Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields

Avas Banerjee; Prasun Roychowdhury; Riju Basak

REVIEW 2 major objections 3 minor 64 references

Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields

T0 review · 2 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims an exact identity relating subcritical and critical weighted Hardy inequalities for radial Baouendi–Grushin vector fields, together with sharp higher-order Rellich constants and explicit $L^2$ remainder identities.

desk verdict Genuinely useful Grushin Hardy–Rellich machinery, but Theorem 1.1's advertised subcritical–critical bridge has a sign error in the log exponent and the Rellich sharpness test function is not admissible. read the letter →

arxiv 2504.19864 v3 pith:4R4AT34A submitted 2025-04-28 math.AP math.CAmath.FA

Avas Banerjee , Riju Basak , Prasun Roychowdhury This is my paper · ORCID

classification math.APmath.CAmath.FA MSC 26D1035H1042B37

keywords Baouendi–GrushinoperatorRellichinequalityHardy–RellichsharpconstantsstabilityofinequalitiescriticalHardyremainderidentitiesextremizeranalysis

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

The paper aims to show that for Baouendi–Grushin vector fields, the subcritical and critical Hardy inequalities are not separate results: Theorem 1.1 asserts an explicit identity that lifts any test function on a lower-dimensional Grushin gauge ball into a test function on a higher-dimensional Grushin space, making the subcritical Hardy deficit on the higher space equal to the critical Hardy deficit on the lower space up to a surface-area constant. On top of that, the paper establishes sharp higher-order Hardy–Rellich inequalities for the radial Grushin operator with explicit best constants, and in the $L^2$ case it replaces inequalities by exact identities whose remainder terms are sums of squared iterated radial operators. A sympathetic reader would care because an exact deficit identity turns a nonattained-constant inequality into a computable quantity, and exact remainders are directly usable in spectral and PDE estimates.

What carries the argument

The machinery is the polar-coordinate calculus attached to the Grushin homogeneous norm $\rho=(|x|^4+4t^2)^{1/4}$, with geometric factor $\psi=|\nabla_G\rho|^2$. The radial Grushin gradient $\nabla_{\rho,G}=\psi^{1/2}\partial_\rho$, the radial Grushin operator $L_{\rho,G}=\psi(\partial_{\rho\rho}+(Q-1)\rho^{-1}\partial_\rho)$, and the logarithmic change of variables $\rho=(\ln(R/\tilde\rho))^{-1/\alpha}$ with $\alpha=(Q-Q'-\beta)/(b-1)$ carry the whole argument: this change of variables is what transfers the logarithmic critical weight on the lower ball into the power-law weight on the higher space. In the $L^2$ remainder identities, the machinery is the family of weighted operators $T_\beta=\psi^{1/2}(\partial_\rho+(Q-\beta-2)/(2\rho))$ and their iterates $R_{\beta,k}=T_\beta\circ T_{\beta+2}\circ\cdots\circ T_{\beta+2k}$, together with spherical-harmonic decomposition.

What would settle it

Take a radial $w$ supported in $B_R^{Q'}(o')$, perform the substitution $\rho=(\ln(R/\tilde\rho))^{-1/\alpha}$ exactly as in the proof of Theorem 1.1, and compare the powers of $\ln(R/\tilde\rho)$ on both sides: the left-hand side has $(\ln(R/\tilde\rho))^{Q'-b}$ while the right-hand deficit has first term $(\ln(R/\tilde\rho))^{b-Q'}$, so the identity fails unless $b=Q'$. A second check is the sharpness sequence of Theorem 4.4: the function $f_\delta$ defined in (3.2) jumps in its first derivative at $\rho=R/2$, so it is not in $W_0^{2,p}$ as the density argument requires.

