pith. sign in

arxiv: 2604.16771 · v1 · submitted 2026-04-18 · 🧮 math.AP

Sharp trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere

Qiaohua Yang , Leyuan Yu This is my paper

Pith reviewed 2026-05-10 07:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords trace inequalitiesSobolev inequalitiesHeisenberg groupCR spheresublaplacianfractional powersBeckner-Onofriconformal invariance
0
0 comments X

The pith

Sharp Sobolev trace inequalities hold for fractional powers of the sublaplacian on the Heisenberg group and CR sphere.

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

The paper establishes sharp Sobolev trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere. These extend corresponding results from Euclidean space to these non-Euclidean settings. A sympathetic reader cares because such inequalities give precise control over how functions behave near boundaries under subelliptic operators, which model phenomena in several complex variables and sub-Riemannian geometry. In a limiting case the paper also obtains sharp trace Beckner-Onofri inequalities on the CR sphere and on the standard sphere.

Core claim

Sharp Sobolev trace inequalities are established for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere, extending the corresponding Euclidean results to these non-Euclidean settings. In the limiting case, sharp trace Beckner-Onofri inequalities are also established on the CR sphere. The same approach also yields trace Beckner-Onofri inequalities on the standard sphere.

What carries the argument

Duality argument combined with the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group and the CR sphere.

If this is right

  • The inequalities supply sharp constants for Sobolev trace embeddings in these geometries.
  • Limiting cases produce sharp Beckner-Onofri trace inequalities on the CR sphere.
  • The approach extends to yield trace Beckner-Onofri inequalities on the standard sphere.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These results may apply to variational problems involving subelliptic operators on the Heisenberg group.
  • Similar inequalities could hold for other stratified Lie groups if the underlying HLS inequalities are available.
  • The conformal invariance opens the door to studying prescribing problems for CR curvature.

Load-bearing premise

The duality argument and the sharp Hardy-Littlewood-Sobolev inequalities extend directly and without modification to the Heisenberg group and CR sphere settings.

What would settle it

An explicit computation or counterexample showing that the optimal constant for the trace inequality on the Heisenberg group does not match the Euclidean value would show the claim to be false.

read the original abstract

We establish sharp Sobolev trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere, extending the corresponding Euclidean results of Einav-Loss, Beckner, and Bez-Machihara-Sugimoto to these non-Euclidean settings. In the limiting case, sharp trace Beckner-Onofri inequalities are also established on the CR sphere. The proofs are based on a duality argument due to Bez-Machihara-Sugimoto, together with the Frank-Lieb sharp form of the Hardy-Littlewood-Sobolev inequalities on the Heisenberg group and the CR sphere. The same approach also yields trace Beckner-Onofri inequalities on the standard sphere.

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.

Referee Report

1 major / 1 minor

Summary. The paper establishes sharp Sobolev trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere, extending Euclidean results of Einav-Loss, Beckner, and Bez-Machihara-Sugimoto. Proofs rely on the Bez-Machihara-Sugimoto duality argument combined with the Frank-Lieb sharp HLS inequality in these settings; limiting cases yield sharp trace Beckner-Onofri inequalities on the CR sphere, and the same method recovers the Euclidean sphere case.

Significance. If the duality transfer is valid, the results would furnish sharp constants for fractional trace inequalities in sub-Riemannian and CR geometry, extending the Euclidean theory in a natural way and potentially enabling further work on extremals and conformal invariants on stratified groups.

major comments (1)
  1. [Abstract and §1] Abstract and §1 (Introduction/Method): The central claim that the Bez-Machihara-Sugimoto duality applies verbatim to obtain sharp constants for the sublaplacian trace inequalities is load-bearing but not verified in detail. The sublaplacian is a sum of squares of left-invariant fields on a stratified nilpotent group; its Riesz potentials and Green's functions possess different homogeneity degrees and lack full rotational symmetry relative to the Euclidean Laplacian. The manuscript must explicitly confirm that the integral representation or pairing identity used in the duality carries over without extra curvature or boundary correction terms, and that the extremal functions remain unchanged. Absent this check, the sharp constants are not justified by the cited Frank-Lieb HLS inequality.
minor comments (1)
  1. [Abstract] The abstract states that the same approach recovers trace Beckner-Onofri inequalities on the standard sphere; this Euclidean recovery should be stated as a corollary with a brief indication of why no new work is required.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address the major concern regarding the applicability of the duality argument in detail below. We agree that an explicit verification strengthens the presentation and will incorporate the requested confirmation in the revised version.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1 (Introduction/Method): The central claim that the Bez-Machihara-Sugimoto duality applies verbatim to obtain sharp constants for the sublaplacian trace inequalities is load-bearing but not verified in detail. The sublaplacian is a sum of squares of left-invariant fields on a stratified nilpotent group; its Riesz potentials and Green's functions possess different homogeneity degrees and lack full rotational symmetry relative to the Euclidean Laplacian. The manuscript must explicitly confirm that the integral representation or pairing identity used in the duality carries over without extra curvature or boundary correction terms, and that the extremal functions remain unchanged. Absent this check, the sharp constants are not justified by the cited Frank-Lieb HLS inequality.

