I'm working on the following assignment:
Define the weighted $L^2$-space $L^2_s$ on $\mathbb{R}^n$ to be the completion of $C_{\text{comp}}^{\infty}(\mathbb{R}^n)$ with the norm
$$
\|f\|_{L^2_s}=\left(\int_{\mathbb{R}^n}(1+|x|^2)^s|f(x)|^2d^nx\right)^{1/2}.
$$
Show that $H^1\subset L^2_s$ is a compact embedding when $s<0$.
Here "compact embedding" means that every uniformly bounded sequence in $H^1$ has a subsequence which is Cauchy in $L^2_s$. But since $L^2_s$ is just $L^2(\mathbb{R}^n,\mu_s)$ with the measure $d\mu_s=(1+x^2)^sd\mathcal{L}^n$ (here $\mathcal{L}^n$ is Lebesgue measure), $L^2_s$ is complete and this is the same as saying every uniformly bounded sequence in $H^1$ has a convergent subsequence in $L^2_s$.
Thoughts
I've spent the last 2-3 days thinking about this problem, but to no avail. It's driving me insane. Here are a couple of my thoughts/observations:
- For small enough $n$ and $|s|$ large enough, $L^2_s=L^2(\mathbb{R}^n,\mu_s)$ is a finite measure space since $s<0$.
- The Fourier transform is a unitary isomorphism from $L^2_s=L^2(\mathbb{R}^n,\mu_s)$ to $H^s$.
- $\|f\|_{L^2_s}\leq\|f\|_{L^2}\leq\|f\|_{H^1}$, so given a uniformly bounded subsequence $\{f_k\}\subset H^1$, $\big\{\|f_k\|_{L^2_s}\big\}$ is uniformly bounded and hence we can find a subsequence $\{f_{k_j}\}$ such that $\big\{\|f_{k_j}\|_{L^2_s}\big\}_{j\in\mathbb{N}}$ converges, but the problem is this doesn't necessarily mean that $\{f_{k_j}\}$ converges in $L^2_s$
- I wanted to use the Rellich compactness theorem, but this only holds when there exists a compact $K\subset\mathbb{R}^n$ such that $\text{supp }f_k\subset K$ for all $k$. I thought using the theorem on closed balls centered at zero whose radii go to infinity and then diagonalizing, but I don't believe this should work.
- For simplicity, I was thinking about the case where $n=1$. In this case I have the Sobolev embedding theorem which shows that $H^1(\mathbb{R}^n)\subset C_0(\mathbb{R}^n)$, where $C_0(\mathbb{R}^n)$ is the family of continuous functions on $\mathbb{R}^n$ which vanish at infinity. Morrey's inequality also shows here that the elements of $H^1$ are H$\ddot{\text{o}}$lder continuous, so a uniformly bounded sequence in $H^1$ would be equicontinuous, but I don't think necessarily bounded, so this ruins trying to use Arzela-Ascoli in any significant way.
I feel like, as with most things in my life, I'm making this more complicated than it needs to be. I feel as though a push in the right direction would hugely beneficial. Any help is greatly appreciated. Please only hints as this is an assignment for class.
Best Answer
Break the integral into $\mathbb{R}^{n}\setminus B(0,R)$ where $R$ is very large and make the integral over $\mathbb{R}^{n}\setminus B(0,R)$ as small as you want using the fact that $\frac1{(1+R^{2})^{-s}}\to 0$ as $R\to\infty$ and your sequence is bounded in $L^2$ with no weights. In $B(0,R)$ use Rellich theorem and use the fact that $\frac{1}{(1+|x|^{2})^{-s}}\leq1$.
PS: I can post the full answer once the homework has been handed in. I already typed it before I read the last part of your post (which I really appreciated. Thanks for the honesty).