Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$

Abstract: We prove stability for the affine Sobolev inequality for exponents $p\geq 2$ with best possible norm and best possible stability exponent. We prove a corresponding results for critical points of the functional in the absence of bubbling.

• The paper demonstrates that the deficit in the affine Sobolev inequality quantitatively controls the distance to the manifold of optimizers modulo affine transformations.
• It employs advanced techniques including nonlinear expansions, concentration-compactness, and spectral gap estimates to achieve sharp, optimal exponents.
• The study extends stability theory to critical points, providing new tools for addressing affine-invariant variational problems in geometric analysis.

Stability for the Affine Sobolev Inequality at $p \ge 2$: Main Results and Methods

Introduction and Context

The affine Sobolev inequalities stand as sharp enhancements over the classical Sobolev inequalities, incorporating affine-invariant geometric averages of Sobolev gradients. Their invariance under the full general linear group yields a profound connection to affine geometry, convex geometry, and analysis. This paper by Frank, Li, and Yang (2607.06415) rigorously advances the stability theory for the affine Sobolev inequality in the $p \ge 2$ regime, establishing quantitative, sharp estimates both near minimizers and near critical points of the associated variational problems.

The results address a central question in geometric analysis: to what extent does proximity to optimality in the affine Sobolev inequality guarantee proximity to the family of all optimizers (modulo affine symmetries), and how does this extend to non-minimizing critical points (critical points of the quotient functional)? The answers involve a highly nontrivial interplay of concentration-compactness, nonlinear expansions, and spectral gap estimates in non-Hilbertian settings.

Affine Sobolev Inequality: Formulation and Symmetry

For $f \in \dot{W}^{1,p}(\mathbb{R}^n)$, the affine Sobolev energy is defined as

$$\mathcal E_p(f) := \left[ \int_{\mathbb S^{n-1}} \|\nabla_\xi f\|_p^{-n} d\xi \right]^{-1/n},$$

where $\nabla_\xi f = \xi\cdot\nabla f$. The sharp affine Sobolev inequality, for $p\in[1,n)$, reads:

$$\mathcal E_p(f) \ge S_{\rm aff} \|f\|_{p^*}, \qquad p^* = \frac{np}{n-p}.$$

The set of optimizers, denoted $\mathcal{M}_{\rm aff}$, is characterized explicitly and is invariant under all invertible affine maps; this contrasts with the strictly smaller symmetry of the classical case.

Main Theorem: Sharp Stability for $p\ge 2$

Theorem 1 [Stability near Minimizers]:

For $p\in[2,n)$, there is $p \ge 2$0 such that for all $p \ge 2$1,

$p \ge 2$2

Both the $p \ge 2$3-power and quadratic terms are shown to be optimal. This result is quantitative: the deficit in the affine Sobolev inequality controls (in the strongest possible norm and exponent) the distance to optimizers, modulo affine transformations.

Methodology Details:

• Compactness and profile decompositions adapted to the affine action;
• Nonlinear expansions of the affine energy functional using negative exponent Lebesgue spaces (handling the underlying nonlinear averaging structure);
• Control of "variance-type" terms arising from the sphere integration, a phenomenon absent in the classical setting;
• New affine-invariant spectral gap inequalities for the linearization at extremals, using spherical harmonic analysis.

Stability for Critical Points: In the Absence of Bubbling

Theorem 2 [Stability at Critical Points]:

For $p \ge 2$4, there are $p \ge 2$5 such that for any positive $p \ge 2$6 with $p \ge 2$7, we have

$p \ge 2$8

where $p \ge 2$9 is the natural affine-invariant residual (measured in the dual homogeneous Sobolev norm) of the Euler-Lagrange equation for the affine quotient. The exponent $f \in \dot{W}^{1,p}(\mathbb{R}^n)$0 is shown to be sharp, in agreement with the optimal behavior for classical Sobolev critical points established recently.

Analytical Techniques:

• Nonlinear expansions of the affine $f \in \dot{W}^{1,p}(\mathbb{R}^n)$1-Laplacian near bubbles;
• Fine asymptotics requiring detailed control over tangent directions and orthogonality to the full affine symmetry group;
• Spectral gap analysis for the linearized operator, with removal of all symmetry-induced degeneracies;
• Modulation arguments similar to the theory for critical (superlinear) elliptic PDEs but adjusted for affine invariance.

Optimality and Explicit Parameterization

The paper provides explicit parametrizations for the manifold $f \in \dot{W}^{1,p}(\mathbb{R}^n)$2 using affine symmetry generators (dilations, translations, volume-preserving linear maps), with the extremizer's canonical form given by

$f \in \dot{W}^{1,p}(\mathbb{R}^n)$3

normalized by $f \in \dot{W}^{1,p}(\mathbb{R}^n)$4.

Sharpness of all exponents and distances is established via explicit test functions (far-separated bubble perturbations) and nonlinear duality estimates.

Theoretical Impact

These results unambiguously resolve the structure of sharp stability in the affine Sobolev setting for $f \in \dot{W}^{1,p}(\mathbb{R}^n)$5, fully matching the best known results in the classical case and extending them in the direction of affine geometrization. The quadratic terms and their necessity constitute a novel phenomenon in the affine context, reflecting deeper structural rigidity due to symmetry enlargement.

The spectral gap constructions on the sphere, the use of negative exponent integration over the sphere, and the handling of all affine trace-free directions are technically subtle and push the limits of current stability techniques. The expansions and quantitative compactness analyses contribute foundational tools for a broader class of functional inequalities with large symmetry groups.

Implications and Future Perspectives

From a geometric analysis viewpoint, this work solidifies the quantitative stability theory for affine-invariant inequalities, which is germane to problems in convex geometry, geometric flows, and PDEs with symmetry. Sharpened deficit controls facilitate a finer understanding of concentration phenomena, eigenvalue problems for nonlocal fully nonlinear operators (the affine $f \in \dot{W}^{1,p}(\mathbb{R}^n)$6-Laplacian), and applications to nonlinear PDEs and isoperimetric inequalities.

As a theoretical stepping stone, these techniques are anticipated to transfer to related inequalities: fractional affine Sobolev, anisotropic, and spectral inequalities. Further, the methods are likely instrumental for quantifying stability (or instability) for minimizers and critical points in variational problems governed by large noncompact symmetry groups—a recurring motif in nonlinear analysis and mathematical physics.

Conclusion

This work establishes the definitive form of sharp, quantitative stability for the affine Sobolev inequality and its critical points in the full affine-invariant setting for $f \in \dot{W}^{1,p}(\mathbb{R}^n)$7 (2607.06415). The analysis not only achieves optimal exponents and distances but also introduces a robust technical apparatus—analytical and spectral—for affine-invariant variational problems. The implications reach into geometric and PDE analysis, with a clear path toward further developments in affine, fractional, and anisotropic settings.