{"id":"710215a6-78f6-4477-b660-985b8f84962c","arxiv_id":"2411.13430","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exponential-power measures on stratified Lie groups, Grushin and Heisenberg-Greiner spaces satisfy q-super-Poincaré and isoperimetric inequalities with optimal exponent r=(α+1)p/(α+1+αp).","lead":"This paper proves sharp super-Poincaré and isoperimetric inequalities for probability measures with density proportional to exp(-N(x)^p) on sub-Riemannian spaces such as Heisenberg and Grushin manifolds. It shows these supergaussian measures obey the isoperimetric profile of a simpler Euclidean exponential-power measure, and the exponent is shown to be optimal.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption (1.7) as printed has the wrong N-exponent: on the Heisenberg/Kaplan example at t=0 it requires 1 ≲ |x|², so the main theorem's hypotheses fail for the paper's own examples; Lemma 1 silently uses |∇N| ≃ |x|^α N^{-α}.","rationale":"The central claim is credible and the overall strategy is coherent, but the most load-bearing weakness is not the unproved background inequalities highlighted by the reader. It is an internal inconsistency in the printed assumptions: (1.7) has N^α where the proof and the examples require N^{-α}. As stated, (1.7) fails for the Heisenberg-Kaplan norm at t=0, so Theorem 1 cannot even be applied to the paper's motivating examples. The proof of Lemma 1 confirms the intended exponent, because the displayed q-term estimate and the cancellation leading to U_q only work with |∇N| ≃ |x|^α N^{-α}. This looks like a sign typo or OCR loss, not a fundamental flaw, and the surrounding argument can probably be repaired by rewriting (1.7) with N^{-α}. The reader's conditional verdict remains appropriate: the paper needs a correction before the theorem can be taken as stated, but rejection is not warranted. A secondary issue is that (1.11) as printed omits the standard radius factor R in the local Poincaré inequality; that too is repairable but should be addressed in revision.","tokens_in":23195,"tokens_out":35712,"duration_ms":359253,"concrete_test":"Verify (1.7) on the Heisenberg group at ξ=(x,0), |x|=ρ: with the printed N^α version, |∇N|=1 but |x|^α N^α=ρ², so the upper bound fails for ρ<1; with N^{-α}, the same point gives ρ/ρ=1 and the estimate holds. Then re-derive Lemma 1 for q=1 using s=-α, t=α with the corrected estimate |∇N| ≃ |x|^α N^{-α}, and check that the q-term is controlled by ε∫|f|U_1 dµ + C_ε∫|∇f| dµ while the negative term is at least c∫|f|U_1 dµ, so (2.2) closes. If the corrected (1.7) is adopted, the theorem's hypotheses and examples become consistent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is in the statement of Theorem 1, equation (1.7), which is printed as |x|^α N^α ≲ |∇N| ≲ |x|^α N^α. For the paper's own H-type example, N=(|x|^4+κ|t|^2)^{1/4} with α=1, take ξ=(x,0) with |x|=ρ small. Since N=|x| at t=0 and the horizontal gradient of the Kaplan norm is a unit vector, |∇N|=1, so the upper bound demands 1 ≲ ρ², which fails as ρ→0. The same issue propagates through Lemma 1: the displayed bound ∫|f|^{q-2}f|∇f||x|^{s+α}N^{t-α}dµ follows from the q-term only if |∇N| ≲ |x|^α N^{-α}, and the negative term needs |∇N|² ≳ |x|^{2α}N^{-2α} to produce U_q. With the printed N^α exponents, the algebra acquires an extra N^{2α} factor and the claimed U-bound (2.2) does not close. Since Lemma 1 is the engine for Proposition 3 and hence Theorem 1, this is a central correctness issue. The intended condition is evidently |x|^α N^{-α} ≲ |∇N| ≲ |x|^α N^{-α}, which matches the cited estimates in [Ing12] and [BDZ21]; with that correction the proof likely works, but the printed assumptions must be fixed before Theorem 1 is usable.