I want to understand the proof of " For s>0 the space $C_0^{\infty}(\mathbb{R}^n)$ of functions with compact support is dense in $W^{s,p}(\mathbb{R}^n)$"
the proof begin like this : For N=1: let $u\in W^{s,p}(\mathbb{R})$ and $\varphi\in D(\mathbb{R})$ such that $\varphi(x)=\begin{cases}1, |x|<1\\ 0, |x|\geq 2\end{cases}, 0\leq \varphi\leq1$. Let $u_n=\varphi(x/n)u(x)$. it is clear that $u_n$ has a compact support with value in $W^{s,p}$ and $u_n\to u$ in $L^p$. now we prove that $$v_n(x,y)=((u_n-u)(x)-(u_n-u)(y))|x-y|^{-s-\frac1p}\to 0\,\text{in} \, L^p(\mathbb{R}^2)$$
For this we have to prove that $$I_n=\int_0^n dx\left(\int_n^{+\infty}|v_n|^p(x,y) dy\right),J_n=\int_n^{+\infty} dx\left(\int_n^{+\infty}|v_n|^p(x,y) dy\right)$$ tends to 0 "
My question is how they found $I_n$ and $J_n$ please.
This is because of the support assumption on $\phi$. Thus $v_n$ is only non-zero when $|x|>n$ or when $|y|>n$. Due to the symmetry in $|v_n(x,y)|=|v_n(y,x)|$, this reduces to two cases (instead of three), either both $|x|>n$ and $|y|>n$ (this is $J_n$), or only one of them $>n$ and the other $\le n$ (this is $I_n$).