On $H^\infty$ on the complement of C^{1+\alpha} curves

TL;DR

This paper characterizes $C^{1+eta}$ curves via Carleson measure conditions on quasiconformal mappings and demonstrates how $H^$ spaces transfer from the upper half-plane to their complements.

Contribution

It establishes a precise Carleson measure criterion on quasiconformal maps that characterizes $C^{1+eta}$ regularity of quasicircles and describes the transfer of $H^$ spaces.

Findings

Carleson measure condition characterizes $C^{1+eta}$ curves.

Transfer of $H^$ spaces from upper half-plane to complements of quasicircles.

Equivalent conditions for quasiconformal maps and curve regularity.

Abstract

Let $\rho$ be a quasiconformal mapping on the plane with complex dilatation $\mu$. We show that if $\mu$ satisfies a certain Carleson measure condition, then one can transfer $H^{\infty}$ on the upper half plane onto the corresponding space in the complement of the quasicircle $\Gamma=\rho(\mathbb{R})$, and that this condition on $\mu$ characterizes $C^{1+\alpha}$ curves.