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Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants

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Pith's one-line read The Sobolev inequality on the Heisenberg group has an asymptotically sharp $1/Q$ stability constant, proved by replacing rearrangement flows with the CR Yamabe flow.

desk verdict Genuinely new proof of the first 1/Q stability bound on the Heisenberg group, but the paper as written has two real gaps (n=1 excluded by the local analysis, and CR Yamabe flow convergence cited for smooth data but applied to S^1 functions) that a serious revision should fix. read the letter →

arxiv 2507.12725 v2 pith:GALOXJBC submitted 2025-07-17 math.AP

Lu Chen , Guozhen Lu , Hanli Tang , Bohan Wang This is my paper · ORCID
classification math.AP MSC 35A2335R0353C1758J35
keywords SobolevinequalityHeisenberggroupstabilityoffunctionalinequalitiesCRYamabeflowbisphericalharmonicsHardy-Littlewood-Sobolevdimension-dependentconstantsasymptoticallysharp
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 proves that the sharp Sobolev inequality on the Heisenberg group is quantitatively stable in the best possible dimension-dependent way: there is a positive constant $\beta_0$, independent of $Q$, such that the Sobolev deficit of any real function $f$ is at least $(\beta_0/Q)\,d(f,M)^2$, where $d(f,M)^2$ is the squared distance to the extremal family. The factor $1/Q$ is optimal because an existing upper bound shows the best stability constant is at most $4/(Q+6)$. Since rearrangement inequalities fail in the Heisenberg setting, the authors replace rearrangement-flow arguments with the CR Yamabe flow on the CR sphere, which decreases Sobolev energy, preserves volume, and converges to an extremal. The same strategy, through a Legendre-duality argument, yields an asymptotically sharp stability estimate for the Hardy–Littlewood–Sobolev inequality at the conformal index $\lambda=Q-2$. The result matters because it supplies sharp quantitative versions of these classical inequalities in a geometry where symmetrization is unavailable.

What carries the argument

The proof rests on three mechanisms. First, the bispherical harmonics $H_{j,k}$ on the CR sphere $S^{2n+1}$ form eigenspaces of the conformal sublaplacian $L$, with eigenvalues $\lambda_{j,k}=((Q-2)/4+j)((Q-2)/4+k)$; the growth of these eigenvalues provides the spectral gap behind the local stability estimate. Second, a perturbation is written as $u=1+r$, and $r$ is cut at two heights into $r=r_1+r_2+r_3$, with the three pieces estimated separately; the factor $1/Q$ enters through $\theta=q-2=4/(Q-2)$. Third, the CR Yamabe flow $\partial\theta/\partial t=(r_\theta-R_\theta)\theta$, written in terms of the conformal factor $u$ as $\partial u/\partial t=(n/2)(r_\theta-R_\theta)u$, monotonically decreases the total Webster scalar curvature (the Sobolev energy), preserves volume, and converges to a constant-Webster-curvature limit; by the classification of extremals of the CR Yamabe equation, that limit is an extremal. This flow takes the role played in Euclidean proofs by rearrangement flows, which are unavailable here because the Heisenberg group lacks the relevant symmetrization inequalities. For the Hardy–Littlewood–Sobolev application, the additional ingredient is the Legendre-duality argument that transfers Sobolev stability into $L^{2Q/(Q+2)}$ distance.

What would settle it

Run the CR Yamabe flow from a finite-energy but non-smooth initial datum on $S^{2n+1}$: if it does not converge in $S^1$ to an extremal, the reduction of global to local stability in Theorem 2.1 collapses. Alternatively, evaluate the quotient $\big(E[u]-S_Q\|u\|^2\big)/\inf_{g\in M^*}E[u-g]$ on a sequence $u_k$ with fixed distance to $M^*$: a decay faster than $\beta_0/Q$ would contradict Theorem 1.1.