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves q-super-Poincaré inequalities (q ∈ [1,2]) and isoperimetric inequalities for probability measures of the form dµ = Z^{-1} e^{-N^p} dξ on stratified Lie groups and related subelliptic settings (Grushin, Heisenberg–Greiner, filiform, anisotropic Heisenberg), where N is a norm satisfying gradient and Laplacian bounds. The main theorem (Theorem 1) yields Iµ ≳ U_r with r = (α+1)p/(α+1+αp) for p ≥ α+1. The proof uses U-bounds, a Hardy inequality, a Caffarelli–Kohn–Nirenberg inequality, and a conversion of a 1-super-Poincaré inequality into a 1-F-Sobolev inequality, followed by an application of the isoperimetric results of [IKZ11].","tokens_in":23563,"tokens_out":11017,"duration_ms":103557,"significance":"If the stated assumptions are corrected as indicated below, the results are significant: they give generically optimal subelliptic isoperimetric inequalities for a class of supergaussian measures, improving earlier spectral and F-Sobolev results in a substantial class of examples. The proof strategy is explicit, with the exponents in Lemmas 1 and 2 displayed, and the paper includes concrete optimality arguments via test functions. The range of examples (H-type groups, Grushin spaces, Heisenberg–Greiner spaces, filiform groups, anisotropic Heisenberg groups) and the careful discussion of the endpoint case are valuable.","major_comments":[{"comment":"The displayed estimates in (1.7) have the wrong exponent on N. For the Kaplan norm N(x,t)=(|x|^4+κ|t|^2)^{1/4} used in Corollary 5, at t=0 one has N=|x| and |∇N|=1, so the upper bound |∇N| ≲ |x|^α N^α = |x|^{2α} fails as |x|→0. The subsequent two estimates in (1.7) fail equally at t=0: ∆N ∼ (n-1)|x|^{-1} while the printed bound gives |x|^{2α}N^{2α+1}=|x|^5, and ∇N·∇|x| = 1 while the printed bound gives |x|^{2α+1}N^{2α+1}=|x|^6. The proof of Lemma 1 in §2.1 uses the corrected forms |∇N| ≲ |x|^α N^{-α}, |∇N|² ≳ |x|^{2α}N^{-2α}, ∆N ≲ |x|^{2α}N^{-2α-1}, and ∇N·∇|x| ≲ |x|^{2α+1}N^{-2α-1}; with the printed exponents the displayed bound after (2.3)–(2.4) does not produce the U_q weight and (2.2) does not close. The intended condition evidently matches the estimates cited from [Ing12] and [BDZ21], but the statement of Theorem 1 must be corrected and all three displayed estimates in (1.7) re-verified against those references.","section":"§2, Proposition 3, equation (2.13)"},{"comment":"The proof asserts (2.13) 'using the elementary lower bound cx + x^{-s} ≳ c^{s/(s+1)}', but that inequality does not by itself yield (2.13). The missing step is a Hölder interpolation between the U-bound (2.2) and the Hardy inequality (2.6): writing N^{q(p-α-1)/(α+1)} = (|x|^{qα}N^{q(p-α-1)})^{1/(α+1)} (|x|^{-q})^{α/(α+1)}, one obtains (2.13) by Hölder's inequality with exponents α+1 and (α+1)/α. Please display this argument explicitly, since (2.13) is used to obtain the growth of β_q(ε) in the main proposition.","section":"§2, Proposition 3, equation (2.13)"}],"minor_comments":[{"comment":"Several LaTeX macros have survived into the text, e.g., '/greaterorsimilar' in the notation paragraph and in Theorem 1, and 'Lata/suppress la' in the bibliography entry [LO00]; these should be cleaned up before publication.","section":"Throughout"},{"comment":"The integration by parts formula (2.8) is stated for a weight ω and vector field h, but the regularity assumptions on ω and h are not specified; please state sufficient conditions so that the boundary terms vanish.","section":"§2, Lemma 2"},{"comment":"The dyadic sets in the proof are written as A_n = {δ^{n+1} > f ≥ δ^n} and f_n = (f − δ^n) ∧ (δ^{n+1} − δ^n); this notation is confusing because δ^{n+1} < δ^n for δ < 1. Please rewrite the dyadic decomposition with consistent ordering.","section":"§2, Lemma 4"},{"comment":"The sentence 'the estimates (1.7) on N are exact by [Ing12, Proposition 2.7 and pp. 20]' should be updated after the correction of (1.7); as printed it is not true of the displayed estimates.","section":"§3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim is plausible and the proof strategy is sound modulo the typo in (1.7). The mistake in the N-exponent is load-bearing because it appears in the statement of the main theorem and is used in Lemma 1, but it is clearly fixable and the examples in §3 are consistent with the corrected form. Please ask the author to check every occurrence of the N-exponent in (1.7) and in the proof of Lemma 1, and to fill in the interpolation argument for (2.13). After these revisions the paper would be a solid contribution to the functional-inequality literature on subelliptic settings."