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It is well known that $H^1(I)$ is compactly embedded in $C(I)$ where $I$ is bounded interval of $\mathbb{R}$, which is not correct for $I$ unbounded.

So, I search about a functional space $Y$ such that the Sobolev space $H^1(0,+\infty)$ is compactly embedded in $Y$.

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    $\begingroup$ You can take a weighted space, for example $C_w(0, \infty)$, the continuous functions on the half line such that $fw$ is bounded, endowed with the sup norm of $fw$. You have to choose $w>0$, continuous and with $\lim_{x \ to \infty}w(x)=0$. $\endgroup$
    – Giorgio Metafune
    Commented Jun 29, 2023 at 9:21
  • $\begingroup$ So, can I choose for example the weight function $w$ which is continuous, and $\lim \sqrt{x}w(x) = 0$ as $x$ goes to $\infty$ to ensure the compact embedding, and add the condition $\dfrac{1}{w}\in L^1(0,+\infty) $ ? I need to this condition. $\endgroup$
    – Isaac
    Commented Jun 29, 2023 at 18:35
  • $\begingroup$ If $w$ tends to zero at infinity, $1/w$ cannot be in $L^1$. $\endgroup$
    – Giorgio Metafune
    Commented Jun 30, 2023 at 6:04

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