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Extended reading notes

Core claim

On the Heisenberg group $\mathbb{H}^n$ with homogeneous dimension $Q=2n+2$, the sharp Sobolev inequality reads $\int_{\mathbb{H}^n}|\nabla_{\mathbb{H}^n}f|^2\,dzdt \ge S_Q\|f\|_{2Q/(Q-2)}^2$, with equality precisely on the family $M$ of real extremals. Theorem 1.1 asserts that there is a constant $\beta_0>0$, independent of $Q$, such that $\int_{\mathbb{H}^n}|\nabla_{\mathbb{H}^n} f|^2\,dzdt - S_Q\|f\|_{2Q/(Q-2)}^2 \ge \frac{\beta_0}{Q}\,d(f,M)^2$ for every $f\in S^1(\mathbb{H}^n)$, where $d(f,M)^2=\inf_{h\in M}\|\nabla_{\mathbb{H}^n}(f-h)\|_2^2$. This is asymptotically sharp: an existing upper bound gives the best possible constant at most $4/(Q+6)$, so the $1/Q$ rate cannot be improved as $Q\to\infty$. The proof passes by Cayley transform to the CR sphere $S^{2n+1}$, establishes an optimal local stability estimate using bispherical harmonics, orthogonality conditions, and a three-level cutting of the perturbation, and then converts local stability into global stability by running the CR Yamabe flow, which decreases the conformal energy, preserves volume, and converges to a constant-Webster-curvature metric. Any such limit is an extremal, so the flow connects every point of the energy space to the extremal manifold. The same machinery, applied through the Legendre-transform dual stability method, gives the analogous asymptotically sharp stability bound for the Hardy–Littlewood–Sobolev inequality on $\mathbb{H}^n$ at $\lambda=Q-2$.

Load-bearing premise

The global-to-local reduction assumes that the CR Yamabe flow, started from an arbitrary finite-energy function on the CR sphere, converges in the energy space to a constant-Webster-curvature extremal; the convergence results cited in Section 2.2 are stated for smooth initial data, and no approximation argument for rough data is given.

Editorial extensions

If this is right

  • Any $f$ whose distance to the extremal family is $\varepsilon$ has Sobolev deficit at least $(\beta_0/Q)\varepsilon^2$, so near-minimizers must lie quantitatively close to a bubble.
  • Because the best possible constant is at most $4/(Q+6)$, the lower bound $\beta_0/Q$ is asymptotically optimal as $Q\to\infty$; no dimension-independent rate better than $1/Q$ can hold.
  • The Hardy–Littlewood–Sobolev inequality on $\mathbb{H}^n$ at $\lambda=Q-2$ inherits a stability estimate with the same $1/Q$ rate and a dimension-free constant.
  • The rearrangement-free strategy extends to fractional Sobolev and HLS inequalities on the Heisenberg group once the corresponding continuous conformal flows are available.
  • On the CR sphere, the local stability statement gives an explicit constant $c_0/Q$ for functions satisfying the orthogonality conditions, sharpening the spectral information associated with the bispherical-harmonic eigenvalues.

Reading between the lines

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

  • A direct next step is to implement the same program for fractional Sobolev and HLS inequalities on $\mathbb{H}^n$; once a fractional flow exists, the monotone-flow reduction should reproduce a dimension-independent constant with rate $1/Q$.
  • The structure suggests that any conformally invariant inequality whose extremal set is a smooth finite-dimensional manifold and whose associated flow is monotone and convergent should exhibit an asymptotically sharp $1/Q$ stability constant, in settings far beyond the Heisenberg group.
  • A technical continuation would be to add a density or regularization argument that extends the convergence of the CR Yamabe flow from smooth initial data to the full energy space, closing the gap between the cited flow theorems and the generality of Theorem 2.1.
  • One can test sharpness by evaluating the quotient (deficit)/(distance squared) on high-frequency bispherical-harmonic perturbations of an extremal; the local analysis predicts saturation at order $1/Q$, matching the global upper bound.
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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 / 5 minor

Summary. The paper proves an asymptotically sharp stability estimate for the Folland-Stein Sobolev inequality on the Heisenberg group, namely a deficit lower bound of order d(f,M)^2/Q with a constant independent of the homogeneous dimension Q. The proof transfers the problem to the CR sphere via the Cayley transform, proves a local stability estimate near the extremal manifold using bispherical harmonics and spectral gap arguments, and then attempts to pass from local to global stability by running the CR Yamabe flow. A dual Legendre-transform argument then yields a stability estimate for the Hardy-Littlewood-Sobolev inequality at the conformal index λ=Q−2.

