{"id":"f1f96db9-ea76-4266-b1ee-ca18d9c85e4c","arxiv_id":"2507.12725","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"On the Heisenberg group, the Sobolev inequality is stable with the asymptotically optimal lower bound β0/Q times the squared distance to extremals.","lead":"This paper proves a sharp quantitative version of the Sobolev inequality on the Heisenberg group, with the optimal dimension-dependent constant 1/Q. The proof uses the CR Yamabe flow to bypass the rearrangement symmetrization that fails in this setting, and as a byproduct gives a sharp stability bound for a Hardy-Littlewood-Sobolev inequality.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Global-to-local reduction via the CR Yamabe flow is applied to S^1 initial data, but the cited convergence theorems require smooth data and no approximation argument is supplied.","rationale":"We read the paper as a serious attempt to prove asymptotically sharp (1/Q) stability on the Heisenberg group without rearrangement. The local spectral analysis (Section 3) and the Legendre-duality application (Section 4) are substantive, and the constants are tracked carefully enough that the 1/Q scaling plausibly survives. The reader's CONDITIONAL verdict is appropriate. Our independent pass identifies the same weakest point: the global-to-local reduction in Section 2.3 applies a convergence theorem for the CR Yamabe flow to initial data of low regularity. This is not a disagreement with consensus; it is an omitted proof step. The cited references [22,23] are for smooth data, and the manuscript itself nowhere proves existence, continuity, or convergence for u0 in S^1. Because every subsequent estimate in Section 2.3 — the crossing time t0, the monotonicity chain, and inequality (2.16) — depends on having a genuine flow curve u(t) with continuous S^1 trajectories, the gap is load-bearing. The n≥2 restriction in Lemma 3.1 is a second, smaller gap: as stated, Theorem 1.1 covers n=1, but the proof of Lemma 3.1 requires q=2Q/(Q−2) ≤ 3, i.e. n≥2, and no separate n=1 argument is given. This affects the statement's full generality but not the asymptotic Q→∞ claim. If the authors supply the missing approximation/regularity argument, the central theorem would be established for n≥2 (and n=1 separately if handled). We see no reason to reject; the verdict remains CONDITIONAL.","tokens_in":23714,"tokens_out":24856,"duration_ms":266517,"concrete_test":"Verify the hypotheses of Ho [22, Thm 1.1] and [23] for the CR Yamabe flow on S^{2n+1}: if smoothness of the initial conformal factor is required, perform the omitted approximation: take u0,ε in C∞(S^{2n+1}) positive with u0,ε→u0 in S^1, run the flow from u0,ε, and check that E[u0,ε]→E[u0], that the monotonicity estimates in Section 2.2 hold uniformly in ε, and that the inequalities (2.16) pass to the limit using lower semicontinuity: Def(u0) ≥ liminf_ε Def(u0,ε) and d^2(u0,M*) ≤ liminf_ε d^2(u0,ε,M*). If either limit inequality fails, Theorem 2.1 is not established for S^1 data.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction (Section 2.3, Eq. (2.15)) starts the CR Yamabe flow from an arbitrary nonnegative u0 in S^1(S^{2n+1}) and asserts a unique limit u∞ in M* with convergence in S^1. The paper cites Ho [22] (and [23]) for convergence, but those theorems are stated for smooth positive initial contact forms/conformal factors. For u0 merely in S^1 — and possibly vanishing on a set of positive measure — neither short-time existence with continuous dependence nor long-time convergence is automatic. The argument that selects t0 with inf_{h in M*} E[u(t0)-h] = δ E[u(t0)] uses continuity of t maps to u(t) in S^1 and continuity of the distance to M*, and the inequality (2.16) needs Def and E to pass through the flow. No density argument is supplied to approximate u0 by smooth positive u0,ε and to show that the final constant survives ε→0 (e.g. that E[u0,ε]→E[u0], d^2(u0,ε,M*)→d^2(u0,M*), and energy monotonicity is uniform in ε). Without this, the quantitative global-to-local link — the main methodological innovation replacing rearrangement flows — is not proved. Secondary: Lemma 3.1 is stated only for n≥2, while Theorem 1.1 claims all n; the case n=1 (q=4) lies outside the range 2≤q≤3 used in Lemma 3.