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Optimal Sobolev inequalities of high order with $L^2$-remainder

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Pith's one-line read Higher-order Sobolev inequalities with the optimal constant are governed by scalar curvature, except in a high-dimensional regime where scalar curvature no longer decides.

desk verdict A genuinely new dichotomy for high-order Sobolev inequalities with L2 remainder, with a real but localized soft spot in the Green's function proof. read the letter →

arxiv 2506.17028 v1 pith:47SUJSC3 submitted 2025-06-20 math.AP

Lorenzo Carletti , Frédéric Robert This is my paper · ORCID
classification math.AP MSC 35J3535J6035B4435J0858J05
keywords optimalSobolevinequalityhigher-orderL2remainderscalarcurvatureblow-upanalysispolyharmonicoperatorGreen'sfunctionclosedRiemannianmanifold
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

For the classical Sobolev inequality on a closed manifold, the optimal Euclidean constant can be attained with an $L^2$ remainder term regardless of geometry. This paper claims that for higher-order versions ($k\ge2$) the situation changes: the optimal inequality $(I_{\mathrm{opt},k})$ holds or fails depending on the manifold's dimension and curvature. The sharpest result is for $k=2$: the inequality holds exactly when $n=5$ or the scalar curvature is non-positive everywhere. For general $k$, negative scalar curvature everywhere guarantees validity, positive scalar curvature somewhere forbids it in high enough dimensions, and in high dimensions scalar curvature alone no longer decides the question. A reader should care because this pinpoints precisely where geometry enters an apparently constant-only problem.

What carries the argument

The argument is carried by a sharp pointwise description of blowing-up solutions. The paper considers sequences $u_\alpha$ of solutions to $\Delta_g^k u_\alpha+\alpha^{2k}u_\alpha=|u_\alpha|^{2^\star-2}u_\alpha$ with energy just below the Euclidean threshold, and proves (Theorem 6) that each such sequence is a Euclidean bubble $U$ plus a small remainder controlled by $$(\mu_\$\alpha$+d(x_\$\alpha$,x))^l\,|\nabla^l(u_\$\alpha$-V_{\$\alpha$,\nu_\$\alpha$})| \le C_{p,\tau}\,\frac{(\$\alpha$(\mu_\$\alpha$+d(x_\$\alpha$,x)))^\tau}{(1+\$\alpha$ d(x_\$\alpha$,x))^p}\,B_{x_\$\alpha$,\mu_\$\alpha$}(x).$$ The fast decay in $\alpha d$ comes from a Green's-function estimate (Theorem 8) for the operator $\Delta_g^k+\alpha^{2k}$. Feeding this control into a Pohozaev identity produces terms whose relative sizes decide validity; separate test-function expansions produce the non-validity obstructions.

What would settle it

Take the flat torus $\mathbb{T}^n$, $n>2k$, and compute the Green's function for $\Delta^k+\alpha^{2k}$ in Fourier series; if the bound $|G_\alpha(x,y)|\le C\,d(x,y)^{2k-n}/(1+\alpha^q d(x,y)^q)$ fails for some $q$, then Theorem 8 is false and the validity proofs built on it collapse, while verifying it for all $q$ would remove the main technical risk.

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Extended reading notes

Core claim

The paper's central discovery is a dichotomy: the inequality $$\left(\int_M |u|^{2^\star}\,dv_g\right)^{2/2^\star} \le K(n,k)\int_M (\$Delta_g^{{k/2}}$u)^2\,dv_g + B\int_M $u^{2}$\,dv_g$$ with the Euclidean optimal constant $K(n,k)$ and an $L^2$ remainder holds on a closed $n$-manifold precisely under geometric conditions. For $k=2$ and $n\ge5$, it holds if and only if $n=5$ or $R_g\le0$ everywhere. More generally, for $n=2k+1$ it always holds; for $n=2k+2$ or $2k+3$, and for $k=2$ with $n\ge6$, it holds exactly when $R_g\le0$; for $n\ge2k+2$ it holds when $R_g<0$ everywhere and fails when $R_g>0$ somewhere. The paper also shows that for $k\ge3$ and $n\ge2k+4$, a Ricci-flat non-flat manifold fails while a flat torus holds, so scalar curvature alone cannot decide validity in high dimensions.

Load-bearing premise

The load-bearing premise is the sharp pointwise estimate (12) for the difference between a solution and its approximating bubble, which relies on the fast decay (69) of the Green's function of $\Delta_g^k+\alpha^{2k}$; if that decay is slower than claimed, the contradiction arguments proving validity close too slowly or not at all.

