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Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere

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Pith's one-line read The reverse Sobolev inequality on the sphere is quantitatively stable across the full admissible parameter range, and in the higher range the sharp stability constant is 1, with no optimizer.

desk verdict Solid extension of the Bianchi–Egnell strategy to reverse Sobolev inequalities, but the headline sharp-constant/no-optimizer claim in the (1,2) case rests on an unpublished preprint that the paper does not reproduce. read the letter →

arxiv 2504.19939 v2 pith:7AOPL66J submitted 2025-04-28 math.AP

Tobias König This is my paper · ORCID
classification math.AP MSC 35A2335B3546E35
keywords reverseSobolevinequalitystabilityexplicitsharpconstantspheresphericalharmonicsmodifieddistanceconformalinvariancequantitative
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 proves a quantitative stability inequality for the reverse Sobolev inequality on the sphere, the extension of the Sobolev inequality to parameters $s>\frac n2$, where the exponent $2n/(n-2s)$ is negative and the inequality still holds for positive functions. The result covers the full admissible range $s-\frac n2\in(0,1)\cup(1,2)$ and says that a function can make the inequality's deficit small only by being close to the known family of optimizers, measured by a modified distance adapted to the non-positive-definite operator. In the range $s-\frac n2\in(1,2)$ the best stability constant is exactly $1$ and is not attained, which the paper identifies as the first Sobolev-type stability inequality with an explicit sharp constant and no optimizer. In the range $(0,1)$ the best constant is positive and bounded above by $4s/(n+2s+2)$, with strict inequality for $n\ge2$. These inequalities matter because they turn uniqueness of optimizers into quantitative control, the kind needed in convergence and blow-up arguments.

What carries the argument

The load-bearing object is the modified distance $d(u)=\inf\{a_{2s}[\rho]: u=h+\rho,\ h\in M,\ \rho\in(T_hM)^\perp\}$, where $M$ is the family of optimizers of the reverse Sobolev inequality and $T_hM$ is the tangent space of $M$ at $h$, with orthogonality taken with respect to the quadratic form $a_{2s}$. In the standard range $s<n/2$ this coincides with the usual distance to the optimizer manifold, but for $s>n/2$, where $A_{2s}$ is not positive definite, the usual distance can be negative or infinite, so the modified distance is needed. The paper proves that the infimum is always attained, that $d(u)>0$ outside $M$, and that locally it agrees with the quadratic form on the orthogonal complement via an implicit-function-theorem construction. This makes the classical two-step stability strategy work: a local expansion near the optimizer family gives the explicit constants $4s/(n+2s+2)$ and $1$, while a compactness result rules out vanishing of the stability quotient away from $M$.

What would settle it

In the range $s-\frac n2\in(1,2)$, take $u_\varepsilon=1+\varepsilon\rho$ for a fixed nonzero spherical harmonic $\rho$ of degree $\ell\ge2$ and compute the stability quotient $E(u_\varepsilon)$; the paper's Proposition 3.1 predicts a lower limit at least $1$, so any $\varepsilon,\ell$ with $E(u_\varepsilon)<1$ would falsify Theorem 1.2(b), while a minimizing sequence with quotient approaching $1$ that converges to a point outside $M$ would falsify the non-attainment claim.

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Extended reading notes

Core claim

The central claim is that for every positive $u\in H^s(S^n)\setminus M$, with $s-\frac n2\in(0,1)\cup(1,2)$, the stability quotient $E(u)=(a_{2s}[u]-S_s\|u\|_{2n/(n-2s)}^2)/d(u)$ is bounded below by a positive constant, where $d(u)$ is the modified distance to the optimizer family $M$ defined through tangent-space orthogonality. The sharp results are asymmetric across the range. For $s-\frac n2\in(1,2)$, Theorem 1.2(b) gives $c_{BE}(s)=1$ and shows the infimum is not attained; the lower bound comes from the stability inequality of the recent preprint [10], and non-attainment follows from its equality conditions in the reverse H\"older inequality, which force the normalized function to be constant and hence in $M$. For $s-\frac n2\in(0,1)$, the paper proves $c_{BE}(s)>0$ with the upper bound $c_{BE}(s)\le 4s/(n+2s+2)$, and Theorem 1.3 upgrades this to a strict inequality in dimension $n\ge2$. The paper thus completes the stability analysis for the full admissible parameter range and identifies the first explicit, non-attained best constant in a Sobolev-type stability inequality.

Load-bearing premise

For the parameter range $s-\frac n2\in(1,2)$, the sharp lower bound $c_{BE}(s)\ge1$ is imported from the unpublished preprint [10], and the nonexistence of an optimizer relies on the equality conditions of that preprint's inequality (3.12), so an error in either would undo the paper's main explicit results.

