Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 119-127

DIRICHLET PROBLEM AND SOKHOTSKI-PLEMELJ JUMP FORMULA ON WEIL-PETERSSON CLASS QUASIDISKS

David Radnell, Eric Schippers and Wolfgang Staubach

Aalto University, Department of Mathematics and Systems Analysis
P.O. Box 11100, FI-00076 Aalto, Finland; david.radnell 'at' aalto.fi

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. We show the solvability of the Dirichlet problem on Weil-Petersson class quasidisks and establish a Sokhotski-Plemelj jump formula for Weil-Petersson class quasicircles. Furthermore we show that the resulting Cauchy projections are bounded. In both cases the boundary data belongs to a certain conformally invariant Besov space. Moreover we show that the WP-class quasicircles are chord-arc curves.

2010 Mathematics Subject Classification: Primary 30F15, 31C05, 31C25; Secondary 30C55, 30C62.

Key words: Dirichlet problem, quasicircles, quasiconformal extension, Poincaré inequality, chord-arc curves, Sokhotski-Plemelj jump decomposition, Cauchy integral, Besov spaces, Weil-Petersson class.

Reference to this article: D. Radnell, E. Schippers and W. Staubach: Dirichlet problem and Sokhotski-Plemelj jump formula on Weil-Petersson class quasidisks. Ann. Acad. Sci. Fenn. Math. 41 (2016), 119-127.

Full document as PDF file

doi:10.5186/aasfm.2016.4108