Watch

Extended reading notes

Core claim

The paper's central claim is an equivalence-with-identity: for Grushin spaces $G=\mathbb{R}^{n+1}$ and $G'=\mathbb{R}^{m+1}$ with homogeneous dimensions $Q>Q'\ge 3$, every $w\in C_0^\infty(B_R^{Q'}(o'))$ can be lifted to $u\in C_0^\infty(\mathbb{R}^{n+1})$ so that the subcritical Hardy deficit on $G$ equals $(\omega_n/\omega_m)((Q-Q'-\beta)/(b-1))^{Q'-1}$ times the critical Hardy deficit on $G'$. The paper further claims that the higher-order Hardy–Rellich constants $C_{k,p,\beta}$ are sharp, that the weighted Grushin Hardy and Rellich deficits are bounded below by explicit distances to extremizer families, and that for $p=2$ the higher-order Hardy–Rellich inequalities are actually identities with explicit positive remainder terms built from iterative radial operators.

Load-bearing premise

The equivalence rests on the change of variables whose logarithmic exponent must come out matching the critical Hardy deficit; the computation in the paper gives the opposite sign unless the parameter $b$ equals the lower ball's dimension $Q'$.

Editorial extensions

If this is right

If Theorem 1.1 is correct, sharpness of the critical Hardy constant on a lower-dimensional Grushin ball transfers to sharpness of the subcritical Hardy constant on the higher-dimensional Grushin space with the explicit multiplicative constant $\omega_n/\omega_m((Q-Q'-\beta)/(b-1))^{Q'-1}$.
The stability estimates from Theorems 1.2 and 1.3 quantify how far a function is from the extremizer family: the deficit controls $\sup_{R>0} d_H(u,R)^p$ and $d_R(u,k,\beta)^2$, so near-equality forces near-invariance under the corresponding Grushin rescaling.
Theorem 4.3 gives the full inductive list of sharp higher-order Rellich and Hardy–Rellich inequalities for radial Grushin operators, and Theorem 4.4 supplies the critical Rellich inequality on gauge balls.
For $p=2$, Theorem 1.4 turns the higher-order inequalities into identities, so the entire deficit is expressed as a finite sum of nonnegative explicit squared remainder terms.
For radial functions, Theorem 5.1 makes the critical/subcritical equivalence two-way and Theorem 5.2 adds a positive remainder term from a weighted interpolation inequality to the critical Hardy inequality.

Reading between the lines

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

The exponent computation in the proof of Theorem 1.1 deserves a stress test: the change of variables produces a logarithmic power $Q'-b$ on the left, whereas the theorem's stated deficit has powers $b-Q'$; if the intended change of variables was $\rho=(\ln(R/\tilde\rho))^{1/\alpha}$ or the identity is symmetric under an exponent flip, the equivalence could survive in a corrected form.
The exact $L^2$ remainder identities plus Lemma 7.1 suggest a decomposition of the full Grushin operator's deficit into a radial remainder plus a spherical-harmonic gap, which would yield an identity for the full operator rather than only its radial part.
The sharp constant ladder $C_{k,p,\beta}$ likely has spectral content: in the Euclidean setting such constants are ground-state energies of polyharmonic operators, so the Grushin versions could be tested numerically as eigenvalues of the radial Grushin operator with inverse-power potentials.
The distance functions $d_H$ and $d_R$ look like conformally invariant metrics; a natural extension is to see whether the stability results survive for $1<p<2$ with a different modulus, by analogy with the $p\ge2$ range treated here.

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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

Here are the two things I'd want you to know. The paper is a real advance on the Grushin side: first extremizer stability theorem for Grushin Hardy inequalities, sharp higher-order Rellich constants via a clean iterative scheme, and explicit L2 remainder identities modelled on Huang–Ye. The polar-coordinate setup is used seriously and the constants are defined carefully. The citation practice is honest; [21] is used for explicit base identities, not for the results being proved.

The soft spots are in two of the advertised headline results. I checked the change of variables in Theorem 1.1 myself. With ρ = (ln(R/ρ~))^{-1/α}, α = (Q-Q'-β)/(b-1), the first transformed integral has log exponent Q'-b, exactly as the proof says, and the second has exponent -(b-1). The theorem statement and the critical deficit (1.6) need b-Q' and b. For b > Q' these are genuinely different, so the equivalence of the subcritical and critical Hardy inequalities is not established as stated. This is not a cosmetic typo: the whole point of Theorem 1.1 is to match the log structure of (1.6). At the borderline b = Q' the statement might survive, but that is a much narrower claim than the one advertised.

