Loss of regularity and the Dirichlet problem for holomorphic functions in spaces with modulus of continuity

Abstract

The goal of this article is to study the Dirichlet problem for holomorphic functions in the unit disk in scales of spaces of modulus of continuity. Using scales, we are able to describe the loss of regularity of the solution in comparison with the given Dirichlet data. Some examples for data which are close to continuous data or close to Lipschitz data complete this article.

Keywords:

• holomorphic functions
• Dirichlet problem
• modulus of continuity
• loss of regularity
• singular integral operators

AMS Subject Classifications::

• 30G20
• 35J55

Acknowledgements

Part of the results are obtained during the stay of A.S. Ilschukov as Euler-fellow at the Faculty of Mathematics and Computer Science of Technical University Bergakademie Freiberg in February/March 2010. All authors thank DAAD for their financial support.

Notes

†Doz. Dr. A. Timofeev passed away on August 19 2011 near Ufa. He has been buried in Syktyvkar.

Additional information

Notes on contributors

A.Y. Timofeev

†Doz. Dr. A. Timofeev passed away on August 19 2011 near Ufa. He has been buried in Syktyvkar.