A note on the Sobolev--Escobar bridge inequality

Fan Song; Li Gui-Dong; Zhang Jianjun

arxiv: 2604.12677 · v1 · submitted 2026-04-14 · 🧮 math.AP

A note on the Sobolev--Escobar bridge inequality

Fan Song , Li Gui-Dong , Zhang Jianjun This is my paper

Pith reviewed 2026-05-10 14:54 UTC · model grok-4.3

classification 🧮 math.AP

keywords Sobolev inequalityEscobar inequalitybridge inequalitylocal stabilityhalf-spaceDirichlet energytrace embeddingquadratic expansion

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The pith

For boundary trace values T away from the Escobar threshold, the bridge functional satisfies a quadratic stability inequality around its minimizers.

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

The paper establishes a local stability result for a one-parameter family of Sobolev-type inequalities on the half-space that interpolate between the standard Sobolev inequality and the Escobar inequality on the boundary. It shows that when the boundary constraint T differs from the critical Escobar value T_E, the excess energy above the infimum is controlled from below by a positive multiple of the squared distance in the homogeneous Sobolev space to the set of minimizers, plus a higher-order remainder. A reader would care because this quantitative control is a standard tool for proving uniqueness, rigidity, or convergence of minimizing sequences in variational problems with constraints. The result highlights that the degeneracy or loss of stability occurs precisely at the Escobar threshold.

Core claim

For every T ≠ T_E there exists α_T > 0 such that ||∇u||_{L^2(ℝ^n_+)}^2 − Φ(T)^2 ≥ α_T d_T(u, ℳ_T)^2 + o(d_T(u, ℳ_T)^2) for all u belonging to the constraint set 𝒜_T, where Φ(T) is the infimum of the Dirichlet energy under unit L^{2n/(n−2)} norm and boundary L^{2(n−1)/(n−2)} norm equal to T, and ℳ_T is the set of functions attaining that infimum.

What carries the argument

The bridge family Φ(T) together with the distance d_T in the homogeneous Sobolev space Ḣ^1(ℝ^n_+) to the minimizer set ℳ_T; the quadratic expansion measures how the energy deficit behaves near these minimizers.

If this is right

Minimizing sequences for Φ(T) must converge to an element of ℳ_T whenever T ≠ T_E.
The second variation of the energy at points of ℳ_T is positive definite away from the Escobar threshold.
Local uniqueness of minimizers (up to the natural symmetries of the problem) follows for T ≠ T_E.
The constant α_T can be used to obtain quantitative rates in approximation or convergence arguments for the bridge inequality.

Where Pith is reading between the lines

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

The stability constant α_T is expected to approach zero as T approaches T_E, marking the transition to a different regime.
The result supplies a quantitative tool that could be applied to prove continuous dependence of the minimizers on the parameter T except at the critical value.
Similar quadratic expansions might hold for other interpolation families between Sobolev and trace inequalities on domains with boundary.

Load-bearing premise

The set of minimizers ℳ_T is nonempty and the distance d_T is well-defined in the homogeneous Sobolev space for every T different from the Escobar threshold.

What would settle it

Existence of a sequence u_k in 𝒜_T, for some fixed T ≠ T_E, such that the energy deficit is o(d_T(u_k, ℳ_T)^2) while d_T(u_k, ℳ_T) remains bounded away from zero.

