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Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group

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Pith's one-line read The paper claims that the optimal stability constant for a critical nonlocal Sobolev inequality on the Heisenberg group is attained and strictly exceeds the local one.

desk verdict Real spectral and two-bubble machinery, but the paper's own text concedes the strict spectral bound (3.4) is unproved, and the proposed repair vanishes by U(1) symmetry, so the attainment theorem does not stand. read the letter →

arxiv 2608.09034 v1 pith:BT4O23ZH submitted 2026-08-10 math.AP

Wenjing Chen , Zexi Wang This is my paper · ORCID
classification math.AP MSC 35B3535P3035A2345E10
keywords HeisenberggroupnonlocalSobolevinequalityquantitativestabilityconstantJerison-LeebubblesHardy-Littlewood-Sobolevcriticalexponentvariationalmethods
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

This paper studies the sharp quantitative stability of the critical nonlocal Sobolev inequality on the Heisenberg group $\mathbb{H}^n$: it asks how much the Sobolev deficit controls the squared distance from a function to the manifold $\mathfrak{M}$ of Jerison--Lee bubbles. Its central aim is to prove that the optimal stability constant $H_{\mathrm{NS}}$ is attained by a function outside $\mathfrak{M}$, and that this value is strictly larger than $H_{\mathrm{BE}}$, the optimal constant for the local Folland--Stein--Sobolev inequality. The proof couples the Hardy--Littlewood--Sobolev interaction with the noncommutative geometry of $\mathbb{H}^n$, diagonalizes the second variation on complex spherical harmonics, and uses two compactness thresholds---a spectral threshold and a two-peaks threshold---to force minimizing sequences to converge. If the argument is right, the nonlocal interaction genuinely raises the optimal lower stability constant, while the universal upper comparison constant remains $1$.

What carries the argument

The load-bearing object is the simultaneous diagonalization of two nonlocal bilinear forms that appear in the second variation after the Cayley transform from $\mathbb{H}^n$ to $S^{2n+1}$. On each bidegree space $H^{n+1}_{i,j}$ of complex spherical harmonics the two forms act by eigenvalues $\nu_{i,j}$ and $\tau_{i,j}$, and the lowest normal mode $(2,0)$ yields the spectral threshold $H_{\mathrm{NS}}^{\mathrm{spec}} = \frac{(4n+8-2\mu)(\mu+4)}{2(n+4)(4n+8-\mu)}$, which equals $1 - \left(\frac{Q^*_\mu-1}{\nu_{2,0}} + \frac{Q^*_\mu}{\tau_{2,0}}\right)$. A separate construction, a two-bubble expansion using Heisenberg inversion and the distance formula, gives the two-peaks threshold $2 - 2^{\frac{2n}{4n+4-\mu}}$. Both thresholds are proved to be strict upper bounds for $H_{\mathrm{NS}}$, and these strict inequalities are the barriers that exclude concentration on the bubble manifold and splitting into two bubbles.

What would settle it

Compute $\int_{S^{2n+1}} \omega_{2,0}(\zeta)^3\,d\zeta$ for the proposed perturbation $\omega_{2,0}=\zeta_1\zeta_2+\zeta_1\zeta_3+\zeta_2\zeta_3$. The rotation $\zeta\mapsto e^{i\theta}\zeta$ multiplies the integrand by $e^{6i\theta}$, so the integral is zero; hence condition (5.2) fails and the strict inequality $H_{\mathrm{NS}} < H_{\mathrm{NS}}^{\mathrm{spec}}$ is not established by the displayed perturbation.

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Extended reading notes

Core claim

The paper's central claim is that the optimal stability constant $H_{\mathrm{NS}}$ for the critical nonlocal Sobolev inequality on the Heisenberg group is attained, not merely approached, under the conditions $\mu\le 4$ or (3.5), and that this attained value is strictly larger than $H_{\mathrm{BE}}$, the optimal constant for the local Folland--Stein--Sobolev inequality. The route is a two-threshold compactness argument: a spectral threshold produced by the lowest coupled eigenmode of the nonlocal second variation, and a two-peaks threshold produced by the interaction of two separated Jerison--Lee bubbles. The paper also establishes that the sharp universal upper comparison constant is $1$, with equality only on the bubble manifold, and that the associated Euler--Lagrange residual quotient obeys a strict single-bubble upper bound. In Section 5.1 the text itself records that the cubic strictness step, as written, does not establish the spectral threshold and that the proposed repair has zero cubic integral; that caveat is the load-bearing open point in the proof.

