$(a)$ Let $\displaystyle L=\sum_{|\alpha|\leq m}a_{\alpha}D^\alpha$ be a differential operator with constant coefficients such that $L$ has a fundamental solution $E$ which is $C^\infty$ on $\mathbb{R}^n\setminus\{0\}$ . Let $\phi\in\mathcal{D}(\mathbb{R}^n)$ with $\phi\equiv1$ in a neighborhood of zero . Show that $P=\phi E$ is a distribution with compact support and such that $L(P)=\delta+\zeta$ for $\zeta\in\mathcal{D}(\mathbb{R}^n)$ .
$(b)$ Deduce that if $T\in\mathcal{D}'(\mathbb{R}^n)$ with $L(T)\in C^\infty(\mathbb{R}^n)$ then $T\in C^\infty(\mathbb{R}^n)$ .
$(c)$ Deduce that every fundamental solution of $L$ is $C^\infty$ on $\mathbb{R}^n\setminus\{0\}$ .
$(a)$ It is easy to prove $\phi E=(\phi E)_f$ for some $f\in\mathcal{D}(\mathbb{R}^n)$ with $f\equiv1$ on $\text{supp}(\phi)$ . Hence $P$ is of compact support . Moreover \begin{align*}L(P)&=\sum_{|\alpha|\leq m}a_{\alpha}D^\alpha(\phi E)\\&=\sum_{|\alpha|\leq m}a_{\alpha}\sum_{k=0}^\alpha{\alpha\choose k}(D^{\alpha-k}\phi)(D^kE)\\&=\sum_{|\alpha|\leq m}a_{\alpha}\sum_{k=0}^{\alpha-1}{\alpha\choose k}(D^{\alpha-k}\phi)(D^kE)+\phi\sum_{|\alpha|\leq m}a_{\alpha}D^\alpha E\\&=\sum_{|\alpha|\leq m}a_{\alpha}\sum_{k=0}^{\alpha-1}{\alpha\choose k}(D^{\alpha-k}\phi)(D^kE)+\delta\end{align*} since $L(E)=\delta$ and $\phi\equiv1$ in a neighborhood of zero . But I don't see why $\displaystyle\sum_{|\alpha|\leq m}a_{\alpha}\sum_{k=0}^{\alpha-1}{\alpha\choose k}(D^{\alpha-k}\phi)(D^kE)\in\mathcal{D}(\mathbb{R}^n)$ so that it may serve for $\zeta$ .
$(b)$ and $(c)$ are not clear to me how to proceed . Any help is appreciated .