Dirichlet级数的Nevanlinna类的复合算子与Carleson嵌入
Composition operators and Carleson embeddings for the Nevanlinna class of Dirichlet series
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中文总结 AI 辅助
本文系统研究Dirichlet级数Nevanlinna类$\mathcal{N}_u$的拓扑与分析性质，证明Littlewood-Paley型恒等式，并利用Carleson测度技术完全刻画诱导有界及紧复合算子的符号，建立Carleson嵌入与几何条件的等价性。
中文摘要 AI 辅助
本文系统研究了由Brevig和Perfekt [Adv.\\ Math., 2021]引入并由Guo等人 [Ann.\\ Inst.\\ Fourier (Grenoble), 2025]进一步发展的Dirichlet级数的Nevanlinna类$\mathcal{N}_u$的结构、分析和算子理论性质。首先，我们考察了$\mathcal{N}_u$的拓扑结构。然后，我们证明了$\mathcal{N}_u$中函数的Littlewood--Paley型恒等式，通过垂直极限函数建立了等价刻画，并证明了该恒等式中极限交换的可行性。此外，我们利用位势理论方法提供了该恒等式的另一种证明。应用这些分析工具，我们研究了作用在$\mathcal{N}_u$上的复合算子$C_\Phi$，并刻画了那些诱导有界复合算子$C_\Phi$的符号$\Phi$。利用半平面和无限维环面上的Carleson测度技术，我们刻画了生成有界和紧复合算子的符号$\Phi$。特别地，我们证明了（消失）Carleson嵌入条件与Carleson正方形上的几何（消失）Carleson条件之间的完全等价性。最后，我们给出了$\mathcal{N}_u$中函数的两个函数论应用。
英文摘要
This paper systematically investigates the structural, analytical, and operator-theoretic properties of the Nevanlinna class $\mathcal{N}_u$ of Dirichlet series, introduced by Brevig and Perfekt [Adv.\ Math., 2021] and further developed by Guo \textit{et al.}\ [Ann.\ Inst.\ Fourier (Grenoble), 2025]. First, we examine the topological structure of $\mathcal{N}_u$. We then prove a Littlewood--Paley type identity for functions in $\mathcal{N}_u$, establish an equivalent characterization via vertical limit functions, and demonstrate that the interchange of limits in this identity is permissible. In addition, we provide an alternative proof of this identity using potential-theoretic approach. Applying these analytical tools, we study composition operators $C_Φ$ acting on $\mathcal{N}_u$ and characterize those symbols $Φ$ that induce bounded composition operators $C_Φ$. Utilizing Carleson measure techniques on half-planes and infinite-dimensional tori, we characterize the symbols $Φ$ that generate bounded and compact composition operators. In particular, we prove the complete equivalence between the (vanishing) Carleson embedding condition and the geometric (vanishing) Carleson condition on Carleson squares. We conclude with two function-theoretic applications to functions in $\mathcal{N}_u$.
发表机构
- Indian Institute of Technology Bhubaneswar（印度技术研究所布巴内斯瓦尔分校）
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