In what the fact that there is a continuous operator $E:W^{1,p}(U)\to W^{1,p}(\mathbb R^n)$ is important?

Question

Let $U$ bounded s.t. $\partial U$ is $\mathcal C^1$ Let $V$ a bounded open set such that $U\subset \subset V$. Then there is a bounded linear operator $$E:W^{1,p}(U)\longrightarrow W^{1,p}(\mathbb R^n),$$ s.t. for all $u\in W^{1,p}(U)$ we have

1) $Eu=u$ a.e. in $U$

2) $Eu$ has support within $V$

3) $\|Eu\|_{W^{1,p}(\mathbb R^n)}\leq C\|u\|_{W^{1,p}(U)}$.

My teacher said that this important is very important, but I don't understand in what it's important, neither what it exactly implies. So any explanation would be appreciated.

Answer 1

It is an important ingredient in proofs of embedding results. One standard way of proving these is:

1. Prove the embedding for $W^{1,p}_0(V)$ using density of $C_c^\infty(V)$ functions.

2. Take $u\in W^{1,p}(U)$, extend it to $Eu\in W^{1,p}_0(V)$. Then use the already established embedding result on $Eu$ and carry it over to $u$.