发表机构
California Polytechnic State University; Guangzhou University（加州州立理工大学; 广州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究加权 Dirichlet 空间上复合算子的闭值域与本质范数，给出闭值域的逆 Carleson 条件刻画，并建立本质范数的双侧估计与精确公式，主要贡献在于统一并推广了相关结果。
AI 中文摘要
我们研究加权 Dirichlet 空间 $\mathcal{D}_{\alpha}$ 上有界复合算子 $C_\varphi$ 的闭值域与本质范数。对于 $-1<\alpha<0$，我们首先考虑其像通过从 $\mathbb{D}$ 的单连通子域中移除一个紧子集而得到的映射。在此情形下，闭值域等价于计数测度 $\mu_{\varphi,\alpha}$ 的逆 Carleson 性质与逆 Carleson 圆盘条件、像的均匀加权面积密度、$\mathbb{T}\subset\overline{\varphi(\mathbb{D})}$，以及像中包含一个外部环形区域。我们还在均匀尾部条件下获得了闭值域刻画。对于所有 $\alpha>-1$，我们建立了以广义 Nevanlinna 计数函数的局部平均表示的双侧本质范数估计，并给出了利用边界尾部积分的精确公式。若归一化计数密度在边界处具有消失振荡，我们则获得以其边界上极限、Berezin 变换和局部平均表示的精确公式。
英文摘要
We study closed range and essential norms of bounded composition operators $C_φ$ on weighted Dirichlet spaces $\mathcal{D}_α$. For $-1<α<0$, we first consider maps whose images are obtained by removing a compact subset from a simply connected subdomain of $\mathbb{D}$. In this setting, closed range is equivalent to the Reverse Carleson property and the Reverse Carleson Disk Condition for the counting measure $μ_{φ,α}$, uniform weighted area density of the image, $\mathbb{T}\subset\overline{φ(\mathbb{D})}$, and the inclusion of an outer annulus in the image. We also obtain a closed range characterization under the Uniform Tail Condition. For all $α>-1$, we establish two-sided essential norm estimates in terms of local averages of the generalized Nevanlinna counting function and an exact formula using boundary tail integrals. If the normalized counting density has vanishing oscillation at the boundary, we obtain exact formulas in terms of its boundary limsup, Berezin transform, and local averages.