Editorial extensions

If this is right

  • For second-order inequalities ($k=2$) the validity question is now completely settled in every dimension $n\ge5$.
  • Every closed manifold of dimension $2k+1$ satisfies the optimal inequality regardless of curvature.
  • Positive scalar curvature at a single point blocks the inequality whenever $n\ge2k+2$ and $k>1$.
  • Negative scalar curvature everywhere permits the inequality for all $k>1$ and $n\ge2k+2$.
  • In high dimensions ($k\ge3$, $n\ge2k+4$) curvature data beyond scalar curvature, such as the Weyl tensor, controls validity.

Reading between the lines

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

  • If the pointwise-control machinery extends to $H^l$ remainders for $1\le l\le k-2$, the same dichotomy should appear with shifted critical dimensions; this is an extension the authors flag but leave unproved.
  • The Ricci-flat failure suggests that on scalar-flat manifolds with non-vanishing Weyl tensor, validity should depend on the vanishing order of the Weyl tensor at the worst point, mirroring known second-order results; this is not established here.
  • A numerical or asymptotic check of the Fourier Green's function on the flat torus could test the sharpness of the decay rate in Theorem 8 and thereby the robustness of the validity proof.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the sharp higher-order Sobolev inequality H^2_k(M^n) -> L^{2n/(n-2k)}(M^n) on closed Riemannian manifolds, with the remainder taken in L^2 instead of H^{k-1}_2. The central results are Theorem 1 and the more precise Theorems 4 and 5: for k>1 the validity of the optimal inequality is no longer independent of geometry, as it is for k=1. In particular, the inequality always holds when n=2k+1; it holds for n=2k+2 and n=2k+3 exactly when the scalar curvature R_g is nonpositive everywhere; for n>=2k+2 it holds when R_g<0 everywhere and fails when R_g>0 somewhere; flat tori satisfy it; and for k>2, n>=2k+4, Ricci-flat non-flat manifolds fail. Theorem 2 gives the complete k=2 classification. The proofs combine test-function estimates for the non-validity part with a pointwise blow-up analysis (Theorems 3, 6, 7) and a Pohozaev-type identity; a central technical ingredient is a uniform Green's function estimate for Delta^k_g + alpha^{2k} (Theorem 8).

Significance. If the results are correct, they are a significant contribution to geometric analysis. They show that the validity of the optimal Sobolev inequality with an L^2-remainder is a genuinely geometric question for k>1, in sharp contrast with the k=1 and with the H^{k-1}_2-remainder case. The complete classification for k=2 (Theorem 2) and the sharpness of the conditions in low dimensions are strong and potentially influential. The paper also introduces a pointwise control theorem for polyharmonic equations that may be useful beyond this problem. However, the proof is heavily dependent on several technical inputs whose full proofs are deferred to other works, including an unpublished note, and one central estimate appears to contain a sign error as printed.

major comments (3)
  1. [§6, Propositions 6.2 and 6.3] Propositions 6.2 and 6.3 are stated without proof, with the phrase "Following the same arguments as in [4,30,31]". These propositions provide the uniform invertibility of the linearized operator L_{\alpha,\nu} and the existence and uniqueness of the correction \phi_{\alpha,\nu}. They are directly used in the proof of Theorem 7 and then in the identification of u_\alpha = V_{\alpha,\nu} + \phi_\alpha in Theorem 6. Since the present setting includes the term \alpha^{2k} and a different kernel than any of the cited papers, the deferred proof is load-bearing: the authors should either include a full proof or point to precise statements (with equation numbers) in [4,30,31] that cover exactly this case.
  2. [§7, Step 4, equations (75) and (77)] The uniform Green's function estimate (69) is a crucial technical ingredient: it enters the pointwise bound (60) in Proposition 6.4 and the estimates of III_\alpha, II_\alpha, IV_\alpha in Section 5. The proof in Section 7, Step 4, asserts the bounds (75) for i > E(n/2) by "Iterating Lemma 7.1" and the identity (77) by "standard cancellation", referring to the unpublished note [32]. The \alpha-power bookkeeping in (75) and the cancellation in (77) are exactly what produces the fast decay in (69). Without a complete proof of these steps, the validity half of Theorems 4 and 5 is not fully established within the manuscript.
  3. [§1, Theorem 3, Eq. (5); §5.1, Eq. (33)] As printed, the pointwise control (5) and its rescaled version (33) contain (1+\alpha d)^p in the numerator. This grows with \alpha and contradicts the announced fast decay at infinity; it is also incompatible with the subsequent use in the proof of Lemma 5.1 and with the estimate IV_\alpha = O(\mu^{n-2k}/\alpha^{2q}) in §5.1. The exponent must be negative. This sign error affects the central estimate of the paper and should be corrected before the proof can be considered rigorous.
minor comments (4)
  1. [§5.4, Lemma 5.1] In the definition of \varphi_\alpha, the notation \exp^{-1}_{x_\alpha} should specify the metric (g or g_{x_\alpha}) to avoid ambiguity.
  2. [§5, equations (42) and (45)] The constants c_n^{(1)} and c_n^{(2)} are not explicitly defined; stating their positivity and independence of \alpha would help the reader.
  3. [§5.1, equation (39)] The definition of \theta_\alpha for n<2k+4 is compact; a one-line derivation of the case n=2k+1, where \theta_\alpha = \mu/\alpha^3, would improve readability.
  4. [§6, equations (46)-(50)] Some "straightforward computations" (e.g., (46) and (47)) are used to justify the size of the error terms; providing a brief indication of the computation would increase verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the L2-remainder dichotomy is not an input to the cited pointwise or Green's-function estimates.

