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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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    I am not exactly sure what you are asking about; but Sobolev embedding in one dimension is essentially just fundamental theorem of calculus + Hölder inequality + (optionally) interpolation of Lebesgue spaces. The statement of the theorem can be easily checked on, say, [Wikipedia](http://en.wikipedia.org/wiki/Sobolev_inequality#Sobolev_embedding_theorem), so I guess you are not just asking for the statement(s). Can you clarify what you mean by "the content"? (What do you seek in an answer?)2012-09-03

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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 $H^k (U)$.

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

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    @corni75 Yeah, I do. [Googling](http://www.google.com) for "Sobolev Embedding theorem" yields [this](http://home.ku.edu.tr/~vkalantarov/math551/SbspPDE(05).pdf), [this](http://www.icmc.usp.br/~andcarva/sobolew.pdf), [this](http://maths.swan.ac.uk/staff/vl/projects/main.pdf), [this](http://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/), [this](http://www-m12.ma.tum.de/foswiki/pub/M12/Allgemeines/SobolevR%E4ume/Kapitel5.pdf), ...2012-08-30