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Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincar\'e Inequalities: a complete characterization of sharp $L^2$ stability and $L^p$ extensions

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Pith's one-line read Weighted Gaussian Poincaré inequalities with singular weights yield sharp stability for the full range of Caffarelli-Kohn-Nirenberg inequalities.

desk verdict The paper claims the first full sharp L2 stability for CKN inequalities over all parameters via a new Laguerre-Kelvin method, but the details need checking. read the letter →

arxiv 2606.08939 v1 pith:VBSEMKO2 submitted 2026-06-08 math.AP math.FA

Anh Xuan Do , Nguyen Lam , Guozhen Lu , Van Hoang Nguyen This is my paper · ORCID
classification math.APmath.FA
keywords Caffarelli-Kohn-NirenberginequalitiesweightedGaussianPoincarésharpstabilityestimatesLaguerrepolynomialexpansionsKelvin-typetransformsphericalharmonicdecompositionssingularweights
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 constructs a new family of weighted Gaussian L²-Poincaré inequalities that admit explicit sharp constants, optimizers, and gradient stability estimates even when the weights are singular. These inequalities extend the classical Gaussian Poincaré inequality and serve as the vehicle for a complete characterization of stability in the L²-Caffarelli-Kohn-Nirenberg inequalities across every admissible parameter tuple, including the stability of the stability statements themselves. The same framework produces weighted L^p-Poincaré inequalities for all p greater than 1 and stability estimates for the L^p-CKN inequalities when p is at least 2, in every regime where sharp constants are already known. Earlier results had covered only isolated special cases of the parameter space.

What carries the argument

The method of generalized Laguerre polynomial expansions combined with spherical harmonic decompositions and a Kelvin-type transform, which produces the sharp constants, optimizers, and stability estimates for the singular weights in every admissible parameter regime.

What would settle it

A concrete counterexample consisting of a specific admissible parameter tuple together with a function for which the derived stability constant is strictly larger than the one claimed, or for which no optimizer attains the constant.

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Extended reading notes

Core claim

The authors introduce a new family of weighted Gaussian L²-Poincaré-type inequalities with explicit sharp constants, optimizers, and corresponding sharp L²-gradient stability estimates. This family substantially extends the classical Gaussian Poincaré inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincaré inequalities do not apply. To overcome this difficulty, they develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, they completely characterize the stability of the L²-Caffarelli-Kohn-Nirenberg inequalities by establishing sharp

Load-bearing premise

The combination of generalized Laguerre polynomial expansions, spherical harmonic decompositions, and Kelvin-type transform is sufficient to produce sharp constants, optimizers, and stability estimates for the singular weights and for every admissible parameter tuple in the CKN family.

Editorial extensions

If this is right

  • Sharp L²-gradient stability estimates hold for the CKN inequalities in every parameter regime.
  • The stability statements themselves satisfy their own stability estimates.
  • Weighted L^p-Poincaré inequalities exist for every p greater than 1.
  • Stability estimates for the L^p-CKN inequalities hold for all p at least 2 whenever sharp constants and optimizers are known.

Reading between the lines

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

  • The same expansion technique may apply to other families of inequalities with radial singular weights that currently lack stability results.
  • Explicit knowledge of the optimizers could be used to test numerical approximations of the inequalities in high-dimensional or non-radial settings.
  • The full-parameter stability may allow quantitative control of the deficit in related Sobolev-type embeddings that rely on CKN as an intermediate step.
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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

0 major / 2 minor

Summary. The paper introduces a new family of weighted Gaussian L²-Poincaré-type inequalities with explicit sharp constants, optimizers, and sharp L²-gradient stability estimates. The approach relies on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform to handle singular weights. As an application, the work completely characterizes the stability of the L²-Caffarelli-Kohn-Nirenberg inequalities throughout the full parameter range (previously limited to special cases) and establishes weighted L^p-Poincaré inequalities for all p>1 together with stability estimates for the L^p-CKN inequalities when p≥2 in the regime where sharp constants and optimizers are known.