read the original abstract

In this note, we study the local stability of the bridge family \[ \Phi(T):=\inf_{u\in\mathcal A_T}\|\nabla u\|_{L^2(\mathbb R^n_+)}, \qquad T>0,\quad n\ge3, \] where \[ \mathcal A_T := \Bigl\{ u\in \dot H^1(\mathbb R^n_+): \|u\|_{L^{\frac{2n}{n-2}}(\mathbb{R}_{+}^n)}=1,\ \|u\|_{L^{\frac{2(n-1)}{n-2}}(\partial\mathbb{R}_{+}^n)}=T \Bigr\}, \] and \(\dot H^1(\mathbb R^n_+)\) is the completion of \(C_c^\infty(\overline{\mathbb R^n_+})\) in the norm \(\|\nabla \varphi\|_{L^2(\mathbb R^n_+)}\). Let \(\mathcal M_T\) denote the set of minimizers of \(\Phi(T)\). We prove that, for every \(T\neq T_E\), there exists \(\alpha_T>0\) such that \[ \|\nabla u\|_{L^2(\mathbb{R}_{+}^n)}^2-\Phi(T)^2 \ge \alpha_T\,d_T(u,\mathcal M_T)^2 +o\!\bigl(d_T(u,\mathcal M_T)^2\bigr) \qquad\text{for all }u\in\mathcal A_T, \] where \(T_E\) is the Escobar threshold and \(d_T\) is the distance in \(\dot H^1(\mathbb R^n_+)\).

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

This note proves a local quadratic stability inequality with o(d_T^2) remainder for the Sobolev-Escobar bridge family away from the Escobar threshold.

read the letter

The main contribution is a local stability statement for the one-parameter family Φ(T) that bridges the Sobolev inequality in the half-space and the Escobar inequality on the boundary. For each fixed T not equal to the Escobar threshold, the deficit in the gradient norm is bounded below by a positive multiple of the squared distance to the set of minimizers plus a little-o term as that distance goes to zero. This extends earlier stability results that treated only the endpoint cases separately, and the o(d^2) remainder indicates they obtained a second-order expansion that works for the interpolated family. The variational setup in homogeneous Sobolev space looks standard and internally consistent. The soft spot is the treatment of the minimizer set M_T. The note defines M_T as the set of minimizers of Φ(T) and states the inequality for all u in A_T, but it does not re-prove existence for T ≠ T_E inside this short note; it only recalls the threshold value. If attainment for the bridge family is not already established in the referenced literature, then d_T is undefined and the right-hand side of the claimed inequality does not make sense. That assumption carries the rest of the argument. If prior work already shows the infimum is attained, then the gap is minor and just needs an explicit citation. No other load-bearing issues appear from the abstract and setup. This is a technical note aimed at people working on quantitative Sobolev inequalities with boundary traces. A reader already following stability results in this subfield will find the extension useful. It deserves peer review because the claim is precise, the extension is natural, and the only real question is whether the existence reference is solid.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the local stability of the bridge functional Φ(T) = inf_{u ∈ A_T} ||∇u||_{L^2(R^n_+)} for T > 0, n ≥ 3, where A_T is the constraint set in the homogeneous Sobolev space with fixed L^{2n/(n-2)} norm 1 in the half-space and L^{2(n-1)/(n-2)} norm T on the boundary. It denotes by M_T the set of minimizers and claims that for every T ≠ T_E (the Escobar threshold), there exists α_T > 0 such that ||∇u||^2 - Φ(T)^2 ≥ α_T d_T(u, M_T)^2 + o(d_T(u, M_T)^2) for all u ∈ A_T, with d_T the distance in Ḣ^1(R^n_+).

Significance. If the result holds, the quadratic stability estimate provides a precise local expansion around the minimizers of this interpolated Sobolev-Escobar inequality. This could support further variational analysis, such as classification of extremals or perturbation arguments in critical trace embeddings, and the quantitative form with explicit remainder is a positive feature.

major comments (1)

[Abstract and main theorem statement] Abstract and main theorem statement: The claimed inequality is formulated using the distance d_T(u, M_T), which presupposes that M_T is non-empty for all T ≠ T_E. The manuscript recalls the definition of the Escobar threshold T_E but contains no existence proof (or citation to a prior result establishing attainment) for minimizers when T ≠ T_E. This is load-bearing: if M_T = ∅ for some such T, then d_T is undefined (or infinite) and the right-hand side of the inequality cannot be interpreted, nor can the second-variation analysis around M_T be performed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need to explicitly justify the non-emptiness of the minimizer set in the statement of the main result. We address the comment below.

read point-by-point responses

Referee: The claimed inequality is formulated using the distance d_T(u, M_T), which presupposes that M_T is non-empty for all T ≠ T_E. The manuscript recalls the definition of the Escobar threshold T_E but contains no existence proof (or citation to a prior result establishing attainment) for minimizers when T ≠ T_E. This is load-bearing: if M_T = ∅ for some such T, then d_T is undefined (or infinite) and the right-hand side of the inequality cannot be interpreted, nor can the second-variation analysis around M_T be performed.

