Distributional harmonicity and Poisson's equation on an open subset of Rn

Let $n\ge 1$ be an integer and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open. Write $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ for smooth real functions with compact support in $\mathrm{\Omega }$. A distribution is a continuous linear functional $T:C_c^{\mathrm{\infty }}(\mathrm{\Omega })\to \mathbb{R}$. Here continuity means that $T({\varphi }_j)\to T(\varphi )$ whenever the supports of ${\varphi }_j$ and $\varphi$ lie in one compact subset of $\mathrm{\Omega }$ and every partial derivative of ${\varphi }_j$ converges uniformly to the corresponding derivative of $\varphi$. Set $({\mathrm{\partial }}_{i}T)(\varphi ):=-T({\mathrm{\partial }}_i\varphi ),\mathrm{\Delta }T:={\sum }_{i=1}^n{\mathrm{\partial }}_i^{2}T\mathrm{.}$ For $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$, $T_f(\varphi ):={\int }_{\mathrm{\Omega }}f\varphi$ is its regular distribution. We say $T$ is distributionally harmonic when $\mathrm{\Delta }T=0$. Given another distribution $F:C_c^{\mathrm{\infty }}(\mathrm{\Omega })\to \mathbb{R}$, we say that $T$ solves the distributional Poisson equation $-\mathrm{\Delta }T=F$ when that equality holds as distributions. These are distinct from the classical conditions on a $C^2$ function in The Laplacian of a $C^2$ function and of a $C^2$ vector field.