半平面上小布洛赫函数和VMOA函数的边界特征
Boundary Characterizations of Little Bloch and $\mathrm{VMOA}$ Functions on the Half-Plane
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中文总结 AI 辅助
该研究将庞梅伦克关于共形映射边界曲线特征描述从单位圆盘扩展到上半平面，通过使用相对于参数化\(g\)的渐近共形性和渐近光滑性的相对版本，证明了\(\log G'\)属于不同函数空间与边界曲线及\(g\)相关性质的等价性。
中文摘要 AI 辅助
我们将庞梅伦克关于共形映射边界曲线根据小布洛赫和VMOA条件的特征描述从单位圆盘\(\mathbb{D}\)扩展到上半平面\(\mathbb{H}\)。设\(G:\mathbb{H}\to\Omega\)是到无界拟圆盘\(\Omega\)且\(G(\infty)=\infty\)的共形映射，其边界扩展为\(g:\mathbb{R}\to\Gamma=\partial\Omega\)。在非紧情形下，边界曲线的欧几里得小性与\(\mathbb{R}\)上参数的小性不一定可比。为克服此困难，我们使用相对于参数化\(g\)的渐近共形性和渐近光滑性的相对版本。我们证明\(\log G'\in B_0(\mathbb{H})\)等价于\(\Gamma\)相对于\(g\)的渐近共形性以及嵌入\(g\)的渐近对称性。还证明\(\log G'\in \mathrm{VMOA}(\mathbb{H})\)等价于\(\Gamma\)相对于\(g\)的渐近光滑性以及\(g\)的渐近光滑性。这些结果提供了庞梅伦克定理的半平面类似物并阐明了无界情形下参数化的作用。
英文摘要
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 $G\colon \mathbb{H}\toΩ$ be a conformal mapping onto an unbounded quasidisk $Ω$ with $G(\infty)=\infty$, and let $g\colon \mathbb{R}\toΓ=\partialΩ$ 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 $Γ$ 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 $Γ$ 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.