Load-bearing premise

The strict spectral upper bound needs a perturbation mode whose cubic integral on the unit sphere is nonzero; the mode displayed in the paper has zero cubic integral, and the paper's own text says the cubic strictness argument does not establish the bound as written.

Editorial extensions

If this is right

  • Under $\mu\le 4$ or condition (3.5), a minimizing sequence can neither approach the bubble manifold (spectral barrier) nor split into two bubbles (two-peaks barrier), so $H_{\mathrm{NS}}$ is attained.
  • The attained minimizer lies outside $\mathfrak{M}$, and the sharp Hardy--Littlewood--Sobolev inequality then forces the strict comparison $H_{\mathrm{NS}} > H_{\mathrm{BE}}$.
  • Every admissible function satisfies deficit $\le \mathrm{dist}(u,\mathfrak{M})^2$, the constant $1$ is optimal, and equality holds exactly on the bubble manifold.
  • The Euler--Lagrange residual quotient has a strict single-bubble upper bound equal to the same spectral constant, giving quantitative stability for critical points independent of functional attainment.
  • Both compactness thresholds for the nonlocal problem exceed the corresponding thresholds of the local Folland--Stein--Sobolev problem, so the nonlocal term changes the quantitative structure despite the common extremal manifold.

Reading between the lines

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

  • A repaired fourth-order expansion or a different choice of $(2,0)$ mode could extend the attainment theorem beyond $\mu\le 4$; the two-peaks barrier already holds for every $0<\mu<Q$.
  • The comparison $H_{\mathrm{NS}}>H_{\mathrm{BE}}$ is a purely nonlocal effect: since both problems share the same bubble manifold, any difference in the optimal constant must come from the Hardy--Littlewood--Sobolev kernel rather than from the extremals.
  • The same simultaneous-diagonalization strategy should apply to analogous critical nonlocal inequalities on other groups with spherical-harmonic decompositions, such as H-type groups or CR spheres of higher dimension.
  • A numerical scan of the sign condition $(4-\mu)(4n+8-\mu)+3\mu(\mu+4)>0$ over integer $n$ and $\mu\in(0,Q)$ would immediately show which parameter ranges survive a corrected perturbation.
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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 / 4 minor

Summary. The paper studies quantitative stability for the critical nonlocal Sobolev inequality on the Heisenberg group, where the deficit of the Hardy-Littlewood-Sobolev-type Sobolev quotient is compared with the squared distance to the manifold of Jerison-Lee bubbles. The main results are: two strict upper bounds for the optimal stability constant H_NS (a spectral bound and a two-peaks bound, Theorem 3.1); attainment of H_NS under either mu <= 4 or the condition (3.5) (Theorem 3.2); the strict comparison H_NS > H_BE with the local Folland-Stein-Sobolev stability constant (Remark 3.1); the sharp universal upper bound 1 with equality characterization (Theorem 3.3); and a strict single-bubble upper bound for a critical-point residual quotient (Theorem 3.4). The proof strategy combines diagonalization of two nonlocal quadratic forms on complex spherical harmonics, a local perturbation around a bubble, a two-bubble interaction expansion, and a Heisenberg-group profile-decomposition compactness argument.

