Duality on harmonic spaces (Q810682)

The following problem from the theory of harmonic spaces is considered. Given a Green function k(x,y), i.e. a system of extremal potentials \(\{k_ y(x)\), \(y\in X\}\) on a harmonic space (X,U), construct a dual harmonic space \((X,U^*)\) such that \(\{k^*_ x(y)=k(x,y)\), \(x\in X\}\) is a system of Green functions on \(U^*\). The problem was solved in the affirmative by Taylor in terms of probability theory. In the present paper a purely potential-theoretic approach is proposed.