On the Riemann-Hilbert problem for analytic functions in circular domains

TL;DR

This paper proves the existence of infinite-dimensional spaces of analytic solutions to the Riemann-Hilbert problem in circular domains with measurable boundary data and coefficients of sigma-finite variation.

Contribution

It establishes the existence of single-valued and multivalent analytic solutions in circular domains with measurable boundary data, expanding the understanding of the Riemann-Hilbert problem.

Findings

Solutions exist in the unit disk and circular domains bounded by finitely many circles.

The solution spaces are infinite-dimensional.

Solutions accommodate coefficients of sigma-finite variation and measurable boundary data.

Abstract

It is proved the existence of single-valued analytic solutions in the unit disk and multivalent analytic solutions in domains bounded by a finite collection of circles for the Riemann-Hilbert problem with coefficients of sigma-finite variation and with boundary data that are measurable with respect to logarithmic capacity. It is shown that these spaces of solutions have the infinite dimension.