Editorial extensions

If this is right

  • In the range $s-\frac n2\in(1,2)$, every positive function satisfies the sharp bound $E(u)\ge1$, and sequences approaching equality must, after normalization and a conformal transformation, have their $L^2$ mass concentrated in asymptotically high spherical-harmonic degrees.
  • In the range $s-\frac n2\in(0,1)$, the strict inequality $c_{BE}(s)<4s/(n+2s+2)$ for $n\ge2$ rules out the local constant as the global optimum and is the step that yields weak precompactness of minimizing sequences.
  • Any minimizing sequence in the range $s-\frac n2\in(0,1)$ with $n\ge2$ admits conformal normalizations that converge weakly to a nonnegative function outside the optimizer family, so loss of compactness can only occur through vanishing or bubble splitting, not by sliding into $M$.
  • The value $c_{BE}(s)=1$ in the range $(1,2)$ gives a precise quantitative benchmark: the deficit $a_{2s}[u]-S_s\|u\|_p^2$ is always at least the modified distance $d(u)$.

Reading between the lines

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

  • Inference: the same tangent-space distance could serve other conformally invariant inequalities whose quadratic form is indefinite, such as higher-order or trace inequalities, though the paper does not claim this.
  • Inference: the non-attainment mechanism, where equality in the underlying reverse H\"older step would force the function into the optimizer family, suggests that explicit sharp constants in related reverse inequalities will typically be unattained as well.
  • Inference: in the range $s-\frac n2\in(0,1)$, the strict inequality in Theorem 1.3 mirrors the mechanism that proves existence of minimizers in the standard range, so one plausible next step is to combine it with a bubble analysis to settle attainment; the author does not take that step here.
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Editorial analysis

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

Referee Report

2 major / 4 minor

Summary. The paper studies quantitative stability for the reverse Sobolev inequality a_{2s}[u] ≥ S_s ||u||_{2n/(n-2s)}^2 on S^n in the admissible range s − n/2 ∈ (0,1) ∪ (1,2). It introduces a modified distance d(u) based on decompositions u = h + ρ with ρ a_{2s}-orthogonal to the tangent space of the optimizer manifold M, proves that this distance is well-defined and positive, and then combines a local spectral expansion near M with a compactness argument in the spirit of Bianchi and Egnell. For s − n/2 ∈ (0,1) the paper obtains c_BE(s) > 0 and c_BE(s) ≤ 4s/(n+2s+2), together with a strict improvement for n ≥ 2; for s − n/2 ∈ (1,2) it claims c_BE(s) = 1 and non-attainment, using an inequality imported from the unpublished preprint [10] together with its equality case.

Significance. If the external input from [10] is correct, Theorem 1.2(b) is a notable result: it would be the first Sobolev-type stability inequality whose sharp constant is explicitly known and which lacks an optimizer. The paper's own contributions are substantial and well executed: the local spectral analysis in Proposition 3.1 is explicit and correct, and the adaptation of the Bianchi–Egnell compactness argument to an indefinite quadratic form through the modified distance d(u) is a genuine technical step. The implicit-function-theorem construction of the local projection (Proposition 3.3) and the proof of positive-definiteness on the orthogonal complement are carefully handled. The principal weakness is that the advertised sharp constant and the non-attainment conclusion are not self-contained: they rest on an unpublished preprint result that is neither proved nor reproduced.

major comments (2)
  1. [Section 3, proof of Theorem 1.2, Eq. (3.1)] The lower bound c_BE(s) ≥ 1 in Theorem 1.2(b) is not proved in the manuscript. The compactness argument establishes only c_BE(s) > 0, and Proposition 3.1(ii) gives the upper bound c_BE(s) ≤ 1; the sharp lower bound c_BE(s) ≥ 1 is imported from [10, Theorem 1.2] in the rephrased form of inequality (3.1), with no proof or verification included. Because (3.1) is an unpublished preprint result and is exactly the step that upgrades an unknown positive constant to the explicit value 1, this is load-bearing for the main claim. The manuscript should either reproduce a proof of (3.1), state and prove the needed consequence directly, or explicitly present Theorem 1.2(b) as conditional on [10] and not independently verified.
  2. [Section 3, proof of Theorem 1.2(b), non-attainment] The non-attainment assertion for s − n/2 ∈ (1,2) is concluded by 'an inspection of the proof of [10, Theorem 1.2]' together with the equality conditions in the reverse Hölder inequality [10, eq. (3.12)]; this argument is not reproduced. The crucial inference that equality in (3.1) forces equality in (3.12), and that this in turn forces u ∈ M, is not demonstrated. Since the claim that c_BE(s) = 1 has no minimizer is one of the two advertised novelties of Theorem 1.2(b), the full equality-case argument should be included in the paper.
minor comments (4)
  1. [Proposition 3.2] In the statement of Proposition 3.2, the phrase 'E(uk) → 0 as n → ∞' should read 'as k → ∞', since the sequence is indexed by k.
  2. [Theorem 1.2, proof] The step saying that the upper bounds in parts (a) and (b) are 'direct consequences of Proposition 3.1' should include the one-line construction of a sequence achieving the asymptotic equality condition; for part (b), for example, one can take ρ_k = ε_k φ_k with φ_k in a high eigenspace and ε_k → 0 so that ∫ ρ_k^2 / a_{2s}[ρ_k] → 0.
  3. [Proposition 2.1, proof] Writing u_Φ − 1 = c + ρ with c ∈ R is confusing, because the constant term of u_Φ is then 1 + c; please rewrite the decomposition as u_Φ = c + ρ with c > 0, or clarify the normalization of the constant.
  4. [References, [5]] Reference [5] is listed with a retraction notice; citing a retracted preprint as support for the plausibility of non-attainment of a stability minimizer should be reconsidered, or the unretracted companion reference [17] should be used instead.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the key lower bound c_BE(s)=1 is imported from the external, non-overlapping preprint [10], not from a self-citation or from a fitted parameter renamed as a prediction.

