Remainder terms, profile decomposition and sharp quantitative stability in the fractional nonlocal Sobolev-type inequality with $n>2s$

In this paper, we study the following fractional nonlocal Sobolev-type inequality \begin{equation*} C_{HLS}\bigg(\int_{\mathbb{R}^n}\big(|x|^{-\mu} \ast |u|^{p_s}\big)|u|^{p_s} dx\bigg)^{\frac{1}{p_s}}\leq\|u\|_{\dot{H}^s(\mathbb{R}^n)}^2\quad \mbox{for all}~~u\in \dot{H}^s(\mathbb{R}^n), \end{equation*} induced by the classical fractional Sobolev inequality and Hardy-Littlewood-Sobolev inequality for $s\in(0,\frac{n}{2})$, $\mu\in(0,n)$ and where $p_{s}=\frac{2n-\mu}{n-2s}\geq2$ is energy-critical exponent. The $C_{HLS}>0$ is a constant depending on the dimension $n$, parameters $s$ and $\mu$, which can be achieved by $W(x)$, and up to translation and scaling, $W(x)$ is the unique positive and radially symmetric extremal function of the nonlocal Sobolev-type inequality. It is well-known that, up to a suitable scaling, \begin{equation*} (-\Delta)^{s}u=(|x|^{-\mu}\ast |u|^{p_s})|u|^{p_s-2}u\quad \mbox{for all}~~u\in\dot{H}^s(\mathbb{R}^n), \end{equation*} is the Euler-Lagrange equation corresponding to the associated minimization problem. In this paper, we first prove the non-degeneracy of positive solutions to the critical Hartree equation for all $s\in(0,\frac{n}{2})$, $\mu\in(0,n)$ with $0<\mu\leq4s$. Furthermore, we show the existence of a gradient type remainder term and, as a corollary, derive the existence of a remainder term in the weak $L^{\frac{n}{n-2s}}$-norm for functions supported in domains of finite measure, under the condition $s\in(0,\frac{n}{2})$. Finally, we establish a Struwe-type profile decomposition and quantitative stability estimates for critical points of the above inequality in the parameter region $s\in(0,\frac{n}{2})$ with the number of bubbles $\kappa\geq1$, and for $\mu\in(0,n)$ with $0<\mu\leq4s$. In particular, we provide an example to illustrate the sharpness of our result for $n=6s$ and $\mu=4s$.