Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-08-25T22:34:23.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive Harmonic Functions and Complete Metrics

Part of: Higher-dimensional theory Two-dimensional theory Geometric function theory

Published online by Cambridge University Press:  20 November 2018

David A. Herron  and
Joel L. Schiff
David A. Herron
Affiliation:
Department of Mathematics University of Cincinnati Cincinnati, Ohio 45221-0025 U.S.A.
Joel L. Schiff
Affiliation:
Department of Mathematics University of Auckland Auckland New Zealand
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce the class of Harnack domains in which a Harnack type inequality holds for positive harmonic functions with bounds given in terms of the distance to the domain's boundary. We give conditions connecting Harnack domains with several different complete metrics. We characterize the simply connected plane domains which are Harnack and discuss associated topics. We extend classical results to Harnack domains and give applications concerning the rate of growth of various functions defined in Harnack domains. We present a perhaps new characterization for quasidisks.

MSC classification

Secondary: 30C20: Conformal mappings of special domains 31A05: Harmonic, subharmonic, superharmonic functions 31A20: Boundary behavior (theorems of Fatou type, etc.) 31B05: Harmonic, subharmonic, superharmonic functions 31B25: Boundary behavior 30C55: General theory of univalent and multivalent functions

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 3 , 01 September 1989 , pp. 286 - 297
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. [AH] Anderson, J. M. and Hinkkanen, A., Positive superharmonic functions and the Holder continuity of conformai mappings, preprint.Google Scholar
2. [BP] Becker, J. and Pommerenke, Ch., Holder continuity of conformai mappings and non-quasiconformal Jordan curves, Comment. Math. Helv. 57 (1982) 221–225.Google Scholar
3. [G] Gehring, F. W., Characteristic Properties ofQuasidisks, Les Presses de l'Université de Montreal, 1982.Google Scholar
4. [GM] and Martio, O., Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985) 203–219.Google Scholar
5. [GP] and Palka, B. P., Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976) 172–199.Google Scholar
6. [HL] Hardy, G. H. and Little wood, J. E., Some properties of fractional integrals II, Math. Z. 34 (1932) 403–439.Google Scholar
7. [Hi] Herron, D. A., The Harnack and other conformally invariant metrics, Kodai Math J. 10 No 1 (1987) 9–19.Google Scholar
8. [H2] Metric boundary conditions for plane domains, Complex Analysis, Proc. 13 Rolf Nevanlinna Colloq., Lecture Notes in Math., No. 1351 Springer-Verlag, (1988), 193–200.Google Scholar
9. [HV] and Vuorinen, M., Positive harmonic functions in uniform and admissible domains, Analysis 8 (1988) 187-206.Google Scholar
10. [J] Jones, P. W., Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980) 41–66.Google Scholar
11. [Ko] Kohn, J., Die Harnacksche Metrik in der Théorie der harmonischen Funktionen, Math Z. 91 (1966) 50–64.Google Scholar
12. [K] Kuran, Û, On positive superharmonic functions in a-admissible domains, J. London Math. Soc. (2) 29 (1984) 269–275.Google Scholar
13. [KS] and Schiff, J. L., A uniqueness theorem for nonnegative superharmonic functions in planar domains, J. Math. Anal. Appl. 93 (1983) 195–205.Google Scholar
14. [P] Pommerenke, Ch., Univalent Functions, Vandenhoeck and Ruprecht, Gôttingen, 1975.Google Scholar
15. [S] Schiff, J. L., A uniqueness theorem for superharmonic functions in Rn, Proc. Amer. Math. Soc. 84 No. 3 (1982) 362–364.Google Scholar