Positive harmonic functions and complete metrics. (English) Zbl 0646.30013

Herron, David A.; Schiff, Joel L.

Can. Math. Bull. 32, No. 3, 286-297 (1989).

A Harnack domain is one 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 which connect Harnack domains with several complete metrics (e.g., the quasihyperbolic metric). We characterize the simply connected plane domains which are Harnack and discuss related topics. We extend some classical function theoretic results to Harnack domains and give applications concerning the rate of growth of various functions defined in Harnack domains. We present an apparently new characterization for the class of quasidisks.

Reviewer: D.A.Herron

MSC:

30C20	Conformal mappings of special domains
31A05	Harmonic, subharmonic, superharmonic functions in two dimensions
31A20	Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
31B25	Boundary behavior of harmonic functions in higher dimensions
30C55	General theory of univalent and multivalent functions of one complex variable
30C62	Quasiconformal mappings in the complex plane

Keywords:

positive harmonic functions; Harnack domains; quasihyperbolic metric; Harnack metric; hyperbolic metric