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

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.