Helmholtz decomposition and the fundamental solution

Question

The Helmholtz decomposition of a vector field $\mathbf{F}$ is given by $\mathbf{F} = -\nabla \Phi + \nabla \times \mathbf{A}$. Wikipedia gives an explicit formula for $\Phi$ and $\mathbf{A}$. They look to be obtained from convolving the divergence and curl of $\mathbf{F}$ by the fundamental solution of the Laplacian. Why is this?