In the book by [Bensoussan and Lions](https://www.amazon.co.uk/Impulse-control-quasi-variational-inequalities-Bensoussan/dp/2040155775), they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that there are classical regularity results based on these spaces.


For example, for each $p\in [1,\infty)$, the weighted $L^p\_\mu(\mathbb{R}^d)$ space on $\mathbb{R}^d$ is defined to be the set of Lebesgue measurable functions such that $f\omega\_\mu(x)\in L^p(\mathbb{R}^n)$, i.e.,
$$\|f\|\_{L^p\_\mu}=\int\_{\mathbb{R}^d}|f|^p\omega^p\_\mu(x)\,dx< \infty,$$ where $\omega\_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for $\mu>0$, and the weighted sobolev space $W^{1,p}\_\mu(\mathbb{R}^d)$ is defined to be the space of functions such that $u\omega\_\mu\in L^p(\mathbb{R}^n)$ and $\partial\_{x\_i} u\omega\_\mu\in L^p(\mathbb{R}^n)$,
where $\partial\_{x\_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}\_\mu(\mathbb{R}^d)$ such that $\partial\_{x\_ix\_j}u\omega\_\mu\in L^p$ for all $i,j$. 


I am interested in a reference on the embedding properties between spaces of different orders. For example, it is pointed out (without a proof) in the book that the injection 
$$
W^{2,p}\_\mu\hookrightarrow W^{1,p}\_\nu \tag{1}$$
 with $\nu>\mu$ is compact. Could you briefly outline or provide a reference for a proof of this statement? Does it follow from the results for the classical sobolev space? 




---


Note that Corollary 3.3 in [Hooton's paper](https://core.ac.uk/download/pdf/82623799.pdf) implies the injection $W^{2,p}\_\mu\hookrightarrow W^{1,p}\_\mu \tag{2}$ 
is not compact.



Embedding of weighted sobolev space with exponential weights
Expanding on my comment, let $u\_i\in W^{2,p}\_\mu(\mathbb{R}^d)$ be a bounded sequence, that is, $$
|u\_i|\_{W^{2,p}\_\mu(\mathbb{R}^d)}\leq C
$$
and consider the balls $B\_k(0)$, $k\in\mathbb{N}$. Let $W^{2,p}(B\_k(0))$ be the usual Sobolev space. It is easy to see that 
$$
|u\_i|\_{W^{2,p}({B\_k(0)})}\leq C\exp(\mu\sqrt{1+k^2})=:C\_k.
$$
By the Rellich-Kochandrov Theorem we may find a subsequence (again labeled $u\_i$) such that $u\_i\to v\_1$ in $W^{1,p}(B\_1(0))$ and pointwise almost everywhere. We may then choose another subsequence such that $u\_i\to v\_2$ to $W^{1,p}(B\_2(0))$. By pointwise convergence it follows that $v\_1=v\_2$ almost everywhere in $B\_1(0)$. Continuing this procedure and taking a diagonal subsequence we find a well-defined function $v$ defined by $v(x)=v\_k(x)$ where $k$ is chosen such that $x\in B\_k(0)$ such that $u\_i\to v$ in $W^{1,p}(B\_k(0))$ for every $k\in\mathbb{N}$ and pointwise almost everywhere. We may also assume that all partial derivatives converge pointwise almost everywhere. By point-wise convergence and Fatou's lemma it follows that $v\in W^{1,p}\_\mu(\mathbb{R}^d)$. Now let $\nu>\mu$ and $\epsilon>0$. As $v\in W^{1,p}\_\mu(\mathbb{R}^d)$ we may choose $K\in\mathbb
{N}$ such that 
$$
\bigg(\int\_{\mathbb{R}^d\setminus B\_K(0)}\omega\_\nu^p(|v|^p+|\nabla v|^p)\bigg)^{1/p}\leq \bigg(\int\_{\mathbb{R}^d\setminus B\_K(0)}\omega\_\mu^p(|v|^p+|\nabla v|^p)\bigg)^{1/p}\leq \frac{\epsilon}{3}.
$$
On the other hand,
$$
\bigg(\int\_{\mathbb{R}^d\setminus B\_K(0)}\omega\_\nu^p(|u\_i|^p+|\nabla u\_i|^p)\bigg)^{1/p}\leq \exp((\nu-\mu)\sqrt{1+K^2})\bigg(\int\_{\mathbb{R}^d\setminus B\_K(0)}\omega\_\mu^p(|u\_i|^p+|\nabla u\_i|^p)\bigg)^{1/p}\leq \frac{\epsilon}{3},
$$
provided $K$ is large enough such that $\exp((\nu-\mu)\sqrt{1+K^2})C\leq \frac{\epsilon}{3}$ (here we use $\nu>\mu$). Finally, as $u\_i\to v$ in $W^{1,p}(B\_K(0))$ it follows that
$$
\bigg(\int\_{B\_K(0)} |u\_i-v|^p+|\nabla u\_i-\nabla v|^p\bigg)\leq \frac{\epsilon}{3} 
$$
for all $i\geq I$, where $I$ depends on $\epsilon$ and $K$. Consequently,
$$
|u\_i-v|\_{W\_\nu^{1,p}(\mathbb{R}^d}\leq 3\frac{\epsilon}{3}=\epsilon.
$$


