In this paper, we study the stability of the fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate: \(\begin{aligned} C_{\textrm{BE}}(n,m,\alpha )\inf _{v\in {\mathcal {M}}_{n,m,\alpha }}\left\Vert {f-v}\right\Vert _{D_\alpha ({{\mathbb {R}}}^n)}^2 \le \left\Vert {f}\right\Vert _{D_\alpha ({{\mathbb {R}}}^n)}^2 - S(n,m,\alpha ) \left\Vert {\tau _mf}\right\Vert _{L^{q}({{\mathbb {R}}}^{n-m})}^2, \end{aligned}\) where \(0\le m< n,\) \(\frac{m}{2}<\alpha <\frac{n}{2}, q=\frac{2(n-m)}{n-2\alpha }\) and \({\mathcal {M}}_{n,m,\alpha }\) denotes the manifold of extremal functions. Additionally, we find an explicit bound for the stability constant \(C_{\textrm{BE}}.\) Furthermore, we establish a compactness result ensuring the existence of minimizers for the Bianchi–Egnell type functional: \(\begin{aligned} S_{\textrm{Tr}}(f):=\frac{\left\Vert {f}\right\Vert _{D_\alpha }^2 - S(n,m,\alpha )\left\Vert {\tau _mf}\right\Vert _{L^q}^2}{\inf \limits _{v\in {\mathcal {M}}_{n,m,\alpha }} \left\Vert {f-v}\right\Vert _{D_\alpha }^2},\quad \text {for }f\in D_\alpha ({{\mathbb {R}}}^n)\backslash {\mathcal {M}}_{n,m,\alpha }. \end{aligned}\) Our stability results extend previous works on the Escobar trace inequality and fractional Sobolev inequality. As a corollary, we derive some improved trace inequalities for functions supported in general domains. Applying a dual scheme, we also obtain a sharp a priori estimate for Neumann problem on the half-space. 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 \(\begin{aligned} {\left\{ \begin{array}{ll} \Delta u= 0 & \text{ in } {{\mathbb {R}}}_+^n, \\ \frac{\partial u}{\partial t}=-|u|^{\frac{2}{n-2}}u & \text{ on } \partial {{\mathbb {R}}}_+^n. \end{array}\right. } \end{aligned}\) where \(\nu =1,n\ge 3\) or \(\nu \ge 2,n=3\) and \({\mathcal {M}}_{\textrm{E}}^\nu \) represents the manifold consisting of \(\nu \) weak-interacting Escobar bubbles. Through some refined estimates, we also give a strict upper bound for \(C_{\textrm{CP}}(n,1),\) which is \(\frac{2}{n+2}.\)
