Sets of harmonicity for finely harmonic functions (Q1876620)

The author establishes the sharpness of a theorem of Fuglede. \textit{B. Fuglede} [Ann. Inst. Fourier 24, No. 4, 77--91 (1974; Zbl 0287.31003)] observed the following result. Let \(U\) be an open set in \({\mathbb R}^n\) (\(n\geq 2\)). If \(u\) is finely harmonic on \(U\), then there is a dense open subset \(V\) of \(U\) on which \(u\) is harmonic. The fine topology of classical potential theory is the coarsest topology on \({\mathbb R}^n\) which makes all superharmonic functions continuous. Let \(V\subseteq U\subseteq{\mathbb R}^n\) (\(n\geq 3\)), where \(U\) and \(V\) are open. The author proves that the following are equivalent: (a) there is a finely harmonic function \(u\) on \(U\) such that \(V\) is the largest open subset of \(U\) on which \(u\) is harmonic; (b) \(V\) is dense in \(U\). This result is obtained by considering fine cluster sets of arbitrary functions.