Significance. If the proof is completed, the result is a significant advance: it gives the first asymptotically optimal dimensional lower bound for stability of Sobolev inequalities on the Heisenberg group, matching the upper bound of Liu-Zhang up to a universal factor. The local spectral analysis is explicit, and the replacement of rearrangement flows by the CR Yamabe flow is a natural and potentially reusable idea. The derivation also correctly identifies the role of the CR sphere and uses sharp external benchmarks (Jerison-Lee, Frank-Lieb, Ho) rather than fitting constants. However, the central global-to-local reduction currently rests on an unproved regularity assertion for the CR Yamabe flow, and the local stability lemma does not cover the case n=1 stated in the main theorem.

major comments (3)
  1. [§2.3, Eq. (2.15)] The CR Yamabe flow is started from an arbitrary nonnegative u0 in S^1(S^{2n+1}), and the argument relies on convergence of this flow to a unique extremal in S^1, citing Ho [22] and [23]. As stated, those convergence theorems are for smooth positive initial data; no approximation argument is supplied for u0 merely in S^1, which may even vanish on a set of positive measure. The existence of a time t0 with inf E[u(t0)−g] = δ E[u(t0)], the continuity of the map t ↦ u(t) in S^1, and the uniformity of the energy monotonicity under approximation are all needed for inequality (2.16). Without a density argument, the quantitative global-to-local reduction — the main methodological innovation of the paper — is not established for the function class in Theorem 1.1.
  2. [Theorem 1.1 vs. Lemma 3.1] Theorem 1.1 is stated for every n, but Lemma 3.1 and the supporting Propositions 3.3–3.5 are proved only for n≥2, where q=2Q/(Q−2)≤3. The case n=1 gives q=4 and is outside the stated range of Lemma 3.2 and the subsequent estimates. No separate argument is given for n=1. This is repairable by invoking Loiudice's positive stability constant [27] for n=1 and absorbing it into β0, since β0 only needs to be independent of Q, but the manuscript does not say this.
  3. [§2.3, Eq. (2.16)] The inequality chain leading to (2.16) uses the monotonicity of the normalized Sobolev deficit along the flow and the preservation of the L^q norm. These properties are derived formally for smooth solutions in §2.2, and their validity for rough S^1 initial data is precisely the missing approximation step described above. This is not a local technicality but the bridge between the local stability lemma and the global theorem.
minor comments (5)
  1. [§3.3] The first sentence of §3.3 says 'we can ensure that I3 ≥ 0', but the subsequent argument concerns I2; this should be corrected.
  2. [§3.3, Eq. (3.15) and following] The exponent in (3.15) should be (k+j)/2 rather than k/2, and a few lines later '3^{K+j}' should read '3^{K+J}'. The intended estimate is clear, but the displayed formulas are inconsistent.
  3. [§2.4] The notation for the extremal set alternates between M^* and M^*_{rea}; the authors should choose one notation and use it consistently.
  4. [§2.1] The new proof for the sphere Sn is a useful exposition, but it is somewhat long relative to its role in the paper; a short remark that the same scheme is used for CR would help the reader.
  5. [§1] In the abstract and introduction, 'Pólya-Szegő' appears with inconsistent spelling; please normalize throughout.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the global-to-local reduction uses the CR Yamabe flow and external convergence/classification theorems, and the local stability proof is a new spectral computation; only non-circular regularity gaps remain.