    Authors: We appreciate the referee's emphasis on this foundational step. The duality argument of Bez-Machihara-Sugimoto relies on an integral pairing between the fractional operator and its dual Riesz potential, which in our setting is realized via the Green's function for the conformally invariant fractional sublaplacian on the Heisenberg group (and its boundary version on the CR sphere). These Green's functions are explicitly known from the literature on subelliptic operators and satisfy the same homogeneity and positivity properties as in the Euclidean case, scaled by the homogeneous dimension Q rather than n. Because the operators are left-invariant and the underlying measure is Haar measure, no additional curvature or boundary correction terms appear in the pairing identity. The extremal functions are the standard conformal bubbles, which remain unchanged by the conformal invariance of the fractional powers. The Frank-Lieb sharp HLS inequality on the Heisenberg group and CR sphere already incorporates the correct homogeneity and lack of full rotational symmetry. Nevertheless, to make the transfer fully explicit, we will add a short subsection in §2 that derives the pairing identity directly from the integral kernel and confirms the absence of extra terms. This revision will also include a brief comparison with the Euclidean sphere case recovered by the same method. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on independent external citations

full rationale

The paper's derivation is presented as an extension of Euclidean results via the duality argument of Bez-Machihara-Sugimoto combined with the Frank-Lieb sharp HLS inequalities on the Heisenberg group and CR sphere. These are treated as pre-existing independent results rather than derived or fitted within the paper itself. No equations, definitions, or steps reduce a claimed prediction or sharp constant to a self-defined input, a fitted parameter renamed as output, or a self-citation chain. The abstract and claimed method contain no self-referential construction that would force the result by definition. This qualifies as a normal non-circular case where the load-bearing steps are externally supported.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all content appears to rest on standard background from cited prior works.

pith-pipeline@v0.9.0 · 5421 in / 1034 out tokens · 28193 ms · 2026-05-10T07:26:07.145778+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    A. G. Ache, S.-Y. A. Chang, Sobolev trace inequalities of order four, Duke Math. J. 166(14)(2017), 2719-2748. 2

    work page 2017

  2. [2]

    Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann

    W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math. (2) 138 (1993), 213-242. 7, 9

    work page 1993

  3. [3]

    Beckner, Functionals for multilinear fractional embedding, Acta Math

    W. Beckner, Functionals for multilinear fractional embedding, Acta Math. Sin. 31 (2015), 1-28. 2, 6

    work page 2015

  4. [4]

    N. Bez, S. Machihara, M. Sugimoto, Extremisers for the trace theorem on the sphere, Math. Res. Lett. 23(2016), No.3, 633-647. 2, 6, 10

    work page 2016

  5. [5]

    T. P. Branson, L. Fontana, C. Morpurgo, Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere, Ann. of Math. (2), 177 (2013), 1-52. 4, 6, 8

    work page 2013

  6. [6]

    L. A. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Equa., 32 (2007), 1245-1260. 1

    work page 2007

  7. [7]

    E. A. Carlen, M. Loss, Competing symmetries of some functionals arising in mathematical physics, Stochastic processes, physics and geometry (Ascona and Locarno, 1988), 277-288, World Sci. Publ., Teaneck, NJ, 1990. 1

    work page 1988

  8. [8]

    J. S. Case, Boundary operators associated with the Paneitz operator. Indiana Univ. Math. J. 67(2018), no. 1, 293-327. 2

    work page 2018

  9. [9]

    J. S. Case, Some energy inequalities involving fractional GJMS operators. Anal. PDE, 10(2)( 2017), 253-280. 2

    work page 2017

  10. [10]

    J. S. Case, Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian, J. Funct. Anal. 279(4)(2020), 108567. 1

    work page 2020

  11. [11]

    J. S. Case S.-Y. A. Chang, On fractional GJMS operators. Comm. Pure Appl. Math. 69(6)(2016), 1017-1061. 2

    work page 2016

  12. [12]

    J. S. Case, W. Luo, Boundary operators associated with the sixth-order GJMS operator, Int. Math. Res. Not. IMRN, 14(2021), 10600-10653. 2

    work page 2021

  13. [13]

    Chatzakou, A

    M. Chatzakou, A. Kassymov, M. Ruzhansky, Logarithmic Sobolev, Hardy and Poincar´ e in- equalities on the Heisenberg group, arXiv:2310.00992 9

    work page arXiv

  14. [14]

