On $\mu$-conformal homeomorphisms and boundary correspondence

TL;DR

This paper investigates the boundary behavior of $$-homeomorphisms of the upper half-plane, providing conditions for continuous extension with specific boundary regularity using modulus estimates related to directional dilatations.

Contribution

It offers new sufficient conditions for boundary extension of $$-homeomorphisms with prescribed regularity, based on modulus estimates of semiannuli.

Findings

Established criteria for boundary extension of $$-homeomorphisms

Connected boundary regularity to modulus estimates and directional dilatations

Provided techniques potentially useful for further boundary behavior analysis

Abstract

We study the boundary correspondence under $\mu$-homeomorphisms $f$ of the open upper half-plane onto itself. Sufficient conditions are given for $f$ to admit a homeomorphic extension to the closed half-plane with prescribed boundary regularity. The proofs are based on the modulus estimates for semiannuli in terms of directional dilatations of $f$ which might be of independent interest.