Optimal subelliptic super-Poincar\'e and isoperimetric inequalities on stratified Lie groups

Yaozhong W. Qiu

REVIEW 2 major objections 4 minor 4 references

Optimal subelliptic super-Poincar\'e and isoperimetric inequalities on stratified Lie groups

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict New and likely correct in spirit, but the printed assumptions in Theorem 1 have a wrong exponent; fix (1.7) before accepting. read the letter →

arxiv 2411.13430 v2 pith:M7I5NG2Y submitted 2024-11-20 math.PR math.FA

Yaozhong W. Qiu This is my paper · ORCID

classification math.PRmath.FA MSC 26D1060J60

keywords super-PoincaréinequalityHardyisoperimetricstratifiedLiegroupsU-boundsGrushinoperatorsHeisenberg-Greinersubellipticfunctionalinequalities

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 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.

What carries the argument

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].

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

Soft 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.

Overall: 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.

Referee Report

2 major / 4 minor

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].

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 (2)

[§2, Proposition 3, equation (2.13)] 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.
[§2, Proposition 3, equation (2.13)] 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.

minor comments (4)

[Throughout] 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.
[§2, Lemma 2] 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.
[§2, Lemma 4] 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.
[§3.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the isoperimetric inequality is derived from stated norm hypotheses and external U-bound/Hardy machinery, not from the conclusion.

full rationale

The derivation is self-contained in the relevant sense. Theorem 1 assumes the norm estimates (1.7)-(1.9) and the background local inequalities (1.10)-(1.11). Lemma 1 obtains the U-bound (2.2) by two integrations by parts using (1.7) and Young's inequality. Lemma 2 derives the L^q Hardy and Caffarelli-Kohn-Nirenberg inequalities from (1.7)-(1.9). Proposition 3 combines these with the local super-Poincare inequality (1.10) to prove the q-super-Poincare inequality (2.12). Lemma 4 converts the q=1 case into a defective logarithmic-Sobolev inequality, and the isoperimetric comparison is obtained by invoking the external theorem [IKZ11, Theorem 4.5], whose function U_q is the isoperimetric profile of the model measure dnu_{q/(q-1)}. Nothing defines N, U_r, or beta_q in terms of I_mu or the target inequality. The cited estimates for the examples ([Ing12], [BDZ21], [VSCC91], [Jer86], [CDG+94], [FGW94]) are prior independent results, not restatements of the conclusion. The only self-citation is [Qiu24], and it occurs in a list of alternative q=2 techniques rather than in the main proof. The printed exponent in (1.7) may be a typo (the H-type example appears to satisfy |grad N| approx |x|^alpha N^{-alpha}, not N^{alpha}), but that is a correctness defect, not a circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim is conditional on structural inequalities for N and on local Sobolev/Poincaré inequalities that are cited, not proved. There are no fitted parameters or invented entities.

assumptions (4)

domain assumption Local 1-super-Poincaré inequality (1.10) holds for Lebesgue measure with growth β̃1(δ) ≲ 1+δ^{-Q}.
Assumed in Theorem 1 and used in Proposition 3 to start the ball decomposition; for the examples it is cited to [VSCC91] and [CDG+94], but it is not proved in this paper.
domain assumption Local 1-Poincaré inequality (1.11) holds on every d-ball B_R.
Assumed in Theorem 1 and cited to [Jer86] and [FGW94]; needed to obtain the Cheeger inequality through [IKZ11].
domain assumption Distributional estimates (1.7) on N, together with (1.8)-(1.9) relating |x|, N and d.
These are structural hypotheses of Theorem 1; they are verified for the Kaplan-type gauges, Grushin, Heisenberg-Greiner, and anisotropic Heisenberg examples, but are not proved generally.
standard math The subgradient vector fields are divergence-free with respect to Lebesgue measure, so integration by parts with no boundary terms is valid.
Used throughout Lemmas 1 and 2 to justify the integration-by-parts identities (2.3)-(2.4) and (2.8); standard in the stratified Lie group setting.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Optimal subelliptic super-Poincar\'e and isoperimetric inequalities on stratified Lie groups." pith.science (2026). https://pith.science/paper/M7I5NG2Y

@misc{pith2026241113430,
  author       = {Pith},
  title        = {Pith review of: Optimal subelliptic super-Poincar\'e and isoperimetric inequalities on stratified Lie groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7I5NG2Y}},
  note         = {Machine review of arXiv:2411.13430}
}

abstract

We prove $q$-super-Poincar\'e inequalities, $q \in [1, 2]$, for a class of exponential power type probability measures defined in terms of a norm in a number of subelliptic settings, primarily on stratified Lie groups but also in the Grushin and Heisenberg-Greiner settings. Our results include generically optimal isoperimetric inequalities for such probability measures.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Sharp defective log-Sobolev inequalities on H-type groups

