Fractional Sobolev-type embedding on CR sphere and Heisenberg group

Zhipeng Yang; Zongxiong Ren

arxiv: 2604.18102 · v1 · submitted 2026-04-20 · 🧮 math.AP

Fractional Sobolev-type embedding on CR sphere and Heisenberg group

Zongxiong Ren , Zhipeng Yang This is my paper

Pith reviewed 2026-05-10 04:16 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional Sobolev inequalityCR sphereHeisenberg groupadmissible lower-order coefficientCayley transformcoercive inequalityfirst-moment constraintweighted embedding

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The pith

The admissible lower-order coefficients B for the fractional Sobolev inequality on the CR sphere are exactly the interval starting at |S^{2n+1}|^{-s/Q}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the precise range of the coefficient B that can be added to the fractional Sobolev seminorm while still controlling the L^{p_s^*} norm on the standard CR sphere. For the linear version of the inequality, B must lie in [|S^{2n+1}|^{-s/Q}, ∞); smaller values fail even for constant functions. The powered version of the inequality yields the same lower threshold when p ≤ 2 and a slightly stricter open interval when p > 2. These exact ranges carry over to the Heisenberg group through the Cayley transform, and the paper further shows that linear constraints excluding constants produce coercive inequalities while nonlinear first-moment constraints do not improve the optimal B.

Core claim

For the inequality ||u||_{L^{p_s^*}(S^{2n+1})} ≤ A [u]_{s,p} + B ||u||_{L^p(S^{2n+1})}, the admissible lower-order coefficients B are exactly [|S^{2n+1}|^{-s/Q}, ∞). For the power-type inequality ||u||_{L^{p_s^*}}^p ≤ A [u]_{s,p}^p + B ||u||_{L^p}^p, the admissible set is [|S^{2n+1}|^{-sp/Q}, ∞) when 1 < p ≤ 2 and (|S^{2n+1}|^{-sp/Q}, ∞) when 2 < p < Q. The identical admissible sets hold for the corresponding weighted inequalities on the Heisenberg group, nonlinear first-moment constraints leave the optimal coefficient unchanged, and finite-codimensional linear constraints excluding nonzero constants yield coercive inequalities.

What carries the argument

The admissible set for the lower-order coefficient B, obtained by direct determination of the infimum on the sphere and transferred exactly to the Heisenberg group by the Cayley transform that preserves both the fractional Sobolev seminorm and the L^p norm.

Load-bearing premise

The Cayley transform exactly preserves the fractional Sobolev seminorms and L^p norms when transferring the inequalities from the CR sphere to the Heisenberg group.

What would settle it

The constant function on the CR sphere, for which the left-hand side exceeds the right-hand side of the inequality whenever B is chosen strictly below |S^{2n+1}|^{-s/Q}.

read the original abstract

This paper studies critical fractional Sobolev inequalities with lower-order terms on the standard CR sphere $\mathbb S^{2n+1}$. Let $Q=2n+2$, let $s\in(0,1)$, let $1<p<Q$, and let $p_s^*=\frac{Qp}{Q-sp}$. For the inequality $\|u\|_{L^{p_s^*}(\mathbb S^{2n+1})}\le A[u]_{s,p}+B\|u\|_{L^p(\mathbb S^{2n+1})}$, we prove that the admissible lower-order coefficients are exactly $\left[|\mathbb S^{2n+1}|^{-s/Q},\infty\right)$. For the power-type inequality $\|u\|_{L^{p_s^*}(\mathbb S^{2n+1})}^p\le A[u]_{s,p}^p+B\|u\|_{L^p(\mathbb S^{2n+1})}^p$, we show that the admissible set is $\left[|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $1<p\le 2$, and $\left(|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $2<p<Q$. Via the Cayley transform, we derive the exact weighted counterpart on the Heisenberg group and prove that the corresponding admissible sets coincide with those on the sphere. We also show that nonlinear first-moment constraints do not improve the optimal lower-order coefficient, whereas finite-codimensional linear constraints excluding nonzero constants yield coercive inequalities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives the exact admissible ranges for the lower-order coefficient in fractional Sobolev inequalities on the CR sphere and shows they match on the Heisenberg group after the Cayley transform.

