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I want to prove that $W^{3,1}$ is continuously embedded into $L^2$. Here is my attempt. ($D^ng$ means that the n-th order partial derivatives)

$$ g \in W^{3,1}\;\;\text{iff}\;\; g \in L^1, Dg \in L^1, D^2 g \in L^1, D^3 g \in L^1.$$ $$ \text{If} \; g \in L^1, \; \text{then} \; g \in L^2. \; $$ $$ \text{Thus} \; W^{3,1} \subset L^2. $$ are these things above right? And here is my question. How can I prove that $$ \text{for any}\;g \in W^{3,1}, \;\exists C \geqslant 0\;\text{s.$\,$t.}\; \|g\|_2 \leqslant C\|g\|_{3,1} $$

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    In general, $L^1$ is not even a subset of $L^2$.2012-04-22
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    @Thomas Then how can i conclude that $W^{3,1} \subset L^2$?2012-04-22
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    You need to use the fact that the weak derivatives are integrable. Is this homework? What did you learn about these spaces?2012-04-22
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    Regarding your last line: You need $ \exists C\;\forall g\; \|g\|_2 \le C\|g\|_{3,1}$.2012-04-22
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    It's $W^{3,1}$ of which open set?2012-04-22

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