What exactly is the content of Sobolev Embedding Theorems (compact for Sobolev spaces and Hölder spaces) when we're looking at functions on the real line?
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The theorem tells you that if $U$ is a bounded open subset of $\mathbb R$ and $k > l + d/2$ then the inclusion $C^\infty (U) \hookrightarrow C^l(U)$ can be continuously extended to $H^k(U) \hookrightarrow C^l(U)$ where $H^k(U)$ is your Sobolev space.
This is the Sobolev embedding theorem for $\mathbb R^d$, so in your question, $d=1$. $C^l(U)$ denotes the set of all continuous functions $f: U \to \mathbb R$ (or $\mathbb C$) such that $f$ has $l$ continuous derivatives.
As a consequence, if $T: C^l \hookrightarrow X$ is a linear operator, you may apply it to $H^k (U)$.