii) reduce to the classical spectral gap of the linearized p-Laplacian (Lemma 4.1), which is cited from Figalli–Neumayer [21] and Figalli–Zhang [22]—external, independent results. Part (iv) for even ℓ≥4 is verified by an explicit computation: λ_{2,p}=1 by direct substitution, and strict monotonicity of {λ_{2k,p}} is proved by computing the ratio λ_{2k+2,p}/λ_{2k,p}−1 and showing its numerator is negative for k≥1, n≥2, p∈[2,n). No self-citation is load-bearing: [18] (Fan–Li–Zhang, fractional) is cited only for the 'affine Hessian viewpoint' as motivation, while [19] (Fan–Li–Zhang, concurrent) is explicitly described as 'concurrent and independent work' and the paper claims to improve upon it. The sharp constant S_aff and optimizer manifold M_aff come from Lutwak–Yang–Zhang [41] (external). The energy expansion (Proposition 3.2), the p-Laplacian expansion (Proposition 3.4), the nonlinear spectral gap (Theorem 4.11), the compactness (Lemma 5.1), and the orthogonality (Lemma 5.2) are all proved within the paper. The sharpness arguments in Section 6 construct explicit test functions and compute deficits independently. No step in the derivation chain reduces to its own inputs by construction.

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

No free parameters are fitted to data. No new physical or mathematical entities are postulated. The affine p-Laplacian and affine Sobolev energy are standard objects from the prior literature. All axioms are external domain results from published literature.

assumptions (5)

domain assumption Sharp affine Sobolev inequality with best constant S_aff and optimizer characterization M_aff (Lutwak–Yang–Zhang [41])
Invoked in Section 1, equation (1.1)–(1.2). External result, not proved here.
domain assumption Classical spectral gap for the linearized p-Laplacian L_U (Figalli–Neumayer [21], Figalli–Zhang [22])
Invoked in Lemma 4.1. Used to handle ℓ=0,1 modes in Proposition 4.6.
domain assumption Profile decomposition in Ẇ^{1,p} (Jaffard [37], Okumura [45])
Invoked in Lemma 2.2 for the compactness argument in Theorem 2.1.
domain assumption Equivalence E_p(f) ~ min_{T∈SL(n)} ||∇(f∘T)||_p (Huang–Li [35])
Invoked in (1.14) and proof of Theorem 2.1 to reduce to bounded sequences.
domain assumption Pointwise inequalities for |x+y|^p (Figalli–Zhang [22, Lemma 2.1])
Invoked in equation (3.17) for the local expansion of the affine energy.

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Pith. "Pith review of Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$." pith.science (2026). https://pith.science/paper/KWPGVB7Q

@misc{pith2026260706415,
  author       = {Pith},
  title        = {Pith review of: Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KWPGVB7Q}},
  note         = {Machine review of arXiv:2607.06415}
}

abstract

We prove stability for the affine Sobolev inequality for exponents $p\geq 2$ with best possible norm and best possible stability exponent. We also show a corresponding result for critical points of the functional in the absence of bubbling. An important ingredient in our proof is the classification of positive energy solutions to the critical affine $p$-Laplace equation.

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