    X. Chen, S. Zhang, A sharp Sobolev trace inequality of order four on three-balls, Journal d’Analyse Math´ ematique, accepted, see avalible arXiv:2403.00380. 2

    work page arXiv

  15. [15]

    Cowling, Unitary and uniformly bounded representations of some simple Lie groups, In: Analysis and Group Representations, C.I.M.E

    M. Cowling, Unitary and uniformly bounded representations of some simple Lie groups, In: Analysis and Group Representations, C.I.M.E. Napoli; Liguori, 1982, 49-128. 3, 10

    work page 1982

  16. [16]

    Einav, M

    A. Einav, M. Loss, Sharp trace inequalities for fractional Laplacians, Proc. Amer. Math. Soc. 140(2012), No.12, 4209-4216. 2, 6

    work page 2012

  17. [17]

    J. F. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J. 37(1988), no. 3, 687-698. 1

    work page 1988

  18. [18]

    Flynn, G

    J. Flynn, G. Lu and Q. Yang, Conformally covariant boundary operators and sharp higher order CR Sobolev trace inequalities on the Siegel domain and complex ball, Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 824 (2025), 39-87. 2

    work page 2025

  19. [19]

    Flynn, G

    J. Flynn, G. Lu and Q. Yang, Conformally covariant boundary operators and sharp higher order Sobolev trace inequalities on Poincar´ e Einstein manifolds, arXiv:2311.10070. 2

    work page arXiv

  20. [20]

    Folland, Harmonic Analysis in Phase Space, Ann

    G.B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud., vol. 122, Princeton University Press, Princeton, NJ, 1989. 4

    work page 1989

  21. [21]

    G. B. Folland, Spherical harmonic expansion of the Poisson-Szeg¨ o kernel for the ball, Proc. Amer. Math. Soc 47 (1975), No.2, 401-408. 4

    work page 1975

  22. [22]

    Frank, M.d.M

    R.L. Frank, M.d.M. Gonz´ alez, D. D. Monticelli, J. Tan, An extension problem for the CR fractional Laplacian, Adv. Math. 270 (2015), 97-137. 2

    work page 2015

  23. [23]

    R. L. Frank, E. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. Math. (2) 176 (2012), No.1, 349-381. 9, 10, 14

    work page 2012

  24. [24]

    Ghosh1, V

    S. Ghosh1, V. Kumar, M. Ruzhansky, Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups, Calc. Var. (2026) 65:28. 9

    work page 2026

  25. [25]

    R. Gong, Q. Yang, S. Zhang, A simple proof of reverse Sobolev inequalities on the sphere and Sobolev trace inequalities on the unit ball, J. Func. Anal., 290 (2026), no. 9, 111380. 2 SHARP TRACE INEQUALITIES FOR FRACTIONAL POWERS OF THE SUBLAPLACIAN 21

    work page 2026

  26. [26]

    E. H. Lieb, M. Loss, Analysis, Second Edition. Graduate Studies in Mathematics, Vol. 14. American Mathematical Society, Providence, RI, 2001. 19

    work page 2001

  27. [27]

    Morpurgo, Sharp inequalities for functional integrals and traces of conformally invariant operators, Duke Math

    C. Morpurgo, Sharp inequalities for functional integrals and traces of conformally invariant operators, Duke Math. J. 114 (3) (2002), 477-553. 18

    work page 2002

  28. [28]

    Onofri, On the positivity of the effective action in a theory of random surfaces, Comm

    E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys. 86 (1982), 321-326. 7

    work page 1982

  29. [29]

    Roncal, S

    L. Roncal, S. Thangavelu, Hardy’s inequality for fractional powers of the sublaplacian on the Heisenberg group, Adv. Math. 302 (2016), 106-158. 3, 10

    work page 2016

  30. [30]

    Stanton, Spectral invariants of CR manifolds, Michigan Math

    N.K. Stanton, Spectral invariants of CR manifolds, Michigan Math. J. 36 (1989), 267-288. 8

    work page 1989

  31. [31]

    Thangavelu, An introduction to the uncertainty principle

    S. Thangavelu, An introduction to the uncertainty principle. Hardy’s theorem on Lie groups, Progress in Mathematics 217. Birkh¨ auser, Boston, MA, 2004. 4

    work page 2004

  32. [32]

    Y. Wang, Q. Yang, Sharp fractional Sobolev and related inequalities on H-type groups, arXiv:2406.16278v2

    work page arXiv

  33. [33]

    Yang, Sharp Sobolev trace inequalities for higher order derivatives, arXiv:1901.03945

    Q. Yang, Sharp Sobolev trace inequalities for higher order derivatives, arXiv:1901.03945. 2 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Peo- ple’s Republic of China Email address:qhyang.math@whu.edu.cn School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Peo- ple’s Republic of China Email address:2023302011095@...

    work page arXiv 1901