The theorems of M. Riesz and Zygmund in several complex variables

TL;DR

This paper extends classical theorems of Riesz and Zygmund to several complex variables, establishing bounds and integrability conditions for holomorphic functions in higher dimensions and on hyperconvex domains.

Contribution

It generalizes fundamental conjugate function theorems to multiple complex variables and introduces new bounds and integrability results in this broader setting.

Findings

Established $L^p$ bounds for conjugate functions in several complex variables.

Derived exponential integrability estimates for bounded functions on hyperconvex domains.

Extended results to Poletsky-Stessin-Hardy spaces.

Abstract

In this note, we extend the well-known theorems of M. Riesz and Zygmund on conjugate functions as follows. Let $\Omega$ be a domain in $\mathbb C^n$. Suppose that $f=u+iv\in \mathcal O(\Omega)$ satisfies $v(z_0)=0$ for some $z_0\in \Omega$. Then $ \|f\|_{p,z_0} \le C_p\, \|u\|_{p,z_0}$ for $1<p<\infty$, where $C_p$ is a constant depending only on $p$ and $\|u\|_{p,z_0}^p$ is defined to be the value at $z_0$ of the least harmonic majorant of $|u|^p$. Moreover, if $|u|\le 1$, then for any $\alpha>1$, there exists $C_\alpha>0$ such that $ \int_{\partial \Omega_t} \frac{\exp\left(\frac{\pi}2 |f| \right)}{(1+|f|)^\alpha}\, d\omega_{z_0,t} \le C_\alpha$ for any exhaustion $\{\Omega_t\}$ of $\Omega$ with $\Omega_t\ni z_0$, where $d \omega_{z_0,t}$ is the harmonic measure of $\Omega_t$ relative to $z_0$. Analogous results for Poletsky-Stessin-Hardy spaces on hyperconvex domains are given.