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In an exercise I am asked to prove the following statement: The embedding $$T:W^{1,1}(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n), u\mapsto u$$ is continuous. Using the Gagliardo-Nierenberg inequality $\|u\|_{W^{1,1}(\mathbb{R}^n)}\geq\|u\|_{L^2(\mathbb{R}^n)}$, $u\in\mathcal{C}^1_c(\mathbb{R}^n)$, I managed to prove this result for the domain $W^{1,1}_0(\mathbb{R}^n)$ where I can approximate by test functions. Does someone know how to approach the general case? Why is the embedding well-defined?

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  • $\begingroup$ Your Gagliardo-Nierenberg inequality is definitely in the wrong direction. Furthermore, $W_0^{1,1}(\Bbb{R}^n) = W^{1,1}(\Bbb{R}^n)$. $\endgroup$ – PhoemueX Oct 24 '15 at 18:41
  • $\begingroup$ Yes, you're right. I've corrected this typo. Thanks! $\endgroup$ – JohnSmith Oct 24 '15 at 18:42
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    $\begingroup$ Do you mean $W^{1,2}(\mathbb{R}^n)$? $W^{1,1}$ does not embed in $L^2$ for $n>2$. $\endgroup$ – Jose27 Oct 24 '15 at 19:01
  • $\begingroup$ Yes, I really mean $W^{1,1}$. It is actually an old exam that I use for revision. Ok, then there seems to be a typo on the sheet. $\endgroup$ – JohnSmith Oct 24 '15 at 19:10
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I am not sure you need this answer or not. But I think @PhoemueX already give you one.

In general, keep in mind that $W^{1,p}=W_0^{1,p}$. (prove it! good exercise). Then the rest is clear.

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