# compact embedding for $V=\{ u \in W^{1,2}(\mathbb{R}) : \int_{\mathbb{R}}{u²(x)x^2dx \lt \infty}\}$

Let $$V=\{ u \in W^{1,2}(\mathbb{R}) : \int_{\mathbb{R}}{\vert u(x)\vert^2 x^2dx }\lt \infty\}$$ with the scalar product $$\langle u,v\rangle = \int_{\mathbb{R}}{ u(x)v(x) (1+x^2)dx}+\int_{\mathbb{R}}{ u'(x)v'(x)dx}$$

Let $$B_1(0)^V$$ denote the unit sphere with respect to the above norm and $$I_N =[-N,N] \subset \mathbb{R}$$

For $$u \in B_1(0)^V$$ . Show that for every $$\epsilon$$ >0, there exists an $$N>0$$ such that $$\Vert u \Vert_{L^2(\mathbb{R^n} \setminus I_N)}\le \epsilon$$ and that the embedding $$V \hookrightarrow L^2(\mathbb{R})$$ is compact.

I am already stuck on the first part of this question. Doesn't $$u \in V$$ imply that $$u$$ vanishes for $$x \rightarrow \infty$$ ? How can I use this to prove the first part of the question. Also on a side note, for the second part I was given the hint that for compact intervals $$I$$ the embedding $$W^{1,2}(I) \hookrightarrow L^2(I)$$ is compact.

Would appreciate any help on this question.

• For the second part I don't know, but for the first one, you have \begin{align} \|u\|_{L^2(\mathbb R \backslash I_N)} &= \left(\int_{\mathbb R \backslash I_N} |u^2(x)| x^2 x^{-2}\right)^{1/2}\\ &\le \|x ~u \|_{L^2(\mathbb R \backslash I_N)} \|1/x\|_{L^2(\mathbb R \backslash I_N)}\\ &= \|1/x\|_{L^2(\mathbb R \backslash I_N)} \to 0 \end{align} for $N \to \infty$. Jan 8, 2022 at 19:41

For the first, note that $$\| u\|_{L^2(\mathbb{R}\setminus I_N)}^2 = \int_{\mathbb{R}\setminus I_N} |u(x)|^2\, dx \leq \int_{\mathbb{R}\setminus I_N} |u(x)|^2 \dfrac{x^2}{N^2}\, dx \leq N^{-2}\| u\|_{V}^2.$$