The $\Delta_{a}$-harmonicity away from $\mathfrak{D}$ follows from (2.7). To see the distributional equation (2.4), it suffices to notice that both sides are $\Delta_{a}$-harmonic away from $\mathfrak{D}$, and the singularities along $\mathfrak{D}$ match up by construction.

The integrability condition (2.5) is equivalent to (2.8). Together with the distributional equation this implies the local existence of the potential required by (2.2). ∎