Some sharp weak-type inequalities for holomorphic functions on the unit ball of $\textbf {C}^ n$

Abstract:

Let ${B^n} = \{ z \in {{\mathbf {C}}^n}:|z| < 1\}$, ${S^n} = \partial {B^n}$ and let ${\sigma _n}$ be the Haar measure on ${S^n}$. Then for all $f \in {H^p}(1 \leqslant p < \infty )$ such that $\operatorname {Im} (f(0)) = 0$ and $t > 0$, \[ {\sigma _n}(\{ z \in {S^n}:|f(z)| \geqslant t\} ) \leqslant {C_p} \cdot \frac {{||\operatorname {Re} f||_p^p}} {{{t^p}}}\] for some constant ${C_p}$ depending only on $p$. The best constant ${C_p}$ is found for $1 \leqslant p \leqslant 2$.