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arxiv: 2605.03668 · v1 · submitted 2026-05-05 · 🧮 math.AP

Quantitative stability for the Trudinger-Moser inequality

Jo\~ao Henrique Andrade , Jos\'e Francisco de Oliveira , Jo\~ao Marcos do \`O , Abiel Costa Macedo , Jesse Ratzkin This is my paper

Pith reviewed 2026-05-07 15:09 UTC · model grok-4.3

classification 🧮 math.AP
keywords Trudinger-Moser inequalityquantitative stabilitydeficitspectral gapoptimizerslinearized operatorstability estimates
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The pith

The deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers if the growth rate is small enough or the domain is a disk.

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

The paper proves quantitative stability estimates for the Trudinger-Moser inequality on smooth bounded domains in the plane. It shows that a small deficit implies the function is close to an optimizer, with the distance controlled quadratically by the deficit. This holds when the exponential growth parameter is below a certain threshold or when the domain is a round disk, and the disk case extends to the critical growth rate. The proof relies on a new spectral gap established for the linearized operator at the optimizers. These estimates provide a way to measure how sharply the inequality is attained and may aid in analyzing related partial differential equations.

Core claim

We establish that the deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers on smooth bounded domains if either the exponential rate is sufficiently small or the domain is a round disk, with the latter valid even in the critical case. Both proofs use a new spectral gap for the linearized operator around the optimizers, which is proved and may be of independent interest. The same stability holds in the nondegenerate case, which occurs generically.

What carries the argument

New spectral gap for the linearized operator around the optimizers of the Trudinger-Moser functional, which yields the quadratic stability estimate.

Load-bearing premise

The new spectral gap for the linearized operator around the optimizers holds and yields quadratic control under the conditions of domain smoothness and either small rate or disk geometry.

What would settle it

Finding a smooth bounded domain that is not a disk and a value of the growth rate where the spectral gap fails to be positive, leading to a function with small deficit but large distance to optimizers.

read the original abstract

We establich quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane. More specifically, we prove that the deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers if either (i) the exponential rate of growth is sufficiently small or (ii) the domain is a round disk. The latter estimate remains valid even in the critical case. Both proofs rely on a new spectral gap that we prove, which may be of independent interest. Additionally we show that the same stability estimate holds in the nondegenerate case, and that this occurs generically.

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 paper establishes quantitative stability estimates for the Trudinger-Moser inequality on smooth bounded domains in R^2. It proves that the deficit in the inequality quadratically controls the distance to the set of optimizers when either the exponential rate of growth is sufficiently small or the domain is a round disk (with the latter holding even in the critical case). Both results rely on a newly proved spectral gap for the linearized operator around the optimizers. The paper also shows that the stability estimate holds in the nondegenerate case, which occurs generically.

Significance. If the spectral gap is established as claimed, the results provide a meaningful quantitative strengthening of the Trudinger-Moser inequality, which is central to 2D Sobolev embeddings and has applications in geometric analysis and nonlinear PDEs. The spectral gap itself may be of independent interest for studying stability in related variational problems. The generic nondegeneracy result broadens the applicability of the stability estimates.

minor comments (2)
  1. [Abstract] Abstract, first sentence: 'establich' is a typo and should read 'establish'.
  2. The main theorems would benefit from a brief forward reference to the precise statement of the spectral gap (e.g., Theorem X.Y) when the stability estimates are introduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript on quantitative stability for the Trudinger-Moser inequality. The recommendation for minor revision is noted, and we will incorporate any necessary adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; stability derived from independently proved spectral gap

full rationale

The paper's central derivation establishes a new spectral gap for the linearized operator around Trudinger-Moser optimizers and then uses this gap to obtain quadratic deficit control for the stability estimates. This gap is proved directly in the manuscript under the stated hypotheses (smooth bounded domains, small exponential rate or disk geometry, including critical case) and is presented as potentially of independent interest. The nondegenerate case is handled separately via a generic argument. No load-bearing step reduces by construction to a fitted parameter, self-referential definition, or prior self-citation chain; the argument chain is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from the authors' previous work as the sole justification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard Sobolev-space theory plus a newly established spectral gap; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard embedding and compactness properties of Sobolev spaces on smooth bounded domains in R^2
    Implicitly used to set up the Trudinger-Moser functional and its optimizers.
  • ad hoc to paper Existence of a spectral gap for the second variation operator at the optimizers that produces quadratic control
    This is the key new ingredient proved in the paper and invoked for both regimes.

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