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Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in \(\mathbb {C}^2\)

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Abstract

Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).

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Acknowledgements

The author is grateful to the referee(s) for careful reading of the paper and valuable suggestions and comments.

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  1. Faculty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City (VNU-HCM), 227 Nguyen Van Cu street, District 5, Ho Chi Minh City, Vietnam

    Ly Kim Ha

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Correspondence to Ly Kim Ha.

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This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under Grant Number C2016-18-17.

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Ha, L.K. Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in \(\mathbb {C}^2\) . Ann Glob Anal Geom 51, 327–346 (2017). https://doi.org/10.1007/s10455-016-9537-x

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