On the representability of harmonic functions as Newtonian potentials (Q1311338)

Let \(L_ n\) be the differential operator in \(\mathbb{R}^ n\) given by \[ L_ n \varphi (x)=\bigl\langle \text{grad} \varphi (x),x \bigr\rangle+{n-2 \over 2} \varphi(x). \] It is shown that a harmonic function \(h\) on the unit ball \(B_ n\) in \(\mathbb{R}^ n\) \((n \geq 3)\) is representable by the Newtonian potential with a positive measure on \(\partial B_ n\) if and only if \(L_ nh \geq 0\) on \(B_ n\). Similarly a function \(h\) harmonic on the complement of \(B_ n\) is representable by such potential if and only if \(L_ nh \leq 0\) and \(\lim_{| x | \to \infty} | x |^{n-2} h(x)\) exists. Some analogous results are given in the case \(n=2\).