full rationale

The derivation chain is not circular. The new distance d(u) in (1.12) is a genuine infimum over orthogonal decompositions, not a fitted quantity, and the stability quotient E(u) is defined independently of the claimed value c_BE(s)=1. Theorem 1.2(a), the positivity c_BE(s)>0, and the upper bound c_BE(s)<=1 in part (b) are proved internally by the local expansion in Proposition 3.1, the compactness/contradiction argument in Proposition 3.2, and the implicit-function-theorem localization in Proposition 3.3. Although Proposition 3.2 invokes [8, Propositions 5 and 6], that is prior published work by the same author establishing the reverse Sobolev inequality and its optimizer compactness; it does not assume the target stability constant. The decisive reverse inequality c_BE(s)>=1 in the range s-n/2 in (1,2) comes from (3.1), quoted from the unpublished, non-overlapping preprint [10] by Gong, Yang, and Zhang, and the non-attainment claim comes from the equality conditions in [10, eq. (3.12)]. This is an external dependency and therefore a verification risk: if [10] contains an error, Theorem 1.2(b) would not be established. But it is not circular, since the paper does not assume c_BE(s)=1 or non-attainment in its own setup, and the step is explicitly attributed rather than self-justifying. The remaining self-citations [13,14,15] are used for auxiliary comparisons and a cubic-integral choice, not as substitutes for the main derivation. No fitted input is relabeled as a prediction, and no load-bearing conclusion reduces to its own definition.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

All constants are explicit expressions in n and s; no numbers are fitted to data and no ad hoc parameters are introduced. The modified distance d(u) is a new mathematical definition rather than a postulated entity; it has no independent falsifiable handle and is treated as part of the proof construction. The main external inputs are the reverse Sobolev inequality and optimizer classification from [8], the stability inequality from [10], and standard tools (Brouwer, implicit function theorem, Sobolev embedding). The author's self-citations [8], [13], [14], [15] provide background results, none of which is equivalent to the target stability theorem.

assumptions (8)
  • domain assumption Reverse Sobolev inequality (1.5) with best constant S_s in (1.3) and optimizer set M in (1.7) for s-n/2 in (0,1) union (1,2)
    The paper proves stability of this inequality and takes the inequality, its constant, and its optimizer classification as given from [8]; if they failed, the stability statements lose their meaning.
  • domain assumption Stability inequality (3.1) from [10] for s-n/2 in (1,2): for every 0<u in H^s(S^n) there exists a conformal map Phi with u_Phi=c+rho, rho in E_{>=2}, and deficit >= a_{2s}[rho]
    Unpublished preprint [10] supplies the lower bound c_BE(s)>=1 and the equality condition used for non-attainment; its proof is not reproduced.
  • standard math Brouwer fixed point theorem
    Used in Lemma 2.2 to obtain the center-of-mass decomposition in Proposition 2.1.
  • standard math Implicit function theorem
    Used in Proposition 3.3 to prove local uniqueness of the decomposition near 1.
  • standard math Fractional Sobolev embedding H^s(S^n) into C(S^n) for s>n/2
    Used to turn H^s convergence into uniform convergence in Propositions 3.1, 3.2 and 1.4.
  • domain assumption [8, Proposition 6] compactness of minimizing sequences for the reverse Sobolev quotient
    Used in Proposition 3.2 to obtain strong convergence after conformal rescaling.
  • domain assumption [8, Proposition 5] lower bound for a_{2s}[1+rho] when inf(1+rho)=0, with equality characterization
    Used in the contradiction argument in Proposition 3.2.
  • domain assumption [14, computation] that rho=omega1 omega2 + omega2 omega3 + omega3 omega1 has positive cube integral on S^n
    Used in Theorem 1.3 to prove c_BE(s) < 4s/(n+2s+2); the integral is not computed here.

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Pith. "Pith review of Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere." pith.science (2026). https://pith.science/paper/7AOPL66J

@misc{pith2026250419939,
  author       = {Pith},
  title        = {Pith review of: Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7AOPL66J}},
  note         = {Machine review of arXiv:2504.19939}
}
abstract

We prove a stability inequality associated to the reverse Sobolev inequality on the sphere $\mathbb S^n$, for the full admissible parameter range $s - \frac{n}{2} \in (0,1) \cup (1,2)$. To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator $A_{2s}$ is not positive definite. As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case $s - \frac{n}{2} \in (1,2)$ remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer.

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