Boundary Characterizations of Little Bloch and $\mathrm{VMOA}$ Functions on the Half-Plane

Abstract

We extend Pommerenke's characterizations of boundary curves of conformal mappings in terms of the little Bloch and $\mathrm{VMOA}$ conditions from the unit disk $\mathbb{D}$ to the upper half-plane $\mathbb{H}$. Let $G\colon \mathbb{H}\to\Omega$ be a conformal mapping onto an unbounded quasidisk $\Omega$ with $G(\infty)=\infty$, and let $g\colon \mathbb{R}\to\Gamma=\partial\Omega$ be its boundary extension. In the non-compact setting, the Euclidean smallness on the boundary curve is not necessarily comparable to the smallness of the parameter on $\mathbb{R}$. To overcome this difficulty, we use relative versions of the asymptotic conformality and the asymptotic smoothness with respect to the parametrization $g$. We prove that $\log G'\in B_0(\mathbb{H})$ is equivalent to the asymptotic conformality of $\Gamma$ relative to $g$, and also to the asymptotic symmetry of the embedding $g$. We further prove that $\log G'\in \mathrm{VMOA}(\mathbb{H})$ is equivalent to the asymptotic smoothness of $\Gamma$ relative to $g$, and also to the asymptotic smoothness of $g$. These results provide half-plane analogues of Pommerenke's theorems and clarify the role of the parametrization in the unbounded case.