Finely holomorphic functions and finely harmonic morphisms (Q1104454)

The following result is well known and easy to prove: If f is a complex valued function on a domain U in the complex plane C with the property that both f and \(f^ 2\) are harmonic in U, then either f or \(\bar f\) is holomorphic in U. An outstanding problem in the theory of finely harmonic functions, posed by the author [18th Scand. Congr. Math., Proc., Aarhus 1980, Prog. Math. 11, 22-38 (1981; Zbl 0462.30004)] is whether the fine version of this holds: If f is a complex valued function on a fine domain W in C with the property that both f and \(f^ 2\) are finely harmonic in W, does it follow that either f or \(\bar f\) is finely holomorphic in W? In this article the author shows that the answer to this question is yes under the extra hypothesis that f is a fine \(C^ 1\)-function. It is still open to what extent this hypothesis can be weakened or dropped entirely.