Sobolev approximation by mollifiers (${\rho }_{\epsilon }\ast u\to u\text{ in }{W}^{k,p}({\mathbb{R}}^n)$)

For $1\le p<\infty$, an integer $k\ge 0$, and a normalized nonnegative mollifier $\rho \in C_c^{\infty }({\mathbb{R}}^n)$, put ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x/\epsilon )$. If $u\in W^{k,p}({\mathbb{R}}^n)$, its mollification is smooth and converges to $u$ in the Sobolev norm. The definition of a weak derivative and Fubini's theorem give ${\partial }^{\alpha }({\rho }_{\epsilon }\ast u)={\rho }_{\epsilon }\ast {\partial }^{\alpha }u$ for every multi-index with $|\alpha |\le k$. The approximate identity theorem in the Lp space $L^p$ applied to these finitely many weak derivatives proves the claim. On a domain with boundary this direct convolution argument applies only away from the boundary unless an extension is first supplied.