Sharp quantitative stability for the fractional Sobolev trace inequality

TL;DR

The paper delivers sharp quantitative stability results for the fractional Sobolev trace inequality in both a functional and a critical-point framework, deriving an explicit best-stability constant tied to the trace constant through $C_{\mathrm{BE}}(n,m,\alpha)=C_{\mathrm{BE}}(n-m,0,\alpha-\tfrac{m}{2})$ and providing minimizer existence and bounds. It introduces a reduction principle linking trace stability to the fractional Sobolev inequality, and extends stability insights to the Escobar-type Neumann problem via Struwe-type and quantitative profile decompositions. In the critical-point setting, it establishes sharp stability estimates for multi-bubble configurations and obtains a concrete bound $C_{\mathrm{CP}}(n,1)\le\tfrac{2}{n+2}$ in the one-bubble case, with a detailed analysis of bubble interactions. The work also yields refined Sobolev embeddings with remainder terms and dual stability results, highlighting the interplay between primal and dual norms in stability analyses and advancing the understanding of stability phenomena in fractional trace inequalities.

Abstract

In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate: $$C_{\mathrm{BE}}(n,m,α)\inf_{v\in\mathcal{M}_{n,m,α}}\left\Vert f-v\right\Vert_{D_α(\mathbb{R}^n)}^2 \leq \left\Vert f\right\Vert_{D_α(\mathbb{R}^n)}^2 - S(n,m,α) \left\Vertτ_mf\right\Vert_{L^{q}(\mathbb{R}^{n-m})}^2,$$ where $0\leq m< n$, $\frac{m}{2}<α<\frac{n}{2}, q=\frac{2(n-m)}{n-2α}$ and $\mathcal{M}_{n,m,α}$ denotes the manifold of extremal functions. Additionally, We find an explicit bound for the stability constant $C_{\mathrm{BE}}$ and establish a compactness result ensuring the existence of minimizers. In the critical point setting, we investigate the validity of a sharp quantitative profile decomposition related to the Escobar trace inequality and establish a qualitative profile decomposition for the critical elliptic equation \begin{equation*} Δu= 0 \quad\text{in }\mathbb{R}_+^n,\quad\frac{\partial u}{\partial t}=-|u|^{\frac{2}{n-2}}u \quad\text{on }\partial\mathbb{R}_+^n. \end{equation*} We then derive the sharp stability estimate: $$ C_{\mathrm{CP}}(n,ν)d(u,\mathcal{M}_{\mathrm{E}}^ν)\leq \left\Vert Δu +|u|^{\frac{2}{n-2}}u\right\Vert_{H^{-1}(\mathbb{R}_+^n)}, $$ where $ν=1,n\geq 3$ or $ν\geq2,n=3$ and $\mathcal{M}_{\mathrm{E}}^ν$ represents the manifold consisting of $ν$ weak-interacting Escobar bubbles. Through some refined estimates, we also give a strict upper bound for $C_{\mathrm{CP}}(n,1)$, which is $\frac{2}{n+2}$.

Theorems & Definitions (41)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Definition 1.7: interaction of Escobar bubbles
• Theorem 1.8
• Remark 1.9
• Remark 1.10