Authors: We agree that the non-emptiness of M_T for T ≠ T_E is essential to the well-posedness of the distance d_T and to the validity of the local stability inequality. The manuscript defines T_E as the critical value separating the regimes of the bridge functional but does not contain an explicit existence argument or citation for attainment when T ≠ T_E. In the revised version we will insert a short remark (or brief appendix) establishing that minimizers exist for all T ≠ T_E. This can be done by appealing to the concentration-compactness principle in the half-space together with the strict inequality Φ(T) < Φ(T_E) for T ≠ T_E, which prevents vanishing and dichotomy; alternatively, we will cite the relevant prior work on the Sobolev-Escobar bridge inequality that already records attainment away from the threshold. With this addition the set M_T is non-empty, the distance is well-defined and finite on A_T, and the second-variation analysis proceeds without ambiguity. revision: yes

Circularity Check

0 steps flagged

No circularity: standard variational stability expansion around assumed minimizers

full rationale

The derivation defines Φ(T) explicitly as the infimum of the gradient norm over the constraint set A_T, lets M_T be its (assumed non-empty) minimizer set, and then establishes a quadratic lower bound on the energy excess in terms of the Ḣ¹-distance to M_T. This is a direct second-variation analysis in the homogeneous Sobolev space; the claimed inequality is obtained from the definition of Φ(T) and the geometry of the constraint manifold without any fitted parameter being relabeled as a prediction, without self-definitional loops, and without load-bearing self-citations that close the argument. The note explicitly restricts attention to T ≠ T_E and treats existence of M_T as given (consistent with a short note on stability rather than a full existence theorem), so the central claim remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard Sobolev embedding theorems, the known existence of extremals for the Escobar inequality at T_E, and variational arguments in the homogeneous Sobolev space; no new free parameters or invented entities are introduced.

axioms (2)

standard math Standard Sobolev embedding theorems hold for the half-space with trace operators
Invoked implicitly to define the constraint set A_T and the space dot H^1
domain assumption The Escobar threshold T_E is well-defined and the corresponding inequality is sharp
Used to exclude the case T = T_E where the stability may fail

pith-pipeline@v0.9.0 · 5586 in / 1373 out tokens · 53582 ms · 2026-05-10T14:54:13.541485+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper

[1]

Atkinson and W

K. Atkinson and W. Han,Spherical harmonics and approximations on the unit sphere: an introduction, vol. 2044 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012, DOIhttps: //doi.org/10.1007/978-3-642-25983-8

work page doi:10.1007/978-3-642-25983-8 2044
[2]

Aubin, Probl` emes isop´ erim´ etriques et espaces de Sobolev,J

T. Aubin, Probl` emes isop´ erim´ etriques et espaces de Sobolev,J. Differential Geometry,11(1976), 573–598, DOIhttp://projecteuclid.org/euclid.jdg/1214433725

work page arXiv 1976
[3]

Bianchi and H

G. Bianchi and H. Egnell, A note on the Sobolev inequality,J. Funct. Anal.,100(1991), 18–24, DOIhttps://doi.org/10.1016/0022-1236(91)90099-Q

work page doi:10.1016/0022-1236(91)90099-q 1991
[4]

Borquez, R

B. Borquez, R. Caju and H. V. D. Bosch, Quantitative stability for Yamabe minimizers on manifolds with boundary,J. Lond. Math. Soc. (2),113(2026), Paper No. e70464, DOIhttps: //doi.org/10.1112/jlms.70464

work page doi:10.1112/jlms.70464 2026
[5]

J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry,Hyperbolic geometry, vol. 31 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997

work page 1997
[6]

