I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2".
The Hardy-Littlewood-Sobolev (HLS) inequality is stated as $$ \| K*u\|_q\le C\|u\|_p $$ where $K=|x|^{-\lambda}$ and $1<p<q<\infty, 0<\lambda<n, \tfrac 1p+\tfrac \lambda n=1+\tfrac 1q$.
When proving the existence of maximizers of HLS inequality, he used his second concentration compactness principle.
(Lemma2.1) Let $u_n$ converge weakly in $L^p(\mathbb R^n)$ to $u$ and assume $|u_n|^p$ is tight. We may assume without loss of generality that $|K*u_n|^q$, $|u_n|^p$ converge weakly (or tightly) in the sense of measure to some bounded nonnegative measures $\nu, \mu$ on $\mathbb R^n$. Then we have: there exist some at most countable set (possibly empty) and two families $(x_j)_{j\in J}$ of distinct points in $\mathbb R^n$, $(\nu_j)_{j\in J}$ in $(0,\infty)$ such that: $$ \nu=|K*u|^q+\sum_{j\in J}\nu_j\delta_{x_j}$$ where $K$ is the kernel $|x|^{-\lambda}$.
To prove this lemma (lemma 2.1 in his paper), he cited his another lemma in his paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 1"
(Lemma 1.2) Let $\mu,\nu$ be two bounded nonnegative measures on $\mathbb R^n$ satisfying for some constant $C_0\ge 0$ $$\left(\int_{\mathbb R^n} |\varphi|^qd\nu\right)^{1/q}\le C_0 \left( \int_{\mathbb R^n} |\varphi|^pd\mu\right)^{1/p}~~~~\text{for any}~\varphi\in C_c^\infty(\mathbb R^n)\tag 1$$
In order to get the inequality (1). After some progress, we can assume that $u=0$, i.e., $u_n\to 0$ weakly in $L^p$.
And he showed that we only need to estimate an upper bound of (See the page 51 of his paper)
$$ v_n(x)=\left|\int_{|y|\le R}\cfrac{\varphi(y)-\varphi(x)}{|x-y|^\lambda}u_n(y)dy \right| $$
He wrote ' Denoting by $R(x,y)=(\varphi(y)-\varphi(x))|x-y|^{-\lambda}$ and observing that $R(x,y) 1_{|y\le R}\in L^r(\mathbb R^n)$ for each $x$ where $r<\frac{n}{\lambda-1}$ if $\lambda>1$, $r\le \infty$ if $\lambda \le 1$, we see that: $v_n\to 0$ a.e. on $\mathbb R^n$'.
My question:
1. It seems that we cannot get that $R(x,y)1_{|y|\le R}\in L^{p'}$ for a.e. $x\in \mathbb R^n$ if we only have $R(x,y)1_{|y|\le R} \in L^r$. Thus we can not use the assumption that $u_n\to 0$ weakly in $L^p$.
I am wondering whether I missed some other important information in his proof and note that if $R(x,y)1_{|y|\le R}$ is not in $L^{p'}$, we don't know whether $v_n$ exists.
2. It seems that we do not need the statement about '$R(x,y)1_{|y|\le R}$'. In his beginning, he got $K*u_n \to K*u$ a.e.. Since $u=0$, we have $$ \int \cfrac{u_n(y)}{|x-y|^{\lambda}}dy\to 0 ~~~a.e.~x\in \mathbb R^n $$ and hence $$ |v_n|\le 2\|\varphi\|_\infty \int \cfrac{u_n(y)}{|x-y|^{\lambda}}dy \to 0. $$ (We can take $u_n$ as nonnegative sequence since $|u_n|$ is also a maximizing sequence)
Does my way work?
Thanks in advance. Thanks for your attention and I hope you can help me with this.