Let $u$ be a compactly supported distribution on $\mathbb{R}^n$. I am trying to estimate the convolution derivative
$$\partial^\alpha(u\ast \phi)(x)=u\ast(\partial^\alpha\phi)(x)=u(\phi(x-\cdot))\ ,\quad\phi\in C^\infty$$
via the seminorm condition for $u$, so that I can show for example whether $u\ast\phi_j\to 0$ in $C^\infty$ whenever $\phi_j\to 0$ in $C^\infty$ (a sequence converges to $0$ on $C^\infty$ iff its derivatives of all order converge uniformly inside every compact). But I am not certain about a step of the argument:
Let $B=\text{supp} u$ compact, and let $A$ be a given compact. I want to show that $\partial^\alpha(u\ast\phi_j)\to 0$ uniformly on $A$ whenever $\phi_j\to 0$ in $C^\infty$. Then $K=A+(-B)$ is a compact, and therefore there exists $c>0$ constant and $k\in\mathbb{Z}_{\geq 0}$ s.t.
\begin{align} |u(\psi)|\leq c\sum_{|\beta\leq k}\sup_{x\in K}|\partial^\beta\psi(x)| \ ,\quad\forall\psi\in C^\infty_c\ ,\mbox{supp}\psi\subseteq K\ . \end{align}
So in order to apply the seminorm estimate above to the function $y\mapsto\phi(x-y)$, where $\phi\in C^\infty$ and $x\in A$, I need to support this function on $K$. For that, I need to support $x\mapsto\phi(x)$ on $A$. Therefore taking a smooth bump function $\rho$ with $\rho=1$ on a neighborhood of $A$, we can write
\begin{align} \sup_{x\in A}|\partial^\alpha(u\ast\phi)| =\sup_{x\in A}|u(\partial^\alpha\phi(x-\cdot))|\\ =\sup_{x\in A}|u(\partial^\alpha(\rho\phi(x-\cdot)))|\\ \leq c'\sum_{|\beta|\leq k+k'}\sup_{x\in K}|\partial^\alpha\phi(x)|\ , \end{align}
where say $|\alpha|\leq k'$, using the Leibniz rule in the last step.
I am not sure about this bump function argument.