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The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates

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Pith's one-line read Sharp constants and stability estimates are proved for weighted L² Caffarelli-Kohn-Nirenberg inequalities on curl-free vector fields and second-order derivatives.

desk verdict This paper adds sharp constants for weighted second-order L2-CKN inequalities and stability estimates for both curl-free and second-order cases, but the spherical-harmonic reduction needs explicit checks that the constraints do not spoil sharpness. read the letter →

arxiv 2606.25692 v1 pith:CGAIEPVJ submitted 2026-06-24 math.AP math.FA

Anh Tuan Duong , Van Hoang Nguyen This is my paper · ORCID
classification math.APmath.FA
keywords Caffarelli-Kohn-NirenberginequalitiesweightedL2curl-freevectorfieldssecondorderderivativessharpconstantsstabilityestimatessphericalharmonicdecomposition
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 establishes a family of sharp weighted second-order L²-Caffarelli-Kohn-Nirenberg inequalities that extend earlier results for curl-free fields and second derivatives. It also derives stability versions of the inequalities for curl-free vector fields and for the new second-order cases, quantifying how near-equality forces closeness to optimizers. The proofs reduce the multidimensional problem by spherical harmonic decomposition to sharpened one-dimensional integral inequalities that remain valid under the given weights. A reader would care because these inequalities bound combinations of derivatives against weighted L² norms, which control many elliptic and parabolic PDEs. The stability refinements supply explicit rates useful for rigidity and approximation arguments.

What carries the argument

Spherical harmonic decomposition together with one-dimensional integral inequalities and their improvements, applied in the weighted setting.

What would settle it

A concrete curl-free vector field or test function in the weighted space for which the ratio of the left-hand side to the right-hand side exceeds the claimed sharp constant, or for which the stability distance bound fails at the predicted rate.

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Extended reading notes

Core claim

In this paper, we study the weighted L²-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives. Firstly, we prove a family of the sharp weighted second order L²-Caffarelli-Kohn-Nirenberg inequalities that complements the results in Cazacu et al. and Duong and Nguyen. Secondly, we establish a stability version of the sharp weighted L²-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields proved by Cazacu, Flynn and Lam. Finally, we prove a stability estimate for the sharp weighted second order L²-Caffarelli-Kohn-Nirenberg inequalities established in this paper. Our approach is based on the spherical harmonic decomposition method, the one d

Load-bearing premise

The spherical harmonic decomposition together with one-dimensional integral inequalities and their improvements remain valid and yield sharp constants when applied to the weighted setting for curl-free fields and second-order derivatives.

Editorial extensions

If this is right

  • The new inequalities attain their sharp constants for the indicated weights and orders.
  • Stability estimates hold with explicit constants for the curl-free case.
  • Stability estimates hold with explicit constants for the second-order inequalities proved here.
  • The reduction via spherical harmonics produces the optimal constants uniformly across the family of weights.

Reading between the lines

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

  • The same decomposition may apply directly to other differential constraints such as divergence-free fields.
  • The stability rates could be used to obtain quantitative convergence in variational problems governed by these inequalities.
  • The method suggests a route to sharp constants for higher-order versions or anisotropic weights.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proves a family of sharp weighted second-order L²-Caffarelli-Kohn-Nirenberg inequalities that extend results of Cazacu-Flynn-Lam and Duong-Nguyen, establishes stability estimates for the weighted L²-CKN inequalities on curl-free vector fields, and derives a stability estimate for the new second-order inequalities. The proofs rely on spherical-harmonic decomposition of the fields (or their derivatives), reduction to a family of one-dimensional weighted integral inequalities, and selection of the admissible mode that attains the best constant.

Significance. If the sharpness claims hold, the work supplies the missing weighted second-order cases and the first stability results in this constrained setting, which are useful for quantitative analysis of equality cases and for applications to elliptic systems. The explicit use of spherical harmonics and improved 1D inequalities is a standard, reproducible technique that, when successful, yields parameter-free constants.

major comments (2)
  1. [Proof of Theorem 1.2 (or equivalent section containing the reduction)] The central claim that the constants obtained after spherical-harmonic reduction remain sharp for curl-free fields and for second-order derivatives is load-bearing. The abstract and the method description do not explicitly verify that the minimizing radial profile for the selected harmonic still satisfies the curl-free condition after reconstruction, nor that the weight |x|^a does not shift the optimal degree once the Hessian or Laplacian terms are present. A concrete check (e.g., verification that the Euler-Lagrange equation for the 1D problem is compatible with the divergence-free or curl-free constraint) is required in the proof of the main theorems.
  2. [Section on stability estimates (likely §4 or §5)] Stability estimates are stated for both the curl-free case and the new second-order inequalities. It is not clear from the outline whether the stability constants depend on the same 1D improvement constants used for sharpness or whether an additional error term arises from the projection onto admissible harmonics; this affects whether the stability is quantitative with explicit constants.
minor comments (2)
  1. [Introduction] Notation for the weight parameters a, b, p, q should be introduced once in the introduction and used consistently; several instances of re-definition appear in the abstract and early statements.
  2. [Introduction] The reference list should include the precise statements of the inequalities from Cazacu et al. (2023) and Duong-Nguyen (2025) that are being complemented, to make the novelty paragraph self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions. The two major comments identify places where additional explicit verification and clarification would strengthen the manuscript. We address each point below and will incorporate the requested checks and explanations in the revised version.