full rationale

The central results, Theorems 1, 2, 4 and 5, are not obtained by assuming (I_opt,k) or a geometrically equivalent statement. Non-validity is proved by independent test-function expansions, (20) and (23), whose inputs are the Euclidean bubble U, conformal normal coordinates, and known expansions of the relevant elliptic operator; none of these inputs is the target inequality. Validity is proved by contradiction: assuming failure produces a sequence u_alpha solving (27), and the Pohozaev-type identities (40)-(41) are then combined with the independent L^2 estimate (45) to force a sign contradiction. The pointwise control (12) used in that blow-up analysis is not a restatement of the Sobolev inequality being classified; it is a statement about solutions of the auxiliary equation (9), established through the Green's function of Delta_g^k + alpha^{2k} in Theorem 8. The proof of Theorem 8 is largely self-contained: the Euclidean Green's function is constructed by Fourier transform in Theorem 9, the approximate Green's function is built in (71)-(73), and the Neumann series estimate (74)-(77) is presented in the paper, even though part of the alpha-power bookkeeping is compressed and referenced to the unpublished note [32]. The skeptical concern about (75)-(77) is a concern about completeness or rigor of a technical estimate, not about circularity: the estimate concerns a different operator, and its failure would break the proof rather than make the conclusion an input. The self-citations to Carletti [4], Robert [33], and the unpublished note [32] are load-bearing as technical tools, but they are not citations of the L2-remainder theorem. In particular, [4] and [39] prove the optimal inequality with an H^{k-1}_2 remainder, which is a different statement; the target L2-remainder dichotomy is not assumed there. Therefore the derivation is not circular by construction, and the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

There are no fitted parameters and no newly postulated entities. The proof rests on standard sharp Sobolev theory, conformal geometry, elliptic regularity, and two background results from the authors' earlier work, one of which is an unpublished note.

assumptions (6)
  • standard math The Euclidean sharp Sobolev constant K(n,k) exists, with explicit extremal U and classification of positive solutions to Delta^k V = V^{2*-1} by Swanson and Wei-Xu.
    Invoked in equations (2) and (6), and in Lemma 4.1 to identify the blow-up limit as the standard bubble U.
  • standard math On any closed manifold, Mazumdar's approximation gives the optimal constant K(n,k) for the H^{k-1}_2-remainder inequality, and Carletti and Zeitler proved the exact inequality (3) with A=K(n,k).
    Used to derive the epsilon-interpolated inequality (4) and to set up the contradiction in Section 5.
  • standard math Conformal normal coordinates exist at every point p with conformal factor phi_p satisfying phi_p(p)=1, gradient(phi_p)(p)=0, Ric_{g_p}(p)=0, and volume form (1+O(|x|^N))dx.
    Equation (7) supplies the test functions and the conformal charts used throughout Sections 3 and 5.
  • standard math GJMS operators P_k^g of order 2k are conformally covariant, and on Einstein manifolds with vanishing Ricci curvature P_k^g = Delta^k_g.
    Equations (14)-(15) and Section 3.2 use these properties for the conformal test-function computations and the Ricci-flat expansion.
  • domain assumption The linear theory from Carletti [4] and Premoselli [30,31] stated in Propositions 6.2 and 6.3 is valid: uniform invertibility of L_{alpha,nu} on K^bot and existence of small fixed-point solutions.
    These propositions are not proved in this paper, and Theorem 7, hence the pointwise control (12), depends on them.
  • domain assumption The Neumann-series construction in the unpublished note [32] correctly gives the Green's function for Delta^k_g + alpha^{2k} and the estimate (69).
    Section 7.4 delegates part of the construction of G_alpha to [32], which is not publicly available; this is the main unverifiable background input.