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23977,"tokens_out":12231,"duration_ms":144399,"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":[{"comment":"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.","section":"§2.3, Eq. (2.15)"},{"comment":"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.","section":"Theorem 1.1 vs. Lemma 3.1"},{"comment":"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.","section":"§2.3, Eq. (2.16)"}],"minor_comments":[{"comment":"The first sentence of §3.3 says 'we can ensure that I3 ≥ 0', but the subsequent argument concerns I2; this should be corrected.","section":"§3.3"},{"comment":"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.","section":"§3.3, Eq. (3.15) and following"},{"comment":"The notation for the extremal set alternates between M^* and M^*_{rea}; the authors should choose one notation and use it consistently.","section":"§2.4"},{"comment":"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.","section":"§2.1"},{"comment":"In the abstract and introduction, 'Pólya-Szegő' appears with inconsistent spelling; please normalize throughout.","section":"§1"}],"recommendation":"major_revision","confidential_remarks":"The central idea is sound and the local analysis is substantial, but the flow-regularity gap in §2.3 is load-bearing and must be fixed by a density/approximation argument or by citing a flow-convergence theorem valid for S^1 initial data. The n=1 omission is easy to repair. I do not see grounds for rejection, but the current version does not fully prove the stated theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is the first asymptotically sharp (1/Q) stability bound for the Sobolev inequality on the Heisenberg group, and the proof mechanism is genuinely new. The authors replace the rearrangement flow, which does not exist on H^n, with the CR Yamabe flow. That is a real idea, and the local bispherical-harmonic computation in Lemma 3.1 is a nontrivial adaptation of the Euclidean proof. The HLS transfer via Legendre duality is clean and standard.\n\nThe result matches Liu-Zhang's upper bound 4/(Q+6), so the dimension rate is optimal. The global-to-local reduction via the Yamabe flow is conceptually nice, and the authors correctly note that this flow keeps the volume fixed and decreases the energy. The paper is honestly written about its debts.\n\nNow the soft spots. Two gaps stand out, and both are real. First, Lemma 3.1 is proved only for n >= 2 (q <= 3), but Theorem 1.1 claims all n. For n = 1 the Sobolev exponent is q = 4, which lies outside the range of Lemma 3.2 and the surrounding estimates. Either the local computation needs to be extended to q = 4, or the main theorem should be restricted to n >= 2. As written, the statement is unjustified.\n\nSecond, the flow step: Section 2.3 starts the CR Yamabe flow from an arbitrary nonnegative u0 in S^1(S^{2n+1}) and asserts convergence in S^1 to an extremal. The cited results of Ho are for smooth initial contact forms (or smooth positive conformal factors). No approximation argument is given to show the constants survive passage from smooth u0,epsilon to u0. The inequality (2.16) needs continuity of the energy and the distance along the flow, so this is a genuine bridge, not a minor footnote. It is probably fixable, since the functionals are continuous on S^1, but the paper does not do it.\n\nMinor quibble: the paper says \"explicit\" constants, but Theorem 1.1 only asserts existence of beta0, and the local parameters are given through epsilon0 without a concrete value. Not a problem for the main result, but the word \"explicit\" is doing more work than the proof supports.\n\nBottom line: this is a serious contribution and deserves a serious referee. The main innovation is sound and the result, if correct, settles a natural open problem. But the two gaps mean the proof, as written, does not cover the stated theorems. I would send it for review with an eye toward a major revision: extend or exclude n=1, and supply the density/approximation argument for the CR Yamabe flow. If those are filled, this will be a nice paper.","headline":"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.","tokens_in":24602,"tokens_out":3233,"would_cite":true,"duration_ms":35340,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A23","35R03","53C17","58J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Sobolev inequality","Heisenberg group","stability of functional inequalities","CR Yamabe flow","bispherical harmonics","Hardy-Littlewood-Sobolev inequality","dimension-dependent constants","asymptotically sharp constants"],"falsifier":"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.","tokens_in":23465,"feed_emoji":"📐","tokens_out":12828,"duration_ms":131769,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sobolev stability on the Heisenberg group is sharp at order 1/Q","feed_subtitle":"Deficit bound matches the best possible rate in high dimension, via a rearrangement-free flow proof.","key_machinery":"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.","core_discovery":"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 $\\lambda=Q-2$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the local-stability template: cutting the perturbation at two heights and reducing global stability to local stability.","marker":"[15]"},{"why":"Provides the sharp Sobolev inequality on the Heisenberg group and the classification of its extremals used as limit objects of the flow.","marker":"[25]"},{"why":"Gives an earlier stability result on Heisenberg-type groups and the upper bound $4/(Q+6)$ that makes the $1/Q$ rate optimal.","marker":"[26]"},{"why":"Introduces the CR Yamabe equation and flow that the paper uses to replace rearrangement flows.","marker":"[24]"},{"why":"States the convergence of the CR Yamabe flow on the CR sphere, on which the global-to-local reduction depends.","marker":"[22]"},{"why":"Establishes sharp Sobolev and HLS inequalities and the eigenvalue formulas for operators on bispherical harmonics used in the local analysis.","marker":"[19]"},{"why":"Proves the earlier positive stability constant for the Sobolev inequality on the Heisenberg group, the baseline the paper refines.","marker":"[27]"},{"why":"Develops the Legendre-duality stability method that transfers Sobolev stability to the HLS inequality.","marker":"[6]"},{"why":"Supplies the refined version of the dual stability argument used for the HLS estimate.","marker":"[7]"},{"why":"The original stability result for the Euclidean Sobolev inequality that set the problem and the local-by-spectrum approach.","marker":"[4]"}],"fun_headline_variants":["Sharp Sobolev stability on Heisenberg group with 1/Q constant","Asymptotically sharp Sobolev stability in Heisenberg setting","Heisenberg Sobolev stability: optimal 1/Q dimension rate","Optimal Heisenberg Sobolev stability via CR flow","Sobolev stability on Heisenberg group sharp as Q grows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp Sobolev stability on Heisenberg group with 1/Q constant","Asymptotically sharp Sobolev stability in Heisenberg setting","Heisenberg Sobolev stability: optimal 1/Q dimension rate","Optimal Heisenberg Sobolev stability via CR flow","Sobolev stability on Heisenberg group sharp as Q grows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000366,"raw_usage":{"total_tokens":2071,"prompt_tokens":1153,"completion_tokens":918,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":769,"completion_tokens_details":{"reasoning_tokens":829}},"tokens_in":769,"tokens_out":918,"duration_ms":9526,"temperature":1.0,"reasoning_tokens":829,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:42:29.397692+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dolbeault, M","cited_arxiv_id":null,"evidence_quote":"Supplies the local-stability template: cutting the perturbation at two heights and reducing global stability to local stability."},{"cited_title":"Jerison, J","cited_arxiv_id":null,"evidence_quote":"Provides the sharp Sobolev inequality on the Heisenberg group and the classification of its extremals used as limit objects of the flow."},{"cited_title":"Liu and A","cited_arxiv_id":null,"evidence_quote":"Gives an earlier stability result on Heisenberg-type groups and the upper bound $4/(Q+6)$ that makes the $1/Q$ rate optimal."},{"cited_title":"Jerison, J","cited_arxiv_id":null,"evidence_quote":"Introduces the CR Yamabe equation and flow that the paper uses to replace rearrangement flows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the convergence of the CR Yamabe flow on the CR sphere, on which the global-to-local reduction depends."},{"cited_title":"Frank and E","cited_arxiv_id":null,"evidence_quote":"Establishes sharp Sobolev and HLS inequalities and the eigenvalue formulas for operators on bispherical harmonics used in the local analysis."},{"cited_title":"Loiudice, Improved Sobolev inequalities on the Heisenberg group , Nonlinear Anal","cited_arxiv_id":null,"evidence_quote":"Proves the earlier positive stability constant for the Sobolev inequality on the Heisenberg group, the baseline the paper refines."},{"cited_title":"Carlen, Duality and stability for functional inequalities , Ann","cited_arxiv_id":null,"evidence_quote":"Develops the Legendre-duality stability method that transfers Sobolev stability to the HLS inequality."},{"cited_title":"Carlen, Stability for the logarithmic Hardy-Littlewood-Sobolev inequality with application to the Keller-Segel equation, J","cited_arxiv_id":null,"evidence_quote":"Supplies the refined version of the dual stability argument used for the HLS estimate."},{"cited_title":"Bianchi and H","cited_arxiv_id":null,"evidence_quote":"The original stability result for the Euclidean Sobolev inequality that set the problem and the local-by-spectrum approach."}],"review_version":1}