Composition operators and Carleson embeddings for the Nevanlinna class of Dirichlet series
Abstract
This paper systematically investigates the structural, analytical, and operator-theoretic properties of the Nevanlinna class 𝒩u 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 𝒩u. We then prove a Littlewood--Paley type identity for functions in 𝒩u, 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 CΦ acting on 𝒩u and characterize those symbols Φ that induce bounded composition operators CΦ. 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 𝒩u.
Related
- Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series
- Weighted Composition-differentiation operators on weighted Dirichlet spaces
- Some Properties of Products of Linear Fractional Composition Operators on Weighted Dirichlet Spaces
- Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces
- Local Nevanlinna embedding for Hardy space of Dirichlet series below critical exponent
BibTeX
@misc{allu2026composition,
title = {{Composition operators and Carleson embeddings for the Nevanlinna class of Dirichlet series}},
author = {Vasudevarao Allu and Dipon Kumar Mondal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17133},
eprint = {2609.17133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17133},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Composition operators and Carleson embeddings for the Nevanlinna class of Dirichlet series
AU - Vasudevarao Allu
AU - Dipon Kumar Mondal
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17133
UR - https://arxiv.org/abs/2609.17133
AB - This paper systematically investigates the structural, analytical, and operator-theoretic properties of the Nevanlinna class 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 𝒩_u. We then prove a Littlewood--Paley type identity for functions in 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 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 𝒩_u.
ER -