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arxiv: 2605.03732 · v1 · submitted 2026-05-05 · 🧮 math.AP

Sharp Stability for the Affine Fractional Sobolev Inequality

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

Pith reviewed 2026-05-07 14:50 UTC · model grok-4.3

classification 🧮 math.AP
keywords affine fractional Sobolev inequalityquantitative stabilityspectral gapaffine Hessianstability constanthomogeneous Sobolev spacefractional Sobolev inequality
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The pith

The affine fractional L2-Sobolev inequality admits a sharp quantitative stability estimate whose optimal global constant is strictly smaller than the local spectral gap.

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

This paper establishes a sharp quantitative stability result for the affine fractional L2-Sobolev inequality on the homogeneous Sobolev space dot H^s(R^n) for 0 < s < 1. It identifies the kernel of the affine Hessian operator and computes the sharp local spectral gap around the equality cases previously found by Haddad-Ludwig. The central new observation is that the best constant controlling the global stability inequality is strictly smaller than the value of this local spectral gap. A reader would care because the result gives a stronger, uniform control on how far any function can be from the equality cases than local linearization alone would suggest.

Core claim

In this paper, we prove a sharp quantitative stability result for the affine fractional L^2-Sobolev inequality in dot H^s(R^n), 0<s<1, introduced by Haddad--Ludwig. In particular, we identify the kernel of the affine Hessian, determine the sharp local spectral gap, and show that the optimal global stability constant is strictly smaller than the corresponding local spectral value.

What carries the argument

The affine Hessian operator, whose kernel consists of the equality cases of the inequality and whose quadratic form yields the local spectral gap used to bound the stability deficit.

If this is right

  • The stability deficit is controlled by a constant strictly smaller than the local spectral gap value.
  • The kernel of the affine Hessian fully characterizes the equality cases.
  • Local stability estimates near equality cases are given by the sharp spectral gap.
  • Global stability holds uniformly for all functions in the space with the improved constant.

Where Pith is reading between the lines

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

  • The strict inequality between global and local constants suggests that the worst-case stability occurs away from the equality cases rather than in the linearized regime.
  • The identification of the kernel may allow direct computation of near-minimizers in related variational problems.
  • The method of comparing local spectral gap to global constant could be tested on other affine or fractional inequalities.

Load-bearing premise

The affine fractional Sobolev inequality holds with equality only for the functions identified by Haddad-Ludwig, and the functions under consideration lie in dot H^s(R^n) so that the affine Hessian is defined.

What would settle it

An explicit test function in dot H^s(R^n) whose ratio of stability deficit to squared distance to the identified kernel falls below the claimed global constant would disprove the result.

read the original abstract

In this paper, we prove a sharp quantitative stability result for the affine fractional \(L^2\)-Sobolev inequality in \(\dot H^s(\mathbb R^n)\), \(0<s<1\), introduced by Haddad--Ludwig (\emph{Math. Ann.} \textbf{388} (2024), 1091--1115). In particular, we identify the kernel of the affine Hessian, determine the sharp local spectral gap, and show that the optimal global stability constant is strictly smaller than the corresponding local spectral value.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a sharp quantitative stability result for the affine fractional L²-Sobolev inequality in the homogeneous Sobolev space Ḣ^s(ℝ^n) for 0 < s < 1. It identifies the kernel of the affine Hessian with the equality cases established by Haddad-Ludwig, computes the sharp local spectral gap via linearization, and establishes that the optimal global stability constant is strictly smaller than the corresponding local spectral value.

Significance. If the result holds, the work supplies a precise stability theory for an affine variant of the fractional Sobolev inequality, extending the equality-case analysis of Haddad-Ludwig. The explicit identification of the kernel, the computation of the sharp local gap, and the strict global-versus-local constant comparison constitute a substantive contribution to quantitative stability theory in geometric analysis and PDEs. The approach relies on standard linearization and spectral techniques applied in the natural space Ḣ^s, which is a strength of the manuscript.

minor comments (2)
  1. [Introduction] The statement of the main theorem in the introduction would benefit from an explicit display of the stability inequality, including the precise form of the constant and the distance functional employed.
  2. [Section 4] In the linearization step around equality cases, the notation for the affine Hessian operator should be introduced with a brief reminder of its definition from the earlier sections to improve readability for readers unfamiliar with the affine setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the contributions on the identification of the affine Hessian kernel, the sharp local spectral gap, and the strict comparison between global and local stability constants have been recognized as substantive.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a quantitative stability result for the affine fractional Sobolev inequality by identifying the kernel of the affine Hessian (matching the equality cases from the external Haddad-Ludwig reference), computing the sharp local spectral gap via linearization around those cases, and proving the global stability constant is strictly smaller than the local spectral value. All steps rely on the natural domain dot H^s(R^n) where the inequality and Hessian are defined, with the base inequality and equality cases taken as given from independent prior work rather than fitted or redefined internally. No self-definitional loops, fitted inputs renamed as predictions, self-citation load-bearing premises, imported uniqueness theorems, smuggled ansatzes, or renamings of known results appear in the derivation chain; the argument remains independent of the target stability constants.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard functional analysis axioms and the prior inequality definition; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math Standard properties of fractional Sobolev space dot H^s(R^n) via Fourier transform
    Defines the space and norm in the inequality.
  • domain assumption Equality case characterization from Haddad-Ludwig
    Stability is built upon this prior result.

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