Sharp quantitative stability for the fractional Sobolev trace inequality

Technical Summary: Sharp Quantitative Stability for the Fractional Sobolev Trace Inequality

Problem Setup
This paper investigates the quantitative stability of the sharp fractional Sobolev trace inequality, originally established by Einav and Loss. The inequality states that for $0 \le m < n$ and $m/2 < \alpha < n/2$, there exists a sharp constant $S(n, m, \alpha)$ such that:
$$S(n, m, \alpha) \|\tau_m f\|_{L^q(\mathbb{R}^{n-m})}^2 \le \|f\|_{D^\alpha(\mathbb{R}^n)}^2$$
where $q = \frac{2(n-m)}{n-2\alpha}$, $\tau_m$ denotes the restriction to the hyperplane $\mathbb{R}^{n-m}$, and $D^\alpha(\mathbb{R}^n)$ is the fractional Sobolev space. The extremal functions form a manifold $M_{n,m,\alpha}$.

The authors analyze the stability of this inequality in two distinct settings:

1. Functional Setting: Quantifying the distance of a function $f$ from the extremal manifold $M_{n,m,\alpha}$ based on the deficit in the inequality.
2. Critical Point Setting: Analyzing the stability of solutions to the associated Euler-Lagrange equation (the Escobar trace equation) on the half-space $\mathbb{R}^n_+$, specifically when the solution is close to a superposition of "Escobar bubbles."

Methodology
The paper employs a combination of reduction principles, spectral analysis, and profile decomposition techniques.

• Reduction Principle: In the functional setting, the authors utilize a reduction principle (Theorem 3.3) established by Einav and Loss. This principle decomposes the fractional Sobolev trace inequality into a composition of a reduction operator $\tau_m$ and a standard fractional Sobolev inequality. This allows the authors to reduce the stability problem of the trace inequality to the known stability results for the standard fractional Sobolev inequality (Bianchi-Egnell type stability).
• Dual Scheme: To derive stability estimates for the Neumann problem (the dual of the trace inequality), the authors apply a dual scheme developed by Carlen. This involves analyzing the Legendre transforms of the relevant functionals and utilizing the convexity properties of the dual functionals.
• Profile Decomposition: In the critical point setting, the authors extend Struwe's global compactness principle. They establish both qualitative and quantitative profile decompositions for sequences of functions that are "almost" solutions to the critical equation. This involves separating the solution into a sum of interacting Escobar bubbles and a remainder term.
• Spectral Gap and Conformal Mapping: To obtain sharp upper bounds for the stability constant in the single-bubble case, the authors employ a conformal mapping between the half-space $\mathbb{R}^n_+$ and the unit ball $B^n$. This transformation links the linearized operator of the problem to the Steklov eigenvalue problem on the ball. By analyzing the spectrum of the Steklov problem, they derive a refined spectral gap inequality.
• Third-Order Expansion: For the upper bound of the stability constant in the single-bubble case, the authors perform a precise third-order expansion of the dual norm, carefully handling the interaction terms and the geometry of the manifold.

Key Contributions and Results

1. Sharp Functional Stability Estimate:
   The authors establish a sharp quantitative stability estimate for the fractional Sobolev trace inequality:
   $$C \inf_{v \in M_{n,m,\alpha}} \|f - v\|_{D^\alpha(\mathbb{R}^n)}^2 \le \|f\|_{D^\alpha(\mathbb{R}^n)}^2 - S(n, m, \alpha) \|\tau_m f\|_{L^q(\mathbb{R}^{n-m})}^2$$
   They prove that the exponent 2 is sharp and provide explicit upper and lower bounds for the optimal stability constant $C_{BE}(n, m, \alpha)$. Crucially, they show that $C_{BE}(n, m, \alpha) = C_{BE}(n-m, 0, \alpha - m/2)$, effectively reducing the trace problem to the standard fractional Sobolev problem. They also prove the existence of minimizers for the stability constant when $n-m \ge 2$.

2. Improved Trace Inequalities and Neumann Stability:
   As a corollary of the gradient stability result, the authors derive improved Sobolev embeddings for functions supported in general domains (including strips) with remainder terms involving weak $L^q$ norms. Furthermore, using the dual scheme, they establish a sharp a priori stability estimate for the Neumann problem on the half-space, linking the deficit in the dual inequality to the distance from the extremal manifold of the Neumann problem.

3. Quantitative Profile Decomposition:
   In the critical point setting, the authors prove a quantitative profile decomposition theorem. They show that if a function $u$ is close to a sum of $\nu$ weakly interacting Escobar bubbles (in the $H^1$ norm), then the distance to the manifold of such bubbles is controlled linearly by the $H^{-1}$ norm of the residual $\|\Delta u + |u|^{p-1}u\|_{H^{-1}}$. This result holds for non-negative functions and extends previous works on the classical Yamabe equation to the Escobar trace setting.

4. Sharp Upper Bound for the Stability Constant:
   For the single-bubble case ($\nu=1$), the authors derive a strict upper bound for the optimal stability constant $CCP(n, 1)$:
   $$CCP(n, 1) < \frac{2}{n+2}$$
   This result is obtained by constructing a specific test function involving a third-order perturbation of the extremal bubble and utilizing the spectral gap inequality derived from the Steklov eigenvalue problem.

Significance
The paper claims to extend previous stability results on the Escobar trace inequality and the fractional Sobolev inequality. By establishing a reduction principle, the authors unify the stability analysis of the trace inequality with that of the standard fractional Sobolev inequality. The derivation of explicit bounds for the stability constants and the proof of the existence of minimizers (in the functional setting) address open questions regarding the quantitative nature of these inequalities.

In the critical point setting, the work provides the first quantitative profile decomposition for the Escobar trace equation, generalizing results known for the classical Yamabe equation. The strict upper bound $2/(n+2)$ for the stability constant in the single-bubble case is a significant refinement, improving upon generic estimates and highlighting the specific spectral properties of the linearized operator in this context. The authors note that the non-degeneracy of the solutions, which is crucial for these quantitative results, is established via the connection to the Steklov eigenvalue problem, a link that may be of independent interest.