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 $\Vert f{\Vert }_{p,z_0}\le C_p\Vert u{\Vert }_{p,z_0}$ for $1<p<\infty$, where $C_p$ is a constant depending only on $p$ and $\Vert u{\Vert }_{p,z_0}^p$ 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.