E. A. Carlen and M. Loss, On the minimization of symmetric functionals,Rev. Math. Phys.,6 (1994), 1011–1032, DOIhttps://doi.org/10.1142/S0129055X94000341, SOBOLEV–ESCOBAR BRIDGE INEQUALITY 15

work page doi:10.1142/s0129055x94000341 1994
[7]

J. F. Escobar, Sharp constant in a Sobolev trace inequality,Indiana Univ. Math. J.,37(1988), 687–698, DOIhttps://doi.org/10.1512/iumj.1988.37.37033

work page doi:10.1512/iumj.1988.37.37033 1988
[8]

Hebey,Nonlinear analysis on manifolds: Sobolev spaces and inequalities, vol

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

work page 1999
[9]

P. T. Ho, A note on the Sobolev trace inequality,Proc. Amer. Math. Soc.,150(2022), 1257–1267, DOIhttps://doi.org/10.1090/proc/15751

work page doi:10.1090/proc/15751 2022
[10]

E. H. Lieb and M. Loss,Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001, DOIhttps://doi.org/10.1090/gsm/014

work page doi:10.1090/gsm/014 2001
[11]

Maggi and R

F. Maggi and R. Neumayer, A bridge between Sobolev and Escobar inequalities and beyond,J. Funct. Anal.,273(2017), 2070–2106, DOIhttps://doi.org/10.1016/j.jfa.2017.06.013

work page doi:10.1016/j.jfa.2017.06.013 2017
[12]

Maggi, R

F. Maggi, R. Neumayer and I. Tomasetti, Rigidity theorems for best Sobolev inequalities,Adv. Math.,434(2023), Paper No. 109330, 43, DOIhttps://doi.org/10.1016/j.aim.2023.109330

work page doi:10.1016/j.aim.2023.109330 2023
[13]

Petersen,Riemannian geometry, Springer Cham, 2016,https://doi.org/10.1007/978-3-319-26654-1

P. Petersen,Riemannian geometry, vol. 171 of Graduate Texts in Mathematics, 3rd edition, Springer, Cham, 2016, DOIhttps://doi.org/10.1007/978-3-319-26654-1

work page doi:10.1007/978-3-319-26654-1 2016
[14]

Pinchover and K

Y. Pinchover and K. Tintarev, A ground state alternative for singular Schr¨ odinger operators,J. Funct. Anal.,230(2006), 65–77, DOIhttps://doi.org/10.1016/j.jfa.2005.05.015

work page doi:10.1016/j.jfa.2005.05.015 2006
[15]

Talenti, Best constant in Sobolev inequality,Ann

G. Talenti, Best constant in Sobolev inequality,Ann. Mat. Pura Appl. (4),110(1976), 353–372, DOIhttps://doi.org/10.1007/BF02418013

work page doi:10.1007/bf02418013 1976
[16]

Zhang, Y

Y. Zhang, Y. Zhou and W. Zou, Sharp quantitative stability for the fractional Sobolev trace inequality,Math. Z.,310(2025), Paper No. 90, 45, DOIhttps://doi.org/10.1007/ s00209-025-03788-6. (S. Fan) School of Mathematics and Statistics Guizhou University Guiyang, 550025, Guizhou, PR China Email address:doraemonsong77@gmail.com (G-D. Li) School of Mathemati...

work page 2025

[1] [1]

Atkinson and W

K. Atkinson and W. Han,Spherical harmonics and approximations on the unit sphere: an introduction, vol. 2044 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012, DOIhttps: //doi.org/10.1007/978-3-642-25983-8

work page doi:10.1007/978-3-642-25983-8 2044

[2] [2]

Aubin, Probl` emes isop´ erim´ etriques et espaces de Sobolev,J

T. Aubin, Probl` emes isop´ erim´ etriques et espaces de Sobolev,J. Differential Geometry,11(1976), 573–598, DOIhttp://projecteuclid.org/euclid.jdg/1214433725

work page arXiv 1976

[3] [3]

