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Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence

Jean Dolbeault, Maria J. Esteban, Alessio Figalli, Rupert L. Frank, Michael Loss

TL;DR

The work addresses sharp quantitative stability for the Sobolev inequality with explicit dimension-dependent constants and demonstrates a dimension-free stability result for the Gaussian log-Sobolev inequality via a large-$d$ limit. The authors combine competing conformal symmetries, a continuous Steiner rearrangement flow, and refined local analysis near Aubin–Talenti optimizers to obtain a constructive local-to-global stability framework. They prove a bound of the form $\\delta_{Sob}(f) \ge {\\beta}/d \, dist(\\nabla f, \\nabla\\mathcal{M})^2$, with $\\beta$ independent of $d$, and show that this sharp-ness persists in the large-d limit, yielding a dimension-free stability constant for the Gaussian log-Sobolev inequality. The large-dimensional analysis also yields a precise passage to the Gaussian setting, linking Sobolev and log-Sobolev theories and providing explicit constants for high-dimensional regimes. Overall, the paper delivers a fully constructive, dimension-sharp stability theory for Sobolev inequalities and a robust bridge to log-Sobolev stability in the Gaussian context.

Abstract

We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions.

Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence

TL;DR

The work addresses sharp quantitative stability for the Sobolev inequality with explicit dimension-dependent constants and demonstrates a dimension-free stability result for the Gaussian log-Sobolev inequality via a large- limit. The authors combine competing conformal symmetries, a continuous Steiner rearrangement flow, and refined local analysis near Aubin–Talenti optimizers to obtain a constructive local-to-global stability framework. They prove a bound of the form , with independent of , and show that this sharp-ness persists in the large-d limit, yielding a dimension-free stability constant for the Gaussian log-Sobolev inequality. The large-dimensional analysis also yields a precise passage to the Gaussian setting, linking Sobolev and log-Sobolev theories and providing explicit constants for high-dimensional regimes. Overall, the paper delivers a fully constructive, dimension-sharp stability theory for Sobolev inequalities and a robust bridge to log-Sobolev stability in the Gaussian context.

Abstract

We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions.
Paper Structure (21 sections, 32 theorems, 288 equations)

This paper contains 21 sections, 32 theorems, 288 equations.

Table of Contents

  1. Introduction and main results
  2. Local stability for nonnegative functions
  3. The Sobolev inequality on the sphere
  4. A stability result for functions close to the manifold of optimizers
  5. The spectral gap inequality
  6. Warm-up: A bound with suboptimal dimension dependence
  7. Cutting r into pieces
  8. A detailed estimate of the deficit
  9. Bound on I1
  10. Bound on I3
  11. Bound on I2
  12. Proof of Theorem \ref{['unifboundclose']}
  13. From a local to a global stability result
  14. Nonnegative functions away from the manifold of optimizers
  15. Competing symmetries
  16. ...and 6 more sections

Key Result

Theorem 1.1

There is an explicit constant $\beta>0$ such that, for all $d\ge3$ and all $f\in\dot{\mathrm H}^1({\mathord{\mathbb R}}^d)$,

Theorems & Definitions (61)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 51 more