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?
Sobolev Embedding Theorems in Dimension One
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real-analysis
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pde
sobolev-spaces
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0I 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 $H^k (U)$.
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0@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