Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 1111–1134

TRANSMISSION OF HARMONIC FUNCTIONS THROUGH QUASICIRCLES ON COMPACT RIEMANN SURFACES

Eric Schippers and Wolfgang Staubach

University of Manitoba, Department of Mathematics
Winnipeg, Manitoba, R3T 2N2, Canada; eric.schippers 'at' umanitoba.ca

Uppsala University, Department of Mathematics
Box 480, 751 06 Uppsala, Sweden; wulf 'at' math.uu.se

Abstract. Let R be a compact surface and let Γ be a Jordan curve which separates R into two connected components Σ₁ and Σ₂. A harmonic function h₁ on Σ₁ of bounded Dirichlet norm has boundary values H in a certain conformally invariant non-tangential sense on Γ. We show that if Γ is a quasicircle, then there is a unique harmonic function h₂ of bounded Dirichlet norm on Σ₂ whose boundary values agree with those of h₁. Furthermore, the resulting map from the Dirichlet space of Σ₁ into Σ₂ is bounded with respect to the Dirichlet semi-norm.

2010 Mathematics Subject Classification: Primary 58J05, 30C62, 30F15.

Key words: Harmonic functions, quasicircles, transmission, Dirichlet spaces, conformally non-tangential limits.

Reference to this article: E. Schippers and W. Staubach: Transmission of harmonic functions through quasicircles on compact Riemann surfaces. Ann. Acad. Sci. Fenn. Math. 45 (2020), 1111–1134.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4559