In the paper <The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1.> by P. L. Lions, I am confused with one inequality (Page 125, Line 8 from below.) as the following. \begin{align} & \left|\left\{\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\bar{u}_{n}(x)\bar{u}_{n}(y)f(x-y)\mathrm{d}x\mathrm{d}y\right\}^{1/2}-\left\{\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}u(x)u(y)f(x-y)\mathrm{d}x\mathrm{d}y\right\}^{1/2}\right|\\ \leq & \left|\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}(\bar{u}_{n}(x)-u(x))(\bar{u}_{n}(y)-u(y))f(x-y)\mathrm{d}x\mathrm{d}y\right|^{1/2} \end{align} where $\bar{u}_{n}$ converges weakly in $L^{\alpha}(\mathbb{R^{N}})$ to u for $1\leq \alpha \leq \frac{p+1}{p}=q$ and $u\in L^{1}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})$. $f\in M^{p}(\mathbb{R}^{N})$ [weak $L^{p}$ space] and $f\geq 0 \ a.e.$
After excluding the two cases: 1.Dichotomy does not occur. 2. Vanishing does not occur. Then we need to verify conclusion, that is "Compactness".
(Compactness):There exists $y_{k}\in \mathbb{R}^{N}$ such that $(\rho_{n_{k}})_{k\geq 1}$ is tight, i.e. $$ \forall \varepsilon>0, there \ exists \ R<\infty, such \ that \ \int_{y_{k}+B_{\mathbb{R}}}\rho_{n_{k}}\mathrm{d}x\geq \lambda-\varepsilon. $$
Here, I add a remark, that $\int_{\mathbb{R}^{N}}\bar{u}_{n}\mathrm{d}x=\int_{\mathbb{R}^{N}}u\mathrm{d}x=\lambda$.
And the inequality that I confused with contributes to the Proof of the "compactness". I want to prove the above inequality rigorously, but I couldn't find the right way. Thank you in advance for your ideas！