Significance. If the main results were established, the paper would be a substantial contribution: it identifies nonlocal-Heisenberg-specific compactness thresholds that differ from both the Euclidean nonlocal problem and the local CR problem, proves existence of minimizers for the stability constant, and upgrades the comparison with the local stability constant to a strict one. The paper has several sound components: the explicit diagonalization of the two bilinear forms in Section 4, the two-bubble expansion in Section 5.2 leading to the bound H_NS < 2 - 2^{1/Q*_mu}, the high-mode argument in Section 7 for the sharp universal upper bound, and the overall profile-decomposition framework. However, the central spectral strictness step (3.4) is explicitly admitted in Section 5.1 to be unproved, and the proposed repair is invalid; since Theorems 3.2 and 3.4 rely on that step, the main claims are not supported in the stated form.

major comments (3)
  1. [Section 5.1, Eq. (5.2) and Theorem 3.1-(3.4)] The proof of the strict spectral upper bound (3.4) reduces to the sign condition (5.2) for a spherical harmonic omega in H^{n+1}_{2,0}. The manuscript itself states that 'the cubic strictness argument in its present form does not establish (3.4)', and the proposed repair omega = zeta1*zeta2 + zeta1*zeta3 + zeta2*zeta3 does not work. Under the rotation zeta -> e^{i theta} zeta on S^{2n+1}, omega^3 transforms by e^{6 i theta}, so the standard rotation-invariant measure forces int_{S^{2n+1}} omega^3 = 0 for every omega in H^{n+1}_{2,0}; in particular the displayed value 6 * int zeta1^2 zeta2^2 zeta3^2 is itself zero because that monomial has holomorphic degree 6. Therefore condition (5.2) fails for every (2,0)-harmonic, and the derivation of (3.4) collapses.
  2. [Section 6, Proposition 6.1 and Theorem 3.2] The mu <= 4 branch of Theorem 3.2 uses Lemma 4.1 together with the strict inequality H_NS < H_spec_NS from (3.4) to exclude minimizing sequences approaching the bubble manifold. Since (3.4) is unproved, the compactness barrier for mu <= 4 is missing, and the attainment conclusion of Theorem 3.2 and the strict comparison H_NS > H_BE in Remark 3.1 are unsupported in that parameter range. The same invalid sign condition (5.2) is invoked in the proof of Theorem 3.4 in Section 8, so the critical-point strict bound is also unsupported.
  3. [Section 5.1, internal consistency] The manuscript explicitly flags the gap in the cubic strictness argument and then attempts to repair it with a computation that contradicts the symmetry statement made on the same page. This internal inconsistency shows that the issue is not a minor omitted detail: the chosen test function cannot produce the required definite sign, and a corrected perturbation scheme or fourth-order expansion would be a substantial new ingredient rather than a local fix.
minor comments (4)
  1. [Section 5.1, proposed test function] For n = 1 the proposed harmonic omega = zeta1*zeta2 + zeta1*zeta3 + zeta2*zeta3 is not defined on S^3, since S^{2n+1} = S^3 has only two complex coordinates; the proof of (3.4) for n = 1 would require a separate argument in any case.
  2. [Section 5.1, exposition] The sentence admitting that the cubic strictness argument does not establish (3.4) should not appear in a submitted proof without a complete resolution; the subsequent 'repair' should either be verified or removed, as the current text contradicts itself.
  3. [Section 7, proof of Theorem 3.3] The limiting argument for the sharp upper bound 1 would be easier to follow if the asymptotic behavior nu_{i,j}, tau_{i,j} -> infinity as i+j -> infinity were stated explicitly before letting i+j tend to infinity.
  4. [Section 3 and Remark 3.1] The paper imports the basic positive lower bound H_NS > 0 from the companion manuscript [12]; this dependence should be stated explicitly in the statement of Theorem 3.1 or the beginning of Section 6, since the reader cannot verify that imported result from the present text alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central thresholds and attainment are self-contained expansions; the self-cited companion result is background, and the admitted gap in (3.4) is a correctness issue, not circularity.