The second issue is the sharpness proof of Theorem 4.4. The test function fδ from (3.2) has a jump in ∂ρ at ρ = δ (and at R/2), so it is not in W^{2,p}. That means the proof that the critical Rellich constant is best possible uses an inadmissible test function as written. This can probably be repaired by mollification, but the repair needs to be written out.

Everything else—the stability estimates, the higher-order Lp iteration, the L2 identities—strikes me as structurally sound, though I did not check every constant. The paper also fails responsibly: it states exactly where the radial-function restriction bites and where the non-radial case is open. That is the behaviour of someone engaged with the subject.

Who is this for? Researchers working on Hardy/Rellich inequalities for degenerate elliptic operators. They should look at the L2 identity section and the stability framework, and they should read Theorem 1.1 with suspicion until the exponent issue is resolved. I would send it to a serious referee rather than desk-reject, because the flaws are isolated and checkable and the surrounding machinery is worth having. If the authors fix the exponent mismatch and the test function, the central claims may well stand.

Referee Report

2 major / 3 minor

Summary. The paper develops Lp Hardy, Rellich, and Hardy–Rellich inequalities for the radial Baouendi–Grushin operator on Grushin spaces. Its main advertised contributions are: an identity (Theorem 1.1) and a radial analogue (Theorem 5.1) connecting the subcritical weighted Hardy inequality on a higher-dimensional Grushin space G with the critical weighted Hardy inequality on a lower-dimensional Grushin space G'; stability estimates for Hardy and higher-order Hardy–Rellich inequalities (Theorems 1.2, 1.3, 6.1, 6.2); sharp higher-order Hardy–Rellich inequalities (Theorem 4.3) and a critical Rellich inequality (Theorem 4.4); and, in the L2 case, exact remainder identities (Theorem 1.4). The paper also contains supporting results such as a two-weight Hardy inequality (Theorem 3.2), a critical Hardy inequality (Corollary 3.1), and a weighted L2 comparison lemma (Lemma 7.1).

Significance. If correct, the subcritical–critical equivalence in Theorem 1.1 would be a substantial extension of the Euclidean result of Sano–Takahashi to Grushin spaces, and the stability estimates and exact identities would be welcome contributions. The paper also demonstrates facility with polar-coordinate machinery for Grushin spaces and provides useful auxiliary inequalities. However, the central identity is not established by the proof as written, and a sharpness proof for the critical Rellich inequality uses an inadmissible test function. These problems bear directly on advertised headline results, so the current version cannot be recommended for publication.

major comments (2)

[Section 5, Theorem 1.1; Theorem 5.1] The proof of Theorem 1.1 changes variables by ρ=(ln(R/~ρ))^{-1/α} with α=(Q−Q'−β)/(b−1) and obtains, on page 22, the logarithmic exponent Q'−b in the first integral, using the identity −(Q−β−1)+(α+1)(Q'−1)=(Q'−b)α. The theorem statement, however, requires the exponent b−Q' in the first term of the bracket, which is precisely the exponent appearing in the critical Hardy inequality (1.6) for p=Q'. Since the hypotheses only impose Q'≤b, for every b>Q' the two exponents differ, and the identity stated in Theorem 1.1 cannot hold as proved. The same mismatch occurs in the radial analogue Theorem 5.1. This is load-bearing: Theorem 1.1 is the advertised equivalence between subcritical and critical Hardy inequalities, and the derivation of Theorem 5.2 relies on Theorem 5.1.
[Section 4, Theorem 4.4] In the sharpness proof of the critical Rellich inequality, the test function fδ from (3.2) with γ=p is declared to lie in W^{2,p}_0(B_R^Q(o)) on page 21. The displayed derivative (3.3) shows that ∂_ρ fδ has jumps at ρ=δ and at ρ=R/2, so the weak second derivative contains singular terms and fδ is not in W^{2,p}. The subsequent density argument therefore does not apply, and the proof that the constant ((Q−2)(p−1)/p)^p is best possible is not valid as written.

minor comments (3)

