Let $H$ be the open half-space of $\Bbb R^n$ defined by $x_n > 0$. Let $f : \overline H \to \Bbb R$ continuous and harmonic on $H$. Define the function $F : \Bbb R^n \to \Bbb R$ by $ F(x_1,\dotsc,x_n) = \begin{cases} f(x_1,\dotsc,x_n) && \text{if $x_n>0$}\\ -f(x_1,\dotsc,-x_n) && \text{if not.} \end{cases}$
$F$ defines a distribution on $\Bbb R^n$ and one computes $ \Delta F = 2 \partial_n( f \sigma),$ where $\sigma$ is the surface distribution of the hyperplane $x_n = 0$.
The conclusion is that the Cauchy problem $f\in C(\overline H), \quad \left\{ \begin{gathered} \Delta f_{|H} = 0 \\ f_{|x_n = 0} = g \end{gathered} \right .$ translates into the distribution equation $ F \in \mathcal D \Bbb R^n\\ \Delta F = 2 \partial_n(g\sigma). $
If $g$ has compact support, it is well-known that the latter equation has a solution (using a convolution with the elementary solution of the laplacian).
Here is my question.
Assume that $g\in C_c(\partial H)$, and that $\Delta F = 2\partial_n(g\sigma)$. How to show that $F$ comes from a solution of the first system ?