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Pith. "Pith review of Optimal Sobolev inequalities of high order with $L^2$-remainder." pith.science (2026). https://pith.science/paper/47SUJSC3

@misc{pith2026250617028,
  author       = {Pith},
  title        = {Pith review of: Optimal Sobolev inequalities of high order with $L^2$-remainder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47SUJSC3}},
  note         = {Machine review of arXiv:2506.17028}
}
abstract

We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal inequality does not hold in general for $k>1$. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when $k=2$ and in small dimensions.

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Works this paper leans on

39 extracted references · 36 canonical work pages

  1. [32]

    Unpublished notes

    Fr´ ed´ eric Robert,Existence et asymptotiques optimales des fonctions de Green des op´ erateurs elliptiques d’ordre deux (Existence and optimal asymptotics of the Green’s functions of second-order elliptic operators) (2010). Unpublished notes

    work page 2010

  2. [1]

    Agmon, A

    S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , Comm. Pure Appl. Math. 12 (1959), 623–727

    work page 1959

  3. [2]

    Thierry Aubin, Espaces de Sobolev sur les vari´ et´ es riemanniennes, Bull. Sci. Math. (2) 100 (1976), no. 2, 149–173 (French)

    work page 1976

  4. [3]

    Thomas Bartsch, Tobias Weth, and Michel Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator , Calc. Var. Partial Differential Equations 18 (2003), no. 3, 253–268

    work page 2003

  5. [4]

    London Math

    Lorenzo Carletti, Attaining the optimal constant for higher-order Sobolev inequalities on manifolds via asymptotic analysis , J. London Math. Soc. 111 (2025), no. 5

    work page 2025

  6. [5]

    Differential Equations 419 (2025), 370–417

    , The Green’s function of polyharmonic operators with diverging coefficients: con- struction and sharp asymptotics , J. Differential Equations 419 (2025), 370–417

    work page 2025

  7. [6]

    Bin Deng, Liming Sun, and Jun-Cheng Wei, Sharp quantitative estimates of Struwe’s decom- position, Duke Math. J. 174 (2025), no. 1, 159–228

    work page 2025

  8. [7]

    Zindine Djadli, Emmanuel Hebey, and Michel Ledoux, Paneitz-type operators and applica- tions, Duke Math. J. 104 (2000), no. 1, 129–169

    work page 2000

Show all 39 references
  1. [8]

    Olivier Druet, The best constants problem in Sobolev inequalities , Math. Ann. 314 (1999), no. 2, 327–346. OPTIMAL SOBOLEV INEQUALITIES OF HIGH ORDER WITH L2−REMAINDER 31

    1999

  2. [9]

    Olivier Druet and Emmanuel Hebey, The AB program in geometric analysis: sharp Sobolev inequalities and related problems, Mem. Amer. Math. Soc. 160 (2002), no. 761, viii+98

    2002

  3. [10]

    Differential Equations 7 (2002), no

    , Asymptotics for sharp Sobolev-Poincar´ e inequalities on compact Riemannian man- ifolds, Adv. Differential Equations 7 (2002), no. 12, 1409–1478

    2002

  4. [11]

    PDE 2 (2009), no

    , Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Anal. PDE 2 (2009), no. 3, 305–359

    2009

  5. [12]

    45, Princeton University Press, Princeton, NJ, 2004

    Olivier Druet, Emmanuel Hebey, and Fr´ ed´ eric Robert,Blow-up theory for elliptic PDEs in Riemannian geometry, Mathematical Notes, vol. 45, Princeton University Press, Princeton, NJ, 2004

    2004

  6. [13]

    Olivier Druet, Emmanuel Hebey, and Michel Vaugon, Optimal Nash’s inequalities on Rie- mannian manifolds: the influence of geometry , Internat. Math. Res. Notices 14 (1999), 735– 779

    1999

  7. [14]

    , Sharp Sobolev inequalities with lower order remainder terms , Trans. Amer. Math. Soc. 353 (2001), no. 1, 269–289

    2001

  8. [15]

    Robin Graham, The ambient metric, Annals of Mathematics Stud- ies, vol

    Charles Fefferman and C. Robin Graham, The ambient metric, Annals of Mathematics Stud- ies, vol. 178, Princeton University Press, Princeton, NJ, 2012

    2012

  9. [16]

    Alessio Figalli and Federico Glaudo, An invitation to optimal transport, Wasserstein dis- tances, and gradient flows, 2nd ed., EMS Textbooks in Mathematics, MS Press, Berlin, [2023] ©2023

