Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$
Is there a way to solve this problem by using potential theory, i.e. if $\Gamma $ is a fundamental solution of $-\Delta$, is it possible to give a precise meaning for the expression $\Gamma\star f=v$ and then prove that $v$ is a weak solution of $(1)$?
Update: Let me simplify things a little bit. Suppose that $f\in H^{-1}(\mathbb{R}^N)$ and $f$ is supported in $\Omega$, i.e. $f$ is a suported distribution.
$f$ being a supported distribution, we have that $f\star\varphi (x)=\langle f, \varphi(x+\cdot)\rangle$ belongs to $C_0^\infty(\mathbb{R}^N)$ for all $\varphi\in C_0^\infty(\mathbb{R}^N)$. Hence, it makes sense to define $f\star\Gamma$ (note that in this case $f\star\Gamma$ solves $(1)$) as distribution acting like this $$\langle f\star\Gamma,\varphi\rangle=\langle\Gamma,f\star\varphi(x)\rangle$$
Can I conclude that $f\star\Gamma\in H^1(\mathbb{R}^N)$?
Thank you