Significance. If the central claims hold, the manuscript delivers a complete characterization of sharp L² stability for the CKN family across the entire admissible parameter regime, a clear advance over prior special-case results. The explicit sharp constants, optimizers, and the stability-of-stability estimates constitute concrete strengths. The new method tailored to singular weights via orthogonal expansions and the Kelvin transform is a technical contribution that directly addresses a gap in the literature on weighted Poincaré inequalities.

minor comments (2)
  1. The phrase 'stability of the stability inequality results' in the abstract is concise but may benefit from a brief parenthetical clarification or forward reference to the relevant section when first introduced in the introduction.
  2. Consider adding a short table or diagram in the introduction that summarizes the admissible parameter ranges for the L² and L^p results (including the previously known special cases) to help readers quickly locate the extension achieved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper derives sharp constants, optimizers, and stability estimates for weighted Gaussian Poincaré inequalities and CKN inequalities via a new method of generalized Laguerre expansions, spherical harmonic decompositions, and Kelvin-type transform. This approach is independent of fitted parameters or self-referential definitions, extends prior special cases without reducing to them by construction, and contains no load-bearing self-citations or ansatz smuggling. The central claims rest on explicit constructions rather than renaming or circular input-output equivalence.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on the assumption that the newly introduced method (Laguerre expansions plus Kelvin transform) works for singular weights without extra restrictions; background facts about orthogonal polynomials and harmonic analysis are treated as standard.

assumptions (2)
  • standard math Generalized Laguerre polynomials and spherical harmonics form a complete orthogonal system suitable for the weighted spaces under consideration
    Invoked to justify the decomposition step in the new method.
  • domain assumption The Kelvin-type transform preserves the form of the weighted inequalities and maps optimizers to optimizers
    Required for the method to produce sharp constants across the singular regime.

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Cite this review

Pith. "Pith review of Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincar\'e Inequalities: a complete characterization of sharp $L^2$ stability and $L^p$ extensions." pith.science (2026). https://pith.science/paper/VBSEMKO2

@misc{pith2026260608939,
  author       = {Pith},
  title        = {Pith review of: Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincar\'e Inequalities: a complete characterization of sharp $L^2$ stability and $L^p$ extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBSEMKO2}},
  note         = {Machine review of arXiv:2606.08939}
}
abstract

We introduce a new family of weighted Gaussian $L^2$-Poincar\'e-type inequalities with explicit sharp constants, optimizers, and corresponding sharp $L^2$-gradient stability estimates. This family substantially extends the classical Gaussian Poincar\'e inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincar\'e inequalities do not apply. To overcome this difficulty, we develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, we completely characterize the stability of the $L^2$-Caffarelli--Kohn--Nirenberg (CKN) inequalities by establishing sharp stability estimates, together with the stability of the stability inequality results, throughout the entire parameter range. Previous results were available only in a few special cases. We further establish weighted $L^p$-Poincar\'e inequalities for all $p>1$, and derive stability estimates for the $L^p$-CKN inequalities for $p\geq 2$ throughout the full parameter regime in which sharp constants and optimizers are known. In contrast, earlier $L^p$ results were restricted to highly limited parameter ranges.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sharp anisotropic $L^2$-Caffarelli-Kohn-Nirenberg inequalities associated with the Minkowski functional

    math.AP 2026-07 conditional novelty 6.0 of 10

    Sharp anisotropic L2-CKN inequalities and Heisenberg uncertainty principles hold for the radial derivative associated with any smooth strictly convex body, with explicit extremals and constants matching the Euclidean theory.

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

    math.AP 2026-06 unverdicted novelty 5.0 of 10

    Establishes sharp weighted second-order L²-CKN inequalities and stability versions for curl-free fields via spherical harmonics and 1D inequalities.

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