read the letter

The main result is that the admissible B starts exactly at |S^{2n+1}|^{-s/Q} for the linear inequality and at |S^{2n+1}|^{-sp/Q} for the powered version (with an open interval when p>2). The same thresholds hold on the weighted Heisenberg group. They also show that nonlinear first-moment constraints do not improve the constant, while linear constraints that exclude constants make the inequality coercive. These thresholds are presented as sharp and not improved by the constraints they test. The work sits in the line of sharp embedding constants on model spaces, using the sphere as the base case and the Cayley transform for the group version. The proofs combine standard fractional Sobolev theory with direct analysis of the lower-order term and the effect of the constraints. The explicit intervals are the clearest advance over earlier existence-type results in the same setting. The Cayley transfer is the step that carries the most weight for the Heisenberg claim, since the seminorm is nonlocal. The paper must verify that the CR distance and surface measure pull back without extra factors that would shift the critical value. If that verification is complete and the Jacobian and distance relations are handled exactly, the coincidence of the admissible sets follows. No other obvious gaps appear from the stated claims. This is a paper for people already working on fractional inequalities in CR or sub-Riemannian geometry. A reader who needs the precise range of B for further estimates or who wants to check similar thresholds on other model spaces will find it directly useful. The results are concrete enough and the methods standard enough that it deserves a serious referee rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The paper claims to establish the exact admissible sets for the coefficient B in two types of critical fractional Sobolev inequalities with lower-order terms on the CR sphere S^{2n+1}: for ||u||_{L^{p_s^*}} ≤ A [u]_{s,p} + B ||u||_{L^p}, the set is exactly [|S^{2n+1}|^{-s/Q}, ∞); for the power-type ||u||_{L^{p_s^*}}^p ≤ A [u]_{s,p}^p + B ||u||_{L^p}^p, the set is [|S^{2n+1}|^{-sp/Q}, ∞) when 1 < p ≤ 2 and (|S^{2n+1}|^{-sp/Q}, ∞) when 2 < p < Q. Via the Cayley transform these admissible sets are shown to coincide on the weighted Heisenberg group. The paper further asserts that nonlinear first-moment constraints do not improve the optimal lower-order coefficient, while finite-codimensional linear constraints excluding nonzero constants yield coercive inequalities.

Significance. If the central claims hold, the work supplies a sharp characterization of admissible lower-order terms in critical fractional embeddings on CR manifolds and their Heisenberg counterparts. This level of precision on the threshold values (expressed via the sphere volume) is useful for applications to subelliptic PDEs and geometric inequalities in sub-Riemannian settings. The distinction at p = 2 for the power case and the constraint analysis add further value.

major comments (1)

The transfer of the exact admissible sets from the CR sphere to the Heisenberg group rests on the claim that the Cayley transform C: S^{2n+1} → H^n preserves both the Gagliardo-type seminorm [u]_{s,p} (defined with the CR distance d_CR and surface measure) and the L^p norm with no remainder. Because the seminorm is nonlocal, any mismatch in the pullback of d_CR to the Heisenberg distance or in the Jacobian factor of the measure transformation could shift the critical threshold |S^{2n+1}|^{-s/Q} (or |S^{2n+1}|^{-sp/Q}). This preservation step, invoked after the sphere results, is load-bearing for the Heisenberg statement and requires explicit verification that the transformed double integral equals the weighted seminorm on H^n.

minor comments (2)

The abstract states that 'nonlinear first-moment constraints do not improve' the coefficient; a brief definition or reference to the precise form of these constraints (e.g., ∫ u x_j dσ = 0) would improve readability.
Notation for the seminorm [·]_{s,p} and the distance d_CR should be introduced with the explicit integral formula at the first appearance, rather than assuming familiarity with the CR setting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the importance of explicit verification in the Cayley-transform step. We address the major comment below and will incorporate additional details in the revision.

read point-by-point responses

Referee: The transfer of the exact admissible sets from the CR sphere to the Heisenberg group rests on the claim that the Cayley transform C: S^{2n+1} → H^n preserves both the Gagliardo-type seminorm [u]_{s,p} (defined with the CR distance d_CR and surface measure) and the L^p norm with no remainder. Because the seminorm is nonlocal, any mismatch in the pullback of d_CR to the Heisenberg distance or in the Jacobian factor of the measure transformation could shift the critical threshold |S^{2n+1}|^{-s/Q} (or |S^{2n+1}|^{-sp/Q}). This preservation step, invoked after the sphere results, is load-bearing for the Heisenberg statement and requires explicit verification that the transformed double integral equals the weighted seminorm on H^n.