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me cut to the chase: this paper is genuinely new and probably correct in spirit, but as printed, assumption (1.7) in Theorem 1 is wrong. For the Kaplan norm on the Heisenberg group with α=1, at t=0 the horizontal gradient of N has absolute value comparable to 1, while |x|^α N^α = |x|^2, so the upper bound fails as |x|→0. This is not a cosmetic typo: Lemma 1 uses (1.7) to derive the U-bound, and with the printed exponents the algebra does not close. The intended condition is almost certainly |x|^α N^{-α} ≲ |∇N| ≲ |x|^α N^{-α}, which matches the cited estimates in [Ing12] and [BDZ21]. The stress-test note is correct on this point.\n\nWhat the paper does well: it proves q-super-Poincaré and generically optimal isoperimetric inequalities for exponential-power measures with Korányi-Folland gauges, covering step-two groups, Grushin spaces, Heisenberg-Greiner, and the anisotropic Heisenberg group. This goes beyond earlier work that used the Carnot–Carathéodory distance as a potential or stopped at F-Sobolev inequalities. The proofs are mostly explicit, with integration by parts and all exponents visible. The endpoint case p=α+1 is handled, and the n1=1 case gets a separate Caffarelli-Kohn-Nirenberg argument.\n\nSoft spots, in proportion: the typo in (1.7) is load-bearing and must be fixed. Proposition 3 compresses the local-to-global step behind an “elementary lower bound” in (2.13); the Hölder interpolation is only sketched. The isoperimetric conclusion imports [IKZ11, Theorem 4.5] as a black box, which is standard here but should be checked. The background local inequalities are cited, not proved; that is acceptable for this audience.\n\nOverall: the paper deserves a serious referee, but not acceptance as-is. Fix the exponent in (1.7), expand the proof of (2.13) by a few lines, and the main theorem likely goes through. If I were an editor, I would send it to a specialist with a request for those revisions. Readers working in functional inequalities on sub-Riemannian spaces will get real value from it, and I would cite it once the correction is made.","headline":"New and likely correct in spirit, but the printed assumptions in Theorem 1 have a wrong exponent; fix (1.7) before accepting.","tokens_in":24098,"tokens_out":10395,"would_cite":true,"duration_ms":95087,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Probability measures with supergaussian tails on stratified Lie groups, Grushin spaces, and Heisenberg-Greiner spaces satisfy the isoperimetric profile of a subgaussian exponential measure, with an exponent that is generically optimal.","keywords":["super-Poincaré inequality","Hardy inequality","isoperimetric inequality","stratified Lie groups","U-bounds","Grushin operators","Heisenberg-Greiner operators","subelliptic functional inequalities"],"falsifier":"For a concrete step-two group with $N_\\kappa=(|x|^4+\\kappa|t|^2)^{1/4}$ and $p=2$, compute the isoperimetric profile of $d\\mu=Z^{-1}e^{-N^p}d\\xi$: exhibiting a family of sets whose profile exceeds $C\\,U_{2p/(p+2)}$ by an unbounded factor would contradict the claimed optimality, while failure of the imported local Poincaré inequality (1.11) on some ball would refute the theorem.","tokens_in":22994,"feed_emoji":"📐","tokens_out":12738,"duration_ms":113556,"temperature":0.7,"pith_summary":"This paper proves super-Poincaré and isoperimetric inequalities for probability measures of the form $d\\mu=Z^{-1}e^{-N^p}d\\xi$ on stratified Lie groups and related subelliptic spaces, where $N$ is a homogeneous norm-like function satisfying the structural estimates (1.7)–(1.9). The headline finding is that these supergaussian-tailed measures obey the isoperimetric profile of a subgaussian exponential measure: $I_\\mu \\gtrsim U_r$ with $r=(\\alpha+1)p/(\\alpha+1+\\alpha p)$, and in the main examples the exponent is generically optimal. The proof works through an $L^1$ analogue of the super-Poincaré inequality rather than curvature assumptions, using U-bounds, a Hardy inequality, and a Caffarelli-Kohn-Nirenberg inequality to convert local embeddings into global ones. A sympathetic reader would care because the result places these subelliptic measures precisely on the scale between exponential and Gaussian isoperimetry and completes a line of spectral and functional-inequality results for H-type groups.","feed_headline":"Supergaussian measures obey subgaussian isoperimetry","feed_subtitle":"On Lie groups and subelliptic spaces, the sharp isoperimetric scale is identified and shown optimal.","key_machinery":"The engine is a chain of inequalities proved for $\\mu$: a U-bound (2.2) with weight $U_q=|x|^{q\\alpha}N^{q(p-\\alpha-1)}$, obtained by integrating by parts against $|x|^sN^t\\nabla N$; an $L^q$ Hardy inequality (2.6) with singularity along $\\{|x|=0\\}$; and the Caffarelli-Kohn-Nirenberg inequality (2.7), which together produce the nondegenerate U-bound (2.13) on the complement of a large ball. The argument then splits space into a ball $B_R$ and its complement: on the ball, the local super-Poincaré inequality (1.10) transfers from Lebesgue measure to $\\mu$ because $V=N^p$ is bounded there, while outside the ball the Hardy and CKN bounds control the $N$-weighted integrals. For $q=1$, the resulting super-Poincaré inequality is converted through the equivalence of [Wan00, Theorem 3.2] into a defective $1$-$F$-Sobolev inequality (1.6), and that, together with a Cheeger inequality obtained from (1.11) and the nondegenerate U-bound, yields the isoperimetric profile by the criterion of [IKZ11].","core_discovery":"The central claim is Theorem 1: under the assumptions (1.7)–(1.11) on $N$—gradient lower and upper bounds, a Laplacian upper bound, comparability of $N$ with the Carnot-Carathéodory distance $d$, and the structure of the Euclidean coordinate $|x|$—every measure $d\\mu=Z^{-1}e^{-N^p}d\\xi$ with $p\\ge\\alpha+1$ satisfies the isoperimetric inequality $I_\\mu \\gtrsim U_r$ for $r=(\\alpha+1)p/((\\alpha+1)+\\alpha p)$. In the step-two group, Grushin, Heisenberg-Greiner, and anisotropic Heisenberg examples, this exponent is generically optimal, meaning it cannot be replaced by anything strictly larger. The same mechanism yields, for each $q\\in[1,2]$, the $q$-super-Poincaré inequality (2.12) with growth $\\beta_q(\\varepsilon)\\lesssim\\exp(C\\varepsilon^{-p(\\alpha+1)/(q(p-\\alpha-1))})$. At the endpoint $p=\\alpha+1$, the isoperimetric inequality becomes linear, $I_\\mu\\gtrsim U_1$, which is exactly a Cheeger-type inequality.","pith_inferences":["Because the proof only uses the local inequalities (1.10)–(1.11) and the norm estimates (1.7)–(1.9), the same strategy should extend to any subelliptic structure whose fundamental-solution norm obeys comparable gradient and Laplacian estimates; the one-dimensional theorem illustrates the transfer when the vector fields contain a Euclidean derivative.","The stability under perturbations noted in the paper suggests that isoperimetric profiles of this shape persist for measures $d\\nu\\propto e^{-W}d\\mu$ when $W$ satisfies suitable growth assumptions, which could be used to build models with prescribed isoperimetric exponents between $U_1$ and $U_2$.","Optimality is proved here for the isoperimetric exponent and for the $q=2$ growth, not for every intermediate $q$; determining whether the $q$-super-Poincaré growth in (2.12) is optimal for $q<2$ would be a natural next step that the paper leaves open."],"forward_implications":["For step-two stratified Lie groups with $N_\\kappa=(|x|^4+\\kappa|t|^2)^{1/4}$, the measure $d\\mu=Z^{-1}e^{-N^p}d\\xi$ with $p\\ge2$ satisfies $I_\\mu\\gtrsim U_{2p/(p+2)}$, and the exponent cannot be increased.","In the Grushin setting with $N_\\eta=(|x|^{2(1+\\eta)}+(1+\\eta)^2|y|^2)^{1/(2(1+\\eta))}$, the profile $I_\\mu\\gtrsim