Sobolev Embedding Theorems in Dimension One

Question

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?

Answer 1

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 $H^k (U)$.

Here are three related threads: 1, 2, 3.