2023/10/24 by Shaolin Chen, Manzi Huang, Chen, Shaolin +3 · 3 citations
Mathematics · #30C62 #30H10 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2310.15452
openalex publication_date 2023/10/24 · openalex created_date 2023/10/26 · openalex updated_date 2026/07/28
The main purpose of this paper is to develop some methods to improve and generalize the main results in a recent paper by Liu and Zhu (Adv. Math., 2023, i.e., \citeL-Z). The paper consists of two parts. In the first part, we discuss the Riesz type theorem in the setting of n-dimensional complex spaces for all n≥ 1. In this part, we first introduce the family of κ-pluriharmonic mappings of the n-dimensional complex unit ball. Then we establish two Riesz type theorems for these mappings, which are the n-dimensional versions of Theorems 1.1 and 1.2 in \citeL-Z, respectively. Furthermore, even when n=1, our first result shows that the assumption of the real parts of the mappings not being negative (or being negative) in \cite[Theorem 1.1]L-Z is redundant; and our second result illustrates that the assumption of "quasiconformality" on the mappings in \cite[Theorem 1.2]L-Z can be replaced by the weaker one of "quasiregularity". In the second part, we investigate the Riesz type theorem in the setting of n-dimensional real spaces for all n≥ 2. In this part, first, we prove a Riesz type theorem for invariant harmonic quasiregular mappings of the unit n-dimensional real ball. Our result indicates that (i) the range of the parameter p discussed in \cite[Theorem 1.3]L-Z can be changed from (1,2) to (1,∞); (ii) the assumption of the first coordinate functions of the mappings being non-zero in \cite[Theorem 1.3]L-Z is redundant. In this way, we complete the discussions carried out in \cite[Theorems 1.3 and 1.4]L-Z. Second, we obtain a Riesz type theorem for harmonic K-quasiregular mappings of the unit n-dimensional real ball. Our result demonstrates that the range of the parameter p discussed in \cite[Theorem 2.1]K-2023 can be changed from (1,2) to (1,∞).