[Section 4, Theorem 4.4 (displayed computation)] The displayed formula for ψ^{-1}L_{ρ,G}fδ contains the coefficient (p^2−1)/p^2 in the second term on the middle interval. A direct differentiation of fδ=(ln(R/ρ))^{(p−1)/p} gives (p−1)/p^2 for that coefficient. The term is O(1) as δ→0, so the sharpness ratio is unaffected, but the computation should be corrected.
[Section 7, Lemma 7.1] In the proof of Lemma 7.1, the phrase 'using the conditions −Q ≤ −2 ≤ α < Q−4' should read 'using the condition −2 ≤ α < Q−4'; the lower bound −Q is not part of the lemma's hypothesis and is irrelevant to the estimate.
[Throughout] Several theorem statements (e.g., Theorems 1.2 and 1.3) use C0∞(R^{n+1}) without requiring the support to avoid the origin, while the proofs use polar-coordinate integration by parts at ρ=0. This is a minor technical point, but specifying C0∞(R^{n+1}\{o}) or a standard density argument would improve precision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained or rests on independently stated identities, not on fitted or definitionally forced outputs.

full rationale

I walked the main derivation chain. Theorem 1.1 is a change-of-variables computation: given w, the paper defines u(ρ)=w(ρ~) with ρ=(ln R/ρ~)^{-1/α}; the subcritical Hardy deficit on G is then computed by polar coordinates and equals a constant times the critical bracket on G'. This is a direct calculation, not an output fitted to the target identity. The only issue I see there is an apparent sign mismatch between the exponent Q'-b produced in the proof and the exponent b-Q' stated in the theorem, which is a correctness or typographical concern, not a circular reduction. The sharp constants in Theorem 4.3 are verified by explicit extremal sequences using Lemma 2.3, and the L2 remainder identities in Theorem 1.4 are derived by induction from the operator identities in Lemmas 7.2-7.5 and Lemma 7.5; the k=1,2 base cases are explicitly attributed to the authors' earlier paper [21], but those are independent, parameter-free identities whose assumptions do not include the higher-order claims being proved, so the self-citation is not load-bearing circularity. No fitted parameter is later relabeled as a prediction, and no ansatz is smuggled in through a citation. I therefore find no step where a claimed prediction or first-principles result reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data fitting is present. The imported constants are sharp constants from prior literature, and the paper introduces no new physical or geometric entities.

assumptions (5)

domain assumption Polar coordinate decomposition for Grushin space is valid, with the stated volume element and radial operators.
All integrations by parts and sharpness computations rely on this coordinate structure from Garofalo and Shen.
domain assumption The Frank-Seiringer inequality gives the sharp lower bound with constant c_p in Lemma 2.2.
The stability theorems 1.2 and 1.3 use it to extract nonnegative remainder terms.
domain assumption The Caffarelli-Kohn-Nirenberg inequality of Song and Li is valid in the stated parameter range.
Proposition 5.1 and Theorem 5.2 use it to add positive remainder terms.
standard math Smooth compactly supported functions are dense in the weighted Sobolev spaces used for truncation and sharpness sequences.
This standard approximation argument is used throughout and is stated in the notation section with reference to [1].
domain assumption The base weighted Hardy and critical Hardy inequalities from [63, Corollary 4.2] and [63, Corollary 4.7] are correct.
The paper builds on these very recent results and compares Theorem 3.2 and Corollary 3.1 with them.

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Pith. "Pith review of Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields." pith.science (2026). https://pith.science/paper/4R4AT34A

@misc{pith2026250419864,
  author       = {Pith},
  title        = {Pith review of: Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4R4AT34A}},
  note         = {Machine review of arXiv:2504.19864}
}

abstract

In this paper, we improve the $L^p$-Rellich and Hardy-Rellich inequalities in the setting of radial Baouendi-Grushin vector fields. We establish an identity relating the subcritical and critical Hardy inequalities, thereby demonstrating their equivalence. Moreover, we obtain improved versions of these inequalities via an analysis of extremizer stability. In the higher-order setting, we derive Hardy-Rellich type inequalities involving all radial operators in the Grushin framework and prove that all resulting constants are sharp. Finally, for the $L^2$-higher-order cases, we compute exact remainder terms by establishing identities rather than inequalities.

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