full rationale

The paper's main chain is: (i) prove local stability on the CR sphere by bispherical harmonics (Section 3); (ii) propagate it to global stability via the CR Yamabe flow (Section 2.3); (iii) transfer to H^n by the Cayley transform and to HLS by Carlen's Legendre duality. None of these steps defines its output in terms of its input. The constant ν*(δ) is an infimum over an explicit set, not a fitted parameter; the flow step uses only monotonicity of the Webster energy, volume preservation, and convergence of the CR Yamabe flow to an extremal, all cited from independent external work (Ho [22,23], Jerison-Lee [24,25], Frank-Lieb [19]). The local stability Lemma 3.1 is proved directly from spectral bounds on H_{j,k} and the pointwise inequality (3.4), with constants chosen by explicit inequalities rather than fitted to the target theorem. The upper bound used for optimality (Liu-Zhang [26]) is also external. The authors' own works [10,11,12] appear only as context for the Euclidean analogues and are not premises of Theorem 1.1. Two technical gaps are noted but are not circularity: the CR Yamabe flow convergence theorem is quoted for smooth data while the flow is started from S^1 data in (2.15) with no approximation argument, and Lemma 3.1 is proved only for n≥2 while Theorem 1.1 states all n. These are rigor gaps that should be fixed, but they do not make the derivation reduce to its own assumptions.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new entities are introduced. The proof relies on cited external theorems (Jerison-Lee, Frank-Lieb, Ho, Duoandikoetxea, Folland) and on hand-chosen auxiliary constants. The only quantity that could be seen as free is epsilon_0, but it is not fitted to the conclusion; the theorem holds for any choice in (0,1/3) after adjusting other parameters.

free parameters (1)
  • epsilon_0 = not specified, in (0,1/3)
    Chosen by hand in Section 3 (Lemma 3.1); all subsequent constants gamma, epsilon_1, epsilon_2, M, delta_tilde are expressed through it. The final beta_0 is therefore existential, not a computed number.
assumptions (5)
  • domain assumption Sharp Sobolev inequality on the Heisenberg group with extremals (Jerison-Lee [25], Frank-Lieb [19])
    Used as the baseline inequality in Theorem 1.1 and to identify the extremal set M* on the CR sphere in Section 2.
  • domain assumption Convergence of the CR Yamabe flow on S^{2n+1} to a constant Webster scalar curvature limit
    Section 2.2 cites Ho [22,23]; the reduction of global stability to local stability requires the flow to converge in S^1 to an extremal for the initial data u0 in S^1.
  • domain assumption Spectral gap of the conformal sublaplacian on bispherical harmonics, lambda_{j,k}=((Q-2)/4+j)((Q-2)/4+k)
    Equation (1.6), used in Section 3.1 to show I1 >= 0 via the projection onto harmonics of degree at least 2.
  • domain assumption L^p bounds for spherical harmonics on S^{2n+1} (Duoandikoetxea [16])
    Proposition 3.5, inequality (3.15) controls projections of the truncation r2 onto high-degree harmonics, a key step for I2 >= 0.
  • domain assumption Classification of solutions of the CR Yamabe equation on S^{2n+1} (Jerison-Lee [25])
    Section 2.2 uses this to identify the limit of the CR Yamabe flow with an extremal of the Sobolev inequality.

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Pith. "Pith review of Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants." pith.science (2026). https://pith.science/paper/GALOXJBC

@misc{pith2026250712725,
  author       = {Pith},
  title        = {Pith review of: Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GALOXJBC}},
  note         = {Machine review of arXiv:2507.12725}
}
read the original abstract

In this paper, we are concerned with the optimal asymptotic lower bound for the stability of Sobolev inequality on the Heisenberg group. We first establish the optimal local stability of Sobolev inequality on the CR sphere through bispherical harmonics and complicated orthogonality technique ( see Lemma 3.1). The loss of rearrangement inequality in the CR setting makes it impossible to use any rearrangement flow technique (either differential rearrangement flow or integral rearrangement flow) to derive the optimal stability of Sobolev inequality on the CR sphere from corresponding optimal local stability. To circumvent this, we will use the CR Yamabe flow to establish the optimal stability of Sobolev inequality on the Heisenberg group with the dimension-dependent constants (see Theorem 1.1). As an application, we also establish the optimal stability of the Hardy-Littlewood-Sobolev (HLS) inequality for special conformal index with the dimension-dependent constants (see Theorem 1.3). Our approach is rearrangement-free and can be used to study the optimal stability problem for fractional Sobolev inequality or HLS inequality on the Heisenberg group once the corresponding continuous flow is established.

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