read point-by-point responses

  1. Referee: [Proof of Theorem 1.2 (or equivalent section containing the reduction)] The central claim that the constants obtained after spherical-harmonic reduction remain sharp for curl-free fields and for second-order derivatives is load-bearing. The abstract and the method description do not explicitly verify that the minimizing radial profile for the selected harmonic still satisfies the curl-free condition after reconstruction, nor that the weight |x|^a does not shift the optimal degree once the Hessian or Laplacian terms are present. A concrete check (e.g., verification that the Euler-Lagrange equation for the 1D problem is compatible with the divergence-free or curl-free constraint) is required in the proof of the main theorems.

    Authors: We agree that an explicit verification strengthens the argument. In the spherical-harmonic decomposition used for curl-free fields (Section 3), the curl-free condition restricts the admissible vector spherical harmonics; only certain angular modes appear, and the radial coefficients are chosen within that subspace. The 1D weighted inequality is then optimized over these admissible modes, so the minimizing radial profile, when reconstructed, automatically satisfies the curl-free condition by construction of the subspace. The same reduction applies to the second-order case, where the Hessian terms are handled via the corresponding scalar or vector harmonics. The weight |x|^a enters the 1D radial integrals directly and does not alter the selection of the optimal angular degree, which is determined by minimizing the resulting 1D constant over admissible l. We will insert a short paragraph in the proof of Theorem 1.2 that records this compatibility with the Euler-Lagrange equation of the selected 1D problem. This addition does not change any constants or statements but makes the argument self-contained. revision: yes

  2. Referee: [Section on stability estimates (likely §4 or §5)] Stability estimates are stated for both the curl-free case and the new second-order inequalities. It is not clear from the outline whether the stability constants depend on the same 1D improvement constants used for sharpness or whether an additional error term arises from the projection onto admissible harmonics; this affects whether the stability is quantitative with explicit constants.

    Authors: The stability constants are precisely the improvement constants obtained from the 1D inequalities; no additional projection error appears. Because the spherical-harmonic decomposition is orthogonal and the curl-free (respectively, second-order) constraint defines an invariant subspace, every admissible field is exactly the sum of its admissible harmonic components. Consequently, the L² distance to the extremal is the sum of the distances in each mode, and the stability inequality passes directly from the 1D improved inequality without cross terms or remainder. The explicit constants therefore remain the same as those in the 1D stability statements (scaled only by the known dimensional factors from the harmonics). We will add one clarifying sentence in the stability sections (§4 and §5) stating that the decomposition is exact on the constrained space and therefore introduces no extra error term. This makes the quantitative nature of the stability estimates fully transparent. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation relies on standard external methods without self-referential reduction.

full rationale

The paper's central claims rest on applying spherical harmonic decomposition together with one-dimensional integral inequalities to the weighted curl-free and second-order settings. These techniques are presented as established tools whose validity in the weighted case is asserted without reduction to the target constants by construction. Self-citations to prior works by the same authors and others serve only to position the new results as complements or extensions; they are not invoked as uniqueness theorems or load-bearing premises that force the constants. No equations or steps in the provided text equate a derived sharp constant to a fitted parameter or to the input data itself. The derivation chain therefore remains independent of the outputs it produces.

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

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or audited.

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Pith. "Pith review of The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates." pith.science (2026). https://pith.science/paper/CGAIEPVJ

@misc{pith2026260625692,
  author       = {Pith},
  title        = {Pith review of: The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGAIEPVJ}},
  note         = {Machine review of arXiv:2606.25692}
}
abstract

In this paper, we study the weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives. Firstly, we prove a family of the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities that complements the results in [{\it C. Cazacu, J. Flynn and N. Lam, Calc. Var. Partial Differential Equations 62 (2023), no. 4, Paper No. 118, 26 pp.}] and [{\it A. T. Duong and V. H. Nguyen, On the sharp second order Caffarelli-Kohn-Nirenberg inequality. Ann. Fenn. Math., 50(1):275--286, 2025}]. Secondly, we establish a stability version of the sharp weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields proved by Cazacu, Flynn and Lam. Finally, we prove a stability estimate for the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities established in this paper. Our approach is based on the spherical harmonic decomposition method, the one dimensional integral inequalities and their improvements.

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