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On (Fejér-)Riesz type inequalities, Hardy–Littlewood type theorems and smooth moduli

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Abstract

The purpose of this paper is to develop some methods to study (Fejér-)Riesz type inequalities, Hardy–Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in Kalaj (Trans Am Math. Soc. 372:4031–4051, 2019). Next, some Hardy–Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established, which give analogies and extensions of a result in Hardy and Littlewood (J Reine Angew Math 167;405–423, 1931). Furthermore, we establish a Fejér–Riesz type inequality on pluriharmonic functions in the Euclidean unit ball in \(\mathbb {C}^n\), which extends the main result in Melentijević and Bo\(\breve{z}\)in (Potential Anal 54:575–580, 2021). Additionally, we also discuss the Hardy–Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions. Consequently, we improve and extend the corresponding results in Dyakonov (Acta Math 178:143–167, 1997), Hardy and Littlewood (Math Z 34:403–439, 1932), Dyakonov (Adv Math 187:146–172, 2004) and Pavlović (Rev Mat Iberoam 23:831–845, 2007).

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Acknowledgements

The authors would like to thank the referee for many valuable suggestions. The research of the first author was partly supported by the National Science Foundation of China (grant no. 12071116), the Hunan Provincial Natural Science Foundation of China (No. 2022JJ10001), the Key Projects of Hunan Provincial Department of Education (grant no. 21A0429), the Double First-Class University Project of Hunan Province (**angjiaotong [2018]469), the Science and Technology Plan Project of Hunan Province (2016TP1020), and the Discipline Special Research Projects of Hengyang Normal University (XKZX21002); The research of the second author was partly supported by JSPS KAKENHI Grant Number JP22K03363.

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  1. College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan, 421002, People’s Republic of China

    Shaolin Chen

  2. Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang, 421002, People’s Republic of China

    Shaolin Chen

  3. Faculty of Science and Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku, Fukuoka, 813-8503, Japan

    Hidetaka Hamada

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Chen, S., Hamada, H. On (Fejér-)Riesz type inequalities, Hardy–Littlewood type theorems and smooth moduli. Math. Z. 305, 64 (2023). https://doi.org/10.1007/s00209-023-03392-6

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