Proof of identity in the sense of distributions

Question

I am trying to prove this:

$\bullet$ Prove that $\Delta(\varrho_\epsilon \star u) = \varrho_\epsilon \star f $ in the sense of distributions, if $\Delta u = f$ in the sense of distributions, $ u \in L^1_{loc}(\mathbb{R}^n)$.

Can anybody help me? Thank you in advance and I take advantage of the situation to wish you a happy new year :D

Answer 1

The only rule you need except for linearity is $\partial_k (u*v) = u * (\partial_k v).$

Answer 2

Suppose that $\Delta u=f$ in the sense of distributions. One can easily verify that $$P (v*u)=(Pv)*u=v*(Pu)$$ in the sense of distributions for $v\in \mathcal{S},u\in\mathcal{S}'$, and any constant-coefficient differential operator $P$. If we take $P=\Delta$, $v=\rho_\epsilon,$ and $u$ to be itself, then it follows directly that
$$ \Delta (\rho_\epsilon*u)=\rho_\epsilon*(\Delta u)=\rho_\epsilon*f$$ in the sense of distributions.