Finely locally injective finely harmonic morphisms (Q1202902)

Let \({\mathcal C}\) be the complex plane. It is known that the unique continuation property holds for usual harmonic functions, finely holomorphic functions but fails for the finely harmonic functions. Fuglede has proved that this property still holds for finely harmonic morphisms, that are complex functions \(f\) defined on a finely open set \({\mathcal U}\) such that: (i) \(f\) is continuous from \({\mathcal U}\) with the fine initial topology to \({\mathcal C}\) with the standard topology. (ii) For any harmonic function \(h\), \(h\circ f\) is finely harmonic. This paper investigates the relationship between these notions and the essential result is the following: Let \(f\) be a finely harmonic morphism. If \(f\) is finely locally injective in \({\mathcal U}\), then either \(f\) or \(\overline {f}\) is finely holomorphic in \({\mathcal U}\).