Exercise 13, § 3

Let $U$ be an open set of $\mathbf{R}^n$ and $j: U\rightarrow \mathbf{R}^n$ the canonical inclusion. For every function $f: U\rightarrow \mathbf{C}$, let $j_!f$ denote the extension of $f$ by zero to $\mathbf{R}^n$, that is to say the function from $\mathbf{R}^n$ into $\mathbf{C}$ which coincides with $f$ in U and is zero outside U.

Let $k\geqslant 0$ be an integer and $p\geqslant 1$ a real number.

a) Let $f\in W^{k,p}(U)$. For every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant k$, one has $\partial^{\alpha}(j_!f) =$ $j_!(\partial^{\alpha}(f))$ in $\mathscr{D}'(\mathbf{R}^n)$.

b) The mapping $f\mapsto j_!f$ induces, by passing to the subspaces, an isometric linear mapping of $W^{k,p}_0$ (U) into $W^{k,p}(\mathbf{R}^n)$.