I have a question concerning Sobolev's embedding.

Let (for simplicity) $\Omega=\left( 0,1\right) $. Then it is well known by Rellich's theorem that $H^{1}\left( \Omega\right) $ is compactly embedded in $L^{2}\left( \Omega\right) $.

The situation becomes unclear to me, if I consider a weighted $H^{1}\left( \Omega\right) $ space by introducing the norm $\left\Vert u\right\Vert _{1,w}:=\left\Vert u\right\Vert _{L^{2}\left( \Omega\right) }+\left\Vert xu^{\prime}\right\Vert _{L^{2}\left( \Omega\right) }$ and letting $H_{w}^{1}\left( \Omega\right) $ be the closure of smooth and compactly support functions in $\Omega$.

Question: Is $H_{w}^{1}\left( \Omega\right) $ compactly embedded in $L^{2}\left( \Omega\right) $ or is the embedding only continuous?


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