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

TL;DR

This paper develops methods to study inequalities and theorems related to holomorphic, pluriharmonic, and harmonic functions in high-dimensional spaces, extending and generalizing previous results in the field.

Contribution

It introduces new sharp Riesz type inequalities and Hardy-Littlewood theorems for functions on high-dimensional domains, broadening existing mathematical frameworks.

Findings

Established sharp Riesz type inequalities for pluriharmonic functions on bounded symmetric domains.

Proved Hardy-Littlewood type theorems for holomorphic and pluriharmonic functions on John domains.

Generalized previous results on smooth moduli and inequalities for harmonic functions.

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).