Interior mollification commutes with weak derivatives

Fix a representative of $u$ and let $\tilde{u}$ be its extension by zero to ${\mathbb{R}}^n$; by [F5], $\tilde{u}\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$, and we define $u_{\epsilon }(x):=(\tilde{u}\ast {\rho }_{\epsilon })(x)={\int }_{{\mathbb{R}}^n}\tilde{u}(y){\rho }_{\epsilon }(x-y)dy$. By [F3] and [F4] applied to $\tilde{u}$, the function $u_{\epsilon }$ is smooth on ${\mathbb{R}}^n$ and for every $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, ${\mathrm{\partial }}^{\alpha }{u}_{\epsilon }(x)={\int }_{{\mathbb{R}}^n}\tilde{u}(y){\mathrm{\partial }}_x^{\alpha }{\rho }_{\epsilon }(x-y)dy\mathrm{.}$

Let $x\in {\mathrm{\Omega }}_{\epsilon }$. If $\mathrm{\Omega }={\mathbb{R}}^n$ this is every point; otherwise $\mathrm{dist}(x,{\mathbb{R}}^n\setminus \mathrm{\Omega })>\epsilon$, so $\bar{B}(x,\epsilon )\subseteq \mathrm{\Omega }$. Since $\mathrm{supp}{\rho }_{\epsilon }\subseteq {\bar{B}}_{\epsilon }(0)$, the integrand $y\mapsto \tilde{u}(y){\mathrm{\partial }}_x^{\alpha }{\rho }_{\epsilon }(x-y)$ is supported in $\bar{B}(x,\epsilon )\subseteq \mathrm{\Omega }$, where $\tilde{u}=u$ almost everywhere; hence ${\mathrm{\partial }}^{\alpha }{u}_{\epsilon }(x)={\int }_{\mathrm{\Omega }}u(y){\mathrm{\partial }}_x^{\alpha }{\rho }_{\epsilon }(x-y)dy\mathrm{.}$

Substitute the sign identity of [F6] in step 2.1: for every $x\in {\mathrm{\Omega }}_{\epsilon }$, ${\mathrm{\partial }}^{\alpha }{u}_{\epsilon }(x)=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}u(y)D_y^{\alpha }{\rho }_{\epsilon }(x-y)dy\mathrm{.}$

For fixed $x\in {\mathrm{\Omega }}_{\epsilon }$ the function $\psi (y):={\rho }_{\epsilon }(x-y)$ lies in $C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ by step 2.1, so the weak identity of [F2] applies with this $\psi$: ${\int }_{\mathrm{\Omega }}u(y)D_y^{\alpha }{\rho }_{\epsilon }(x-y)dy=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}(D^{\alpha }u)(y){\rho }_{\epsilon }(x-y)dy\mathrm{.}$

Combining steps 3.1 and 3.2 and cancelling the two signs, which multiply to $+1$, gives for every $x\in {\mathrm{\Omega }}_{\epsilon }$ ${\mathrm{\partial }}^{\alpha }{u}_{\epsilon }(x)={\int }_{\mathrm{\Omega }}(D^{\alpha }u)(y){\rho }_{\epsilon }(x-y)dy=({\rho }_{\epsilon }\ast E_0{D}^{\alpha }u)(x),$ the last expression being the convolution of the zero-extended derivative class with ${\rho }_{\epsilon }$.

The right-hand side of step 4.1 is a smooth function of $x$ on ${\mathrm{\Omega }}_{\epsilon }$ by [F4] applied to the locally integrable extension $E_0{D}^{\alpha }u$; thus ${\mathrm{\partial }}^{\alpha }{u}_{\epsilon }$ extends the smooth function ${\rho }_{\epsilon }\ast E_0{D}^{\alpha }u$ restricted to ${\mathrm{\Omega }}_{\epsilon }$. By [F7] applied on the open set ${\mathrm{\Omega }}_{\epsilon }$, this classical derivative is the weak derivative of $u_{\epsilon }$ there, for every $\mathrm{\mid }\alpha \mathrm{\mid }\le k$; in particular $D^{\alpha }({\rho }_{\epsilon }\ast u)={\rho }_{\epsilon }\ast (D^{\alpha }u)\text{on }{\mathrm{\Omega }}_{\epsilon }\mathrm{.}$ Finally, the construction does not depend on the chosen representative of $u$: changing it on a null set changes $\tilde{u}$ on a null set only, hence leaves both sides unchanged as almost-everywhere classes by [F5]. The case $k=0$ is the single identity $D^0({\rho }_{\epsilon }\ast u)={\rho }_{\epsilon }\ast u$, which is smoothness of $u_{\epsilon }$; the case $\mathrm{\Omega }={\mathbb{R}}^n$ has ${\mathrm{\Omega }}_{\epsilon }={\mathbb{R}}^n$ by definition; and complex scalars are handled by the bilinear pairing componentwise.