[ACG24] Antonelli, G., M. Calzi, and M. Gordina (2024). “Sha rp defective log-Sobolev inequal- ities on H-type groups”. In: arXiv preprint arXiv:2410.15566 (cit. on p. 13). [BR20] Badreddine, Z. and L. Riﬀord (2020). “Measure contrac tion properties for two-step analytic sub-Riemannian structures and Lipschitz Carnot g roups”. In: Annales de l’Institut Fo...

work page Pith review arXiv 2024
[5]

Bakry-Emery Calculus For Diffusion With Additional Multiplicative Term

De Gruyter, pp. 1147– 1158 (cit. on p. 16). [KP25] Konstantopoulos, T. and I. Papageorgiou (2025). “Th e log-Sobolev inequality for spin systems with higher order interactions”. In: Journal of Mathematical Physics 66.1 (cit. on p. 4). [LO00] Lata/suppress la, R. and K. Oleszkiewicz (2000). “Between sobolev and poincar´ e”. In:Geometric Aspects of Function...

work page Pith review arXiv 2025
[70]

Functional inequalities for Mar kov semigroups

6, pp. 2303–2330 (cit. on p. 13). REFERENCES 21 [Bak04] Bakry, D. (2004). “Functional inequalities for Mar kov semigroups”. In: Probability measures on groups. Tata Institute of Fundamental Research, Mubai, pp. 91–147 (cit. on p. 2). [BCG08] Bakry, D., P. Cattiaux, and A. Guillin (2008). “Rate of convergence for ergodic contin- uous Markov processes: Lyap...

work page 2004
[103]

Spectral Gap Inequalities on Nilpotent Lie Groups in Infinite Dimensions

Springer (cit. on pp. 1, 2, 4). [BB12] Baudoin, F. and M. Bonnefont (2012). “Log-Sobolev in equalities for subelliptic opera- tors satisfying a generalized curvature dimension inequality”. In: Journal of Functional Analysis 262.6, pp. 2646–2676 (cit. on pp. 2, 12). [BBG14] Baudoin, F., M. Bonnefont, and N. Garofalo (2014). “ A sub-Riemannian curvature- di...

work page Pith review arXiv 2012

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Sharp defective log-Sobolev inequalities on H-type groups

[ACG24] Antonelli, G., M. Calzi, and M. Gordina (2024). “Sha rp defective log-Sobolev inequal- ities on H-type groups”. In: arXiv preprint arXiv:2410.15566 (cit. on p. 13). [BR20] Badreddine, Z. and L. Riﬀord (2020). “Measure contrac tion properties for two-step analytic sub-Riemannian structures and Lipschitz Carnot g roups”. In: Annales de l’Institut Fo...

work page Pith review arXiv 2024

[2] [5]

Bakry-Emery Calculus For Diffusion With Additional Multiplicative Term

De Gruyter, pp. 1147– 1158 (cit. on p. 16). [KP25] Konstantopoulos, T. and I. Papageorgiou (2025). “Th e log-Sobolev inequality for spin systems with higher order interactions”. In: Journal of Mathematical Physics 66.1 (cit. on p. 4). [LO00] Lata/suppress la, R. and K. Oleszkiewicz (2000). “Between sobolev and poincar´ e”. In:Geometric Aspects of Function...

work page Pith review arXiv 2025

[3] [70]

Functional inequalities for Mar kov semigroups

6, pp. 2303–2330 (cit. on p. 13). REFERENCES 21 [Bak04] Bakry, D. (2004). “Functional inequalities for Mar kov semigroups”. In: Probability measures on groups. Tata Institute of Fundamental Research, Mubai, pp. 91–147 (cit. on p. 2). [BCG08] Bakry, D., P. Cattiaux, and A. Guillin (2008). “Rate of convergence for ergodic contin- uous Markov processes: Lyap...

work page 2004

[4] [103]

Spectral Gap Inequalities on Nilpotent Lie Groups in Infinite Dimensions

Springer (cit. on pp. 1, 2, 4). [BB12] Baudoin, F. and M. Bonnefont (2012). “Log-Sobolev in equalities for subelliptic opera- tors satisfying a generalized curvature dimension inequality”. In: Journal of Functional Analysis 262.6, pp. 2646–2676 (cit. on pp. 2, 12). [BBG14] Baudoin, F., M. Bonnefont, and N. Garofalo (2014). “ A sub-Riemannian curvature- di...

work page Pith review arXiv 2012