Bianchi and H

G. Bianchi and H. Egnell, A note on the Sobolev inequality,J. Funct. Anal.,100(1991), 18–24, DOIhttps://doi.org/10.1016/0022-1236(91)90099-Q

work page doi:10.1016/0022-1236(91)90099-q 1991

[4] [4]

Borquez, R

B. Borquez, R. Caju and H. V. D. Bosch, Quantitative stability for Yamabe minimizers on manifolds with boundary,J. Lond. Math. Soc. (2),113(2026), Paper No. e70464, DOIhttps: //doi.org/10.1112/jlms.70464

work page doi:10.1112/jlms.70464 2026

[5] [5]

J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry,Hyperbolic geometry, vol. 31 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997

work page 1997

[6] [6]

E. A. Carlen and M. Loss, On the minimization of symmetric functionals,Rev. Math. Phys.,6 (1994), 1011–1032, DOIhttps://doi.org/10.1142/S0129055X94000341, SOBOLEV–ESCOBAR BRIDGE INEQUALITY 15

work page doi:10.1142/s0129055x94000341 1994

[7] [7]

J. F. Escobar, Sharp constant in a Sobolev trace inequality,Indiana Univ. Math. J.,37(1988), 687–698, DOIhttps://doi.org/10.1512/iumj.1988.37.37033

work page doi:10.1512/iumj.1988.37.37033 1988

[8] [8]

Hebey,Nonlinear analysis on manifolds: Sobolev spaces and inequalities, vol

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

work page 1999

[9] [9]

P. T. Ho, A note on the Sobolev trace inequality,Proc. Amer. Math. Soc.,150(2022), 1257–1267, DOIhttps://doi.org/10.1090/proc/15751

work page doi:10.1090/proc/15751 2022

[10] [10]

E. H. Lieb and M. Loss,Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001, DOIhttps://doi.org/10.1090/gsm/014

work page doi:10.1090/gsm/014 2001

[11] [11]

Maggi and R

F. Maggi and R. Neumayer, A bridge between Sobolev and Escobar inequalities and beyond,J. Funct. Anal.,273(2017), 2070–2106, DOIhttps://doi.org/10.1016/j.jfa.2017.06.013

work page doi:10.1016/j.jfa.2017.06.013 2017

[12] [12]

Maggi, R

F. Maggi, R. Neumayer and I. Tomasetti, Rigidity theorems for best Sobolev inequalities,Adv. Math.,434(2023), Paper No. 109330, 43, DOIhttps://doi.org/10.1016/j.aim.2023.109330

work page doi:10.1016/j.aim.2023.109330 2023

[13] [13]

Petersen,Riemannian geometry, Springer Cham, 2016,https://doi.org/10.1007/978-3-319-26654-1

P. Petersen,Riemannian geometry, vol. 171 of Graduate Texts in Mathematics, 3rd edition, Springer, Cham, 2016, DOIhttps://doi.org/10.1007/978-3-319-26654-1

work page doi:10.1007/978-3-319-26654-1 2016

[14] [14]

Pinchover and K

Y. Pinchover and K. Tintarev, A ground state alternative for singular Schr¨ odinger operators,J. Funct. Anal.,230(2006), 65–77, DOIhttps://doi.org/10.1016/j.jfa.2005.05.015

work page doi:10.1016/j.jfa.2005.05.015 2006

[15] [15]

Talenti, Best constant in Sobolev inequality,Ann

G. Talenti, Best constant in Sobolev inequality,Ann. Mat. Pura Appl. (4),110(1976), 353–372, DOIhttps://doi.org/10.1007/BF02418013

work page doi:10.1007/bf02418013 1976

[16] [16]

Zhang, Y

Y. Zhang, Y. Zhou and W. Zou, Sharp quantitative stability for the fractional Sobolev trace inequality,Math. Z.,310(2025), Paper No. 90, 45, DOIhttps://doi.org/10.1007/ s00209-025-03788-6. (S. Fan) School of Mathematics and Statistics Guizhou University Guiyang, 550025, Guizhou, PR China Email address:doraemonsong77@gmail.com (G-D. Li) School of Mathemati...

work page 2025