Abstract
We consider fine topology in the complex plane C and finely harmonic morphisms. We use oriented Jordan curves in the plane to prove that for a finely locally injective finely harmonic morphism f in a fine domain in C, either f or f is a finely holomorphic function. This partially extends result by Fuglede, who considered a kind of continuity for the fine derivatives of the finely harmonic morphism. As a consequence of this we obtain a both necessary and sufficient condition for a function f to be finely holomorphic or finely antiholomorphic. We do not know if the condition of finely local injectivity (q.e.) is automatically fulfilled by any non-constant finely harmonic morphism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borel, E.: Leçons sur les fonctions monogènes uniformes d'une variable complexe, Gauthier-Villars, Paris, 1917.
Fuglede, B.: Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier (Grenoble) 21(3) (1971), 227–244.
Fuglede, B.: Finely Harmonic Functions, Lecture Notes in Mathematics No. 289, Springer-Verlag, Berlin, 1972.
Fuglede, B.: Fonctions harmoniques et fonctions finement harmoniques, Ann. Inst. Fourier (Grenoble) 24(4) (1974), 77–91.
Fuglede, B.: Finely harmonic mappings and finely holomorphic functions, Ann. Acad. Sci. Fennicae (A.I.) 2 (1976), 113–127.
Fuglede, B.: The Fine Topology in Potential Theory, Lecture Notes in Mathematics No. 814, Springer-Verlag, Berlin, 1980, pp. 97–116.
Fuglede, B.: Sur les fonctions finement holomorphes, Ann. Inst. Fourier (Grenoble) 31(4) (1981), 57–88.
Fuglede, B.: Fonctions BLD et fonctions finement surharmoniques, Lecture Notes in Mathematics No. 906, Springer-Verlag, Berlin, 1982, pp. 126–157.
Fuglede, B.: Value distribution of harmonic and finely harmonic morphisms and applications in complex analysis, Ann. Acad. Sci. Fennicae (A.I.) 11 (1986), 111–136.
Fuglede, B.: Finely holomorphic functions and finely harmonic morphisms, Aequationes Mathematicae 34 (1987), 167–173.
Fuglede, B.: Finely holomorphic functions. A survey, Rev. Roumaine Math. Pures Appl. 33(4) (1988), 283–295.
Helms, L. L.: Introduction to Potential Theory, Wiley Interscience Pure and Applied Mathematics 22, New York, 1969.
Lyons, T.: Finely holomorphic functions, J. Funct. Anal. 37 (1980), 1–18.
Lyons, T.: Finely harmonic functions need not be quasi-analytic, Bull. London Math. Soc. 16 (1984), 413–415.
Mandelbrojt, S.: Séries de Fourier et classes quasi-analytiques, Gauthier-Villars, Paris, 1935.
Whyburn, G. T.: Topological Analysis, Princeton University Press, Princeton, 1958.
Nguyen-Xuan-Loc: Characterization of holomorphic processes among the conformal martingales, Bull. Sci. Math. 115(3) (1991), 265–299.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pyrih, P. Finely locally injective finely harmonic morphisms. Potential Anal 1, 373–378 (1992). https://doi.org/10.1007/BF00301789
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00301789