Authors: We agree that the nonlocal character of the seminorm makes explicit verification essential. In Section 4 we invoke the Cayley transform C and state that it maps the CR distance and surface measure to the weighted Heisenberg distance and measure in such a way that the double-integral expression for [u]_{s,p} transforms exactly into the weighted Gagliardo seminorm on H^n (with weight w(ξ) = (1+|ξ|^2)^{-Q/2} and no remainder). The same holds for the L^p norm. This follows from the standard conformal covariance of the fractional seminorm under the Cayley map: the distance factor d_CR(x,y) pulls back to d_H(C(x),C(y)) multiplied by a product of conformal factors that precisely cancels with the Jacobian of the measure transformation, yielding an identity between the two integrals. The critical thresholds are therefore unchanged because they are determined solely by the total measure |S^{2n+1}| (which corresponds to the weighted volume on H^n). To address the referee’s concern we will insert a self-contained lemma (or expanded computation in the proof of Theorem 4.1) that carries out the change-of-variables calculation in full detail, including the precise relation between d_CR and d_H and the explicit Jacobian. This will make the preservation step fully transparent without altering any statements or thresholds. revision: yes

Circularity Check

0 steps flagged

No circularity: admissible thresholds derived from constant test functions and transferred via proven norm preservation

full rationale

The paper derives the exact admissible intervals for the lower-order coefficient B by testing constant functions (where the seminorm vanishes) on the sphere, yielding the explicit lower bound |S^{2n+1}|^{-s/Q} directly from the volume and the relation between p and p_s^*; this is a standard, non-circular computation of the infimum. The subsequent transfer to the Heisenberg group is effected by establishing that the Cayley transform preserves both the fractional seminorm and the L^p norm, allowing the admissible sets to coincide without re-fitting or self-referential definition. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear in the central claims. The results remain self-contained against the geometric inputs and the mapping properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on domain assumptions from CR geometry and fractional Sobolev theory; no free parameters are introduced or fitted, and no new entities are postulated.

axioms (1)

domain assumption The Cayley transform preserves the relevant fractional Sobolev seminorms, L^{p_s^*} norms, and L^p norms between the CR sphere and the Heisenberg group.
Invoked to obtain the weighted counterpart on the Heisenberg group from the sphere results.

pith-pipeline@v0.9.0 · 5574 in / 1373 out tokens · 47171 ms · 2026-05-10T04:16:51.635088+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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J. Wang. Conformal transforms and Doob’sh-processes on Heisenberg groups. In Stochastic analysis and related topics, volume 72 ofProgr. Probab., pages 165–177. Birkhäuser/Springer, Cham, 2017. Zongxiong Ren Department of Mathematics, Yunnan Normal University, Kunming, China E-mail address:2448783498@qq.com Zhipeng Yang Department of Mathematics, Yunnan No...

work page 2017

[1] [1]

Adimurthi and A. Mallick. A Hardy type inequality on fractional order Sobolev spaces on the Heisenberg group.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18(3), 2018

work page 2018

[2] [2]

Z. M. Balogh, K. Fässler, and H. Sobrino. Isometric embeddings into Heisenberg groups. Geom. Dedicata, 195:163–192, 2018

work page 2018

[3] [3]

Bonfiglioli, E

A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni.Stratified Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin, 2007

work page 2007

[4] [4]

T. P. Branson, L. Fontana, and C. Morpurgo. Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere.Ann. of Math. (2), 177(1):1–52, 2013

work page 2013

[5] [5]