U_{p(1+\\eta)/(p\\eta+1+\\eta)}$ holds for $p\\ge1+\\eta$, interpolating from near-linear to near-Gaussian isoperimetry as $p$ grows.","The $q$-super-Poincaré inequality (2.12) with the stated exponential growth implies, through [Wan00], that the associated operator has empty essential spectrum and satisfies the corresponding defective $q$-$F$-Sobolev inequalities; at $q=2$ this confirms the spectral conjecture for these H-type measures.","At the endpoint $p=\\alpha+1$, the isoperimetric inequality is linear, $I_\\mu\\gtrsim U_1$, parallel to the Euclidean statement for $d\\nu_1=e^{-|x|}dx$.","For the anisotropic Heisenberg group $H_n(1/2,1)$, the same methods give $I_\\mu\\gtrsim U_{2p/(p+2)}$ for $p\\ge2$, improving previously known log-Sobolev-type results and removing dimensional restrictions."],"supporting_citations":[{"why":"Supplies the super-Poincaré framework and the equivalence to defective F-Sobolev inequalities used to pass from (1.5) to (1.6).","marker":"[Wan00]"},{"why":"Introduces the U-bound method and the spectral test-function argument used to prove optimality of the stated exponents.","marker":"[HZ10]"},{"why":"Provides the degenerating U-bound along $\\{|x|=0\\}$, prior spectral results on H-type groups, and the test function used in the optimality argument.","marker":"[Ing12]"},{"why":"Supplies the bridge from U-bounds and Cheeger inequalities to isoperimetric profiles.","marker":"[IKZ11]"},{"why":"Gives the L1-Sobolev inequality on groups that implies the local 1-super-Poincaré inequality (1.10).","marker":"[VSCC91]"},{"why":"Gives the local 1-Poincaré inequality (1.11) for vector fields satisfying Hörmander's condition.","marker":"[Jer86]"},{"why":"Verifies the estimates (1.7) for step-two Carnot groups and provides related U-bounds.","marker":"[BDZ21]"},{"why":"Supplies the geometric Sobolev embedding for Grushin-type vector fields used for (1.10) in that setting.","marker":"[CDG+94]"},{"why":"Supplies the weighted Sobolev-Poincaré inequality for Grushin operators used for (1.11) in that setting.","marker":"[FGW94]"},{"why":"Gives the explicit Kaplan norm formula used in the step-two examples.","marker":"[Kap80]"}],"fun_headline_variants":["Optimal isoperimetric inequalities on stratified Lie groups","Sharp isoperimetric scale for supergaussian laws on subelliptic spaces","Super-Poincaré plus optimal isoperimetry on Lie groups","Generic optimal isoperimetry for exponential power measures","Subelliptic isoperimetric sharpness for supergaussian measures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the local inequalities (1.10) and (1.11) hold for Lebesgue measure on subelliptic balls with the stated polynomial growth; these are imported from earlier works rather than proved here, and the conversion of local to global super-Poincaré bounds in Proposition 3 depends on them at every scale.","fun_headline_variants_meta":{"raw":{"variants":["Optimal isoperimetric inequalities on stratified Lie groups","Sharp isoperimetric scale for supergaussian laws on subelliptic spaces","Super-Poincaré plus optimal isoperimetry on Lie groups","Generic optimal isoperimetry for exponential power measures","Subelliptic isoperimetric sharpness for supergaussian measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00077,"raw_usage":{"total_tokens":3376,"prompt_tokens":875,"completion_tokens":2501,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":2414}},"tokens_in":491,"tokens_out":2501,"duration_ms":17289,"temperature":1.0,"reasoning_tokens":2414,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:26:36.965983+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete step-two group with $N_\\kappa=(|x|^4+\\kappa|t|^2)^{1/4}$ and $p=2$, compute the isoperimetric profile of $d\\mu=Z^{-1}e^{-N^p}d\\xi$: exhibiting a family of sets whose profile exceeds $C\\,U_{2p/(p+2)}$ by an unbounded factor would contradict the claimed optimality, while failure of the imported local Poincaré inequality (1.11) on some ball would refute the theorem.","supporting_citations":[],"review_version":1}