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We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define $$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \right\|}_p} = {{\left( {\int_0^1 {{x^\gamma }{{\left| {v\left( x \right)} \right|}^p}{\text{d}}x} } \right)}^{\frac{1}{p}}} < \infty } \right\},$$

$$W_\gamma ^{1,2}\left( {0,1} \right) = \left\{ {v \in L_\gamma ^2\left( {0,1} \right):{v_x} \in L_\gamma ^2\left( {0,1} \right)} \right\},$$ with $${\left\| v \right\|_{W_\gamma ^{1,2}}} = \sqrt {\left\| v \right\|_2^2 + \left\| {{v_x}} \right\|_2^2} .$$ With some elementary inequalities, I can prove that $W_\gamma ^{1,2}\left( {0,1} \right) \hookrightarrow L_\gamma ^p\left( {0,1} \right)$ is continuous with some $p \in \left[ {1,{\gamma ^*}} \right)$. But I don't have any idea to prove the compact embedding from $W_\gamma ^{1,2}\left( {0,1} \right)$ to $L_\gamma ^p\left( {0,1} \right)$. How can we do it?

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    $\begingroup$ If you have continuity and just need to add compactness you could try something like using compactness on each $ [\epsilon,1]$ by usual results and then use diagonal argument. This would give compactness unless stuff concentrates and zero and so maybe you just need to rule that out... $\endgroup$
    – Math604
    Commented Jan 3, 2021 at 8:30
  • $\begingroup$ another comment... might we worth looking at a scalilng $ \phi(\lambda x)$ and sending $ \lambda \rightarrow \infty$ before doing anything...to get an idea of what the result should be. $\endgroup$
    – Math604
    Commented Jan 3, 2021 at 8:46
  • $\begingroup$ Thank you for your help. We consider a bounded sequence $\left\{ {{u_n}} \right\} \subset W_\gamma ^{1,2}$. We have ${H^1}\left( {\varepsilon ,1} \right) \hookrightarrow \hookrightarrow C\left( {\left[ {\varepsilon ,1} \right]} \right)$. If we use diagonal argument, we can find $u \in C\left( {\left( {0,1} \right]} \right)$ such that ${\left\| {{u_{{n_k}}} - u} \right\|_{C\left( {\left[ {\varepsilon ,1} \right]} \right)}} \to 0$. But I cant prove that $u \in L_\gamma ^p\left( {0,1} \right)$ and ${\left\| {{u_{{n_k}}} - u} \right\|_{L_\gamma ^p}} \to 0$. Can you help me? $\endgroup$
    – Trần Quang Minh
    Commented Jan 3, 2021 at 14:55
  • $\begingroup$ whats $\gamma^*$? $\endgroup$
    – Math604
    Commented Jan 5, 2021 at 2:53
  • $\begingroup$ We have $W_\gamma ^{1,2}\left( {0,1} \right) \hookrightarrow L_\gamma ^p\left( {0,1} \right)$ is continuous provided by $p \in \left( {2,{\gamma _*}} \right)$. $\endgroup$
    – Trần Quang Minh
    Commented Jan 5, 2021 at 13:13

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