# Compactness of an embedding between weighted spaces

I read somewhere that if:

• $N\geq 2$ is an integer,
• $p\in ]1,N[$, $r>N/p$,
• $m\in L^r(0,a)$ (with $a>0$) and $m>0$ a.e. in $(0,a)$,

then the weighted Sobolev space $W^{1,p^\prime} ((0,a),m^{-1/(p-1)})$, defined as the completion of $C_c^1(0,a)$ w.r.t. the norm: $$\begin{split} \| u\|_{W^{1,p^\prime}((0,a),m^{-1/(p-1)})} &:= \| u\|_{L^{p^\prime}((0,a),m^{-1/(p-1)})} +\| \dot{u}\|_{L^{p^\prime}((0,a),m^{-1/(p-1)})}\\ &= \left( \int_0^a |u|^{p^\prime}\ \frac{1}{m^{1/(p-1)}}\ \text{d} s\right)^{1/p^\prime} + \left( \int_0^a |\dot{u}|^{p^\prime}\ \frac{1}{m^{1/(p-1)}}\ \text{d} s\right)^{1/p^\prime}\; , \end{split}$$ compactly embeds into $L^{p^\prime} ((0,a), m^{-1/(p-1)})$... But I didn't succeed in doing the right computations to get the correct Sobolev inequality.

Does anyone knows how to prove $W^{1,p^\prime} ((0,a),m^{-1/(p-1)}) \hookrightarrow L^{p^\prime} ((0,a), m^{-1/(p-1)})$ compactly?

Any hint will be appreciated.

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I assume you want $m > 0$? – Willie Wong Jun 6 '12 at 15:23
@WillieWong : I forgot to mention it, but yes: $m>0$ a. e. in $(0,a)$. – Pacciu Jun 7 '12 at 15:34
@DavideGiraudo : In this context $p^\prime$ is the Hölder conjugate of $p$, i.e. $p^\prime := \frac{p}{p-1}$. I thought it was a standard notation... – Pacciu Jun 25 '12 at 20:32