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

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