full rationale

The paper's main claims—the spectral threshold H_spec_NS, the two-peaks threshold, attainment of H_NS, the strict comparison H_NS > H_BE, and the sharp universal upper bound H_UB = 1—are derived from direct second-variation expansions around Jerison–Lee bubbles, the Funk–Hecke diagonalization (2.6)–(2.8), the two coupled eigenvalue problems in Section 4, and a two-bubble asymptotic expansion. No parameter is fitted to a target constant, and no quotient is normalized in a way that makes a predicted quantity equal to an input by construction. The only self-citation is the authors' companion paper [12], invoked in Section 3 to recall an already-known positive lower bound H_NS > 0 for mu <= 4; this bound is not used as a compactness threshold and does not enter the proofs of Theorems 3.1–3.4, so it is not load-bearing. The cited nondegeneracy and spectral results of Yang–Zhang [45], Zhang–Xu–Wang [49], and Tang–Zhang–Zhang [43] are external inputs whose stated assumptions do not include the target constant H_NS; citing them is independent support, not circularity. The paper itself flags in Section 5.1 that 'the cubic strictness argument in its present form does not establish (3.4); a corrected perturbation scheme or a fourth-order expansion is needed.' That is an admitted proof gap affecting the mu <= 4 branch of Theorem 3.2 and Theorem 3.4, but it is a mathematical correctness failure, not a self-referential reduction: the claimed strict inequality does not follow by construction from its hypotheses. Per the review rules, such a correctness risk is not circularity. Therefore no circular step is exhibited.

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

No data-fitting parameters appear. The paper rests on external spectral and nondegeneracy results and on the authors' own companion stability theorem [12]; the decisive missing step is not an axiom but an unproved strictness claim, which is recorded in red_flags.

assumptions (5)
  • domain assumption Sharp HLS inequality on H^n, classification of extremals of (2.2), and nondegeneracy of bubbles (Lemmas 2.1-2.2)
    These external theorems define the bubble manifold M and the Euler-Lagrange structure used throughout; cited from [24,45,48].
  • standard math Funk-Hecke diagonalization (2.6)-(2.8) for the HLS kernel on S^{2n+1}
    Used in Section 4 to obtain the spectra ν_{i,j} and τ_{i,j} that define the spectral threshold.
  • standard math Nonlocal Brezis-Lieb splitting and Lions concentration-compactness on the Heisenberg group
    Used in Section 6 to decompose minimizing sequences; quoted from [25,30,34].
  • standard math Korányi-Reimann inversion formula for the Heisenberg distance
    Used in Proposition 5.1 to compute the two-bubble interaction expansion.
  • domain assumption Positive quantitative stability lower bound H_NS > 0 from the companion paper [12, Theorem 1.1]
    Invoked as the starting point of (3.1); not reproved here and is the authors' own result.

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Pith. "Pith review of Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group." pith.science (2026). https://pith.science/paper/BT4O23ZH

@misc{pith2026260809034,
  author       = {Pith},
  title        = {Pith review of: Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BT4O23ZH}},
  note         = {Machine review of arXiv:2608.09034}
}
abstract

We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{nS} S_{HL}(Q,\mu)\left(\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}} \frac{|u(\xi)|^{Q^{\ast}_{\mu}}|u(\eta)|^{Q^{\ast}_{\mu}}} {|\eta^{-1}\xi|^{\mu}}\,d\xi d\eta\right)^{\frac{1}{Q^{\ast}_{\mu}}} \leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}}u|^{2}d\xi, \qquad u\in S^{1,2}(\mathbb{H}^{n}), \end{equation} where $Q=2n+2$, $n\geq1$, $0<\mu<Q$, and $Q^{\ast}_{\mu}=(2Q-\mu)/(Q-2)$. Let $\mathfrak{M}$ denote the manifold of Jerison-Lee bubbles, and let $H_{NS}$ be the infimum of the quotient between the deficit in \eqref{nS} and $\mathrm{dist}(u,\mathfrak{M})^{2}$. The central issue is that the Euclidean nonlocal problem and the local Folland-Stein-Sobolev problem each possess their own spectral and compactness structures, whereas the present problem couples the HLS interaction with the noncommutative conformal geometry of $\mathbb{H}^{n}$. Under the stated conditions on $(n,\mu)$, these strict thresholds, together with a Heisenberg-group profile decomposition and a nonlocal Br{e}zis-Lieb splitting, imply that $H_{NS}$ is attained. The minimizer then yields the strict comparison $H_{NS}>H_{BE}$ with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of $H_{NS}$.

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    2025

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.