Sharp quantitative stability for the fractional Sobolev trace inequality
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 $\frac{2}{n+2}$.