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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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    It's $W^{3,1}$ of which open set?2012-04-22

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