Q: For $u\in\mathcal{D}'(\mathbb{R}^3)$ and $f\in\mathcal{D}'(\mathbb{R}^3)$, if they satisfy the equation $$\triangle u=f,$$ show that $$u\in C^{\infty}(\mathbb{R}^3\setminus \mathrm{sopp}(f)).$$
We first consider the version for those $f\in\mathcal{E}'(\mathcal{R}^3)$.
we can use the fundamental solution $$E=-\frac{1}{|\mathbb{S}^2|\cdot |x|}$$ of the Laplace equation. Observe that,
Lemma. For $c\in\mathcal{E}'(\mathbb{R}^3)$, $\frac{1}{|x|}*c$ is smooth on $\mathbb{R}^3\setminus\mathrm{supp}(c)$.
proof. We choose a smooth function $\chi_{\varepsilon}$ satisfying $$\mathrm{support}(\chi_{\varepsilon})\subset B(0,2\varepsilon),\ \chi_{\varepsilon}\big |_{B(0,\varepsilon)}\equiv 1.$$ We have $$\frac{1}{|x|}* c=\frac{\chi_{\varepsilon}}{|x|}*c+\frac{1-\chi_{\varepsilon}}{|x|}*c.$$ The support of the first part is contained in $B(0,\varepsilon)+\mathrm{supp}(c)$, and the second part is smooth. Let $\varepsilon\to0+$, we will have done.
Since smoothness is a local property, we just need to show that, for every $x_0\in\mathbb{R}^3\setminus \mathrm{supp}(f)$, there is a neighborhood of $x_0$ on which the distribution $u$ is smooth. So, we choose $\varepsilon>0$, s.t. $B(x_0,2\varepsilon)\cap \mathrm{supp}(f)=\emptyset$ (this can be done since $\mathrm{supp}(f)$ is compact), and choose a smooth function $\theta$ satisfying $$0\leq \theta\leq 1, \ \mathrm{supp}(\theta)\subset B(x_0,2\varepsilon), \ \theta\big |_{B(x_0,\varepsilon)}\equiv 1.$$
We have \begin{align*} \theta\cdot u&=\delta_0*(\theta\cdot u)\\ &=\triangle E*(\theta\cdot u) \\ &=E*\triangle(\theta\cdot u) \end{align*}
and \begin{align*} \triangle(\theta\cdot u)&=\triangle \theta+\triangle u+2\sum_{i=1}^3 \frac{\partial \theta}{\partial x_i}\cdot \frac{\partial u}{\partial x_i} \\ &=f+\underbrace{\triangle \theta+2\sum_{i=1}^3 \frac{\partial \theta}{\partial x_i}\cdot \frac{\partial u}{\partial x_i}}_{=:g}. \end{align*}
We can see that $\triangle(\theta\cdot u)$ has compact support, and $$g\big|_{B(0,\varepsilon)}\equiv0, $$ so that $$B(x_0,\varepsilon)\cap \mathrm{supp}(\triangle (\theta\cdot u))=\emptyset. $$
Then we have $\theta\cdot u$ is smooth on $B(x_0,\varepsilon)$ by the lemma.
On the proof above, we have used the compactness of $\mathrm{supp}(f)$, and what should we do to prove the version for $f\in\mathcal{D}'(\mathbb{R}^3)$ ?