On Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli

Abstract: The purpose of this paper is to develop some methods to study 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 (\textit{Trans. Amer. Math. Soc.} {\bf 372} (2019)~ 4031--4051). Furthermore, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established. Additionally, we also discuss the Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions. Consequently, we improve and generalize the corresponding results in (\textit{Acta Math.} {\bf 178} (1997)~ 143--167) and (\textit{Adv. Math.} {\bf 187} (2004)~ 146--172).