    2023

  10. [17]

    Robin Graham, Ralph Jenne, Lionel J

    C. Robin Graham, Ralph Jenne, Lionel J. Mason, and George A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence , J. London Math. Soc. (2) 46 (1992), no. 3, 557–565

    1992

  11. [18]

    5, New York University, Courant Institute of Mathemat- ical Sciences, New York; American Mathematical Society, Providence, RI, 1999

    Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathemat- ical Sciences, New York; American Mathematical Society, Providence, RI, 1999

    1999

  12. [19]

    , Sharp Sobolev inequalities of second order, J. Geom. Anal. 13 (2003), no. 1, 145–162

    2003

  13. [20]

    Emmanuel Hebey and Michel Vaugon, Le probl` eme de Yamabe ´ equivariant, Bull. Sci. Math. 117 (1993), no. 2, 241–286 (French, with English and French summaries)

    1993

  14. [21]

    , The best constant problem in the Sobolev embedding theorem for complete Riemann- ian manifolds, Duke Math. J. 79 (1995), no. 1, 235–279

    1995

  15. [22]

    Khuri, Fernando C

    Marcus A. Khuri, Fernando C. Marques, and Richard M. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009), no. 1, 143–196

    2009

  16. [23]

    Lee and Thomas H

    John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91

    1987

  17. [24]

    YanYan Li and Jingang Xiong, Compactness of conformal metrics with constant Q-curvature. I, Adv. Math. 345 (2019), 116–160

    2019

  18. [25]

    Differential Equations 261 (2016), no

    Saikat Mazumdar, GJMS-type operators on a compact Riemannian manifold: best constants and Coron-type solutions, J. Differential Equations 261 (2016), no. 9, 4997–5034

    2016

  19. [26]

    Saikat Mazumdar and J´ erˆ ome V´ etois,Existence results for the higher-order Q-curvature equation, Calc. Var. Partial Differential Equations 63 (2024), no. 6, Paper No. 151, 29

    2024

  20. [27]

    Francesco Nobili and Davide Parise, Quantitative stability of Sobolev inequalities on compact Riemannian manifolds, Int. Math. Res. Not. IMRN 1 (2025), Paper No. rnae269, 20

    2025

  21. [28]

    Bruno Premoselli and Fr´ ed´ eric Robert,One-bubble nodal blow-up for asymptotically critical stationary Schr¨ odinger-type equations, J. Funct. Anal. 288 (2025), no. 6, Paper No. 110808, 43

    2025

  22. [29]

    Bruno Premoselli, A pointwise finite-dimensional reduction method for a fully coupled system of Einstein-Lichnerowicz type, Commun. Contemp. Math. 20 (2018), no. 6, 1750076, 72

    2018

  23. [30]

    , Towers of Bubbles for Yamabe-Type Equations and for the Br´ ezis-Nirenberg Problem in Dimensions n≥ 7 32 (2022), no. 3

    2022

  24. [31]

    , A priori estimates for finite-energy sign-changing blowing-up solutions of critical elliptic equations, Int. Math. Res. Not. IMRN 6 (2024), 5212–5273

    2024

  25. [33]

    , Localization of bubbling for high order nonlinear equations , 2024.arXiv:2501.00531

    2024 arXiv

  26. [34]

    , Critical dimensions for polyharmonic operators: the Pucci–Serrin conjecture for solutions of bounded energy, Calc. Var. Partial Differential Equations 64 (2025), no. 3, Paper No. 100. 32 LORENZO CARLETTI AND FR ´ED´ERIC ROBERT

    2025

  27. [35]

    Richard Schoen and Dong Zhang, Prescribed scalar curvature on the n-sphere, Calc. Var. Partial Differential Equations 4 (1996), no. 1, 1–25

    1996

  28. [36]

    Swanson, The best Sobolev constant , Appl

    Charles A. Swanson, The best Sobolev constant , Appl. Anal. 47 (1992), no. 4, 227–239

    1992

  29. [37]

    Juncheng Wei and Xingwang Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999), no. 2, 207–228

    1999

  30. [38]

    Shing-Tung Yau, On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge-Amp` ere equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411

    1978

  31. [39]

    Samuel Zeitler, A sharp higher order Sobolev inequality on Riemannian manifolds , J. Funct. Anal. 289 (2025), no. 6, Paper No. 111001. Lorenzo Carletti, Universit´e Libre de Bruxelles, Service d’analyse, CP 213, Boule- vard du Triomphe, B-1050 Bruxelles, Belgique. Email addres...

    2025

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