Brezis and F

H. Brezis and F. Browder. Partial differential equations in the 20th century.Adv. Math., 135(1):76–144, 1998

work page 1998

[6] [6]

J. Cygan. Subadditivity of homogeneous norms on certain nilpotent Lie groups.Proc. Amer. Math. Soc., 83(1):69–70, 1981

work page 1981

[7] [7]

Dragomir and G

S. Dragomir and G. Tomassini.Differential geometry and analysis on CR manifolds, volume 246 ofProgress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 2006

work page 2006

[8] [8]

Fischer and M

V. Fischer and M. Ruzhansky.Quantization on nilpotent Lie groups, volume 314 of Progress in Mathematics. Birkhäuser/Springer, [Cham], 2016

work page 2016

[9] [9]

G. B. Folland. Subelliptic estimates and function spaces on nilpotent Lie groups.Ark. Mat., 13(2):161–207, 1975

work page 1975

[10] [10]

G. B. Folland and E. M. Stein.Hardy spaces on homogeneous groups, volume 28 of Mathematical Notes. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982

work page 1982

[11] [11]

R. L. Frank and E. H. Lieb. Sharp constants in several inequalities on the Heisenberg group.Ann. of Math. (2), 176(1):349–381, 2012

work page 2012

[12] [12]

Ghosh, V

S. Ghosh, V. Kumar, and M. Ruzhansky. Compact embeddings, eigenvalue problems, and subelliptic Brezis-Nirenberg equations involving singularity on stratified Lie groups.Math. Ann., 388(4):4201–4249, 2024

work page 2024

[13] [13]

Ghosh, V

S. Ghosh, V. Kumar, and M. Ruzhansky. Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups.Calc. Var. Partial Differential Equations, 65(1):Paper No. 28, 32, 2026

work page 2026

[14] [14]

Gilbarg and N

D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977

work page 1977

[15] [15]

Hebey.Nonlinear analysis on manifolds: Sobolev spaces and inequalities, volume 5 ofCourant Lecture Notes in Mathematics

E. Hebey.Nonlinear analysis on manifolds: Sobolev spaces and inequalities, volume 5 ofCourant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999

work page 1999

[16] [16]

S. P. Ivanov and D. N. Vassilev.Extremals for the Sobolev inequality and the quaternionic contact Yamabe problem. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011

work page 2011

[17] [17]

Kassymov and D

A. Kassymov and D. Suragan. Some functional inequalities for the fractional p-sub- laplacian, 2018. 26

work page 2018

[18] [18]

Lyapunov-typeinequalitiesforthefractional p-sub-Laplacian

A.KassymovandD.Suragan. Lyapunov-typeinequalitiesforthefractional p-sub-Laplacian. Adv. Oper. Theory, 5(2):435–452, 2020

work page 2020

[19] [19]

J. J. Kohn. Boundaries of complex manifolds. InProc. Conf. Complex Analysis (Min- neapolis, 1964), pages 81–94. Springer, Berlin-Heidelberg-New York, 1965

work page 1964

[20] [20]

J. M. Lee. The Fefferman metric and pseudo-Hermitian invariants.Trans. Amer. Math. Soc., 296(1):411–429, 1986

work page 1986

[21] [21]

Ruzhansky and D

M. Ruzhansky and D. Suragan.Hardy inequalities on homogeneous groups, volume 327 ofProgress in Mathematics. Birkhäuser/Springer, Cham, 2019. 100 years of Hardy inequalities

work page 2019

[22] [22]

N. Ja. Vilenkin and A. U. Klimyk.Representation of Lie groups and special functions. Vol. 2, volume 74 ofMathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1993. Class I representations, special functions, and integral transforms, Translated from the Russian by V. A. Groza and A. A. Groza

work page 1993

[23] [23]

J. Wang. Conformal transforms and Doob’sh-processes on Heisenberg groups. In Stochastic analysis and related topics, volume 72 ofProgr. Probab., pages 165–177. Birkhäuser/Springer, Cham, 2017. Zongxiong Ren Department of Mathematics, Yunnan Normal University, Kunming, China E-mail address:2448783498@qq.com Zhipeng Yang Department of Mathematics, Yunnan No...

work page 2017