Compact embedding of $W^{k,p}(\Omega)$ into $W^{k-1,p*}(\Omega)$, where $p^*$ is the Sobolev conjugate

by vaoy   Last Updated June 22, 2018 07:20 AM

How do I see that the embedding of $$W^{k,p}(\Omega) \hookrightarrow W^{k-1,p^*}(\Omega)$$ for $1 \leq p < \infty$, the Sobolev conjugate $\frac{1}{p^*}=\frac{1}{p} - \frac{1}{n}$, and $\Omega$ open and bounded in $\mathbb{R}^n$ is a compact operator? I think the embedding is just the identity map right? But I do not see how it is a compact operator, i.e. that every bounded sequence in $W^{k,p}(\Omega)$ contains a converging sequence in $W^{k-1,p^*}(\Omega)$. Does anyone know how to prove this?

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