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

• Published in: Mathematische Zeitschrift Vol. 305; no. 4
• Main Authors: Chen, Shaolin, Hamada, Hidetaka
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2023; Springer Nature B.V
• Subjects: Harmonic functions; Inequalities; Mathematical analysis; Mathematics; Mathematics and Statistics; Theorems; 46E15; 31C10; Riesz type inequality; Primary 42A50; 42B30; Hardy–Littlewood type theorem; Smooth moduli; 32U05; Secondary 31B05
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-023-03392-6

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 C n , which extends the main result in Melentijević and Bo 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).