Zero sets of holomorphic functions in the unit ball: non-radial growth characteristics

Abstract

Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint $|f|\leq \exp M$ on $\mathbb B$, where $M\not\equiv \pm \infty$ is $\delta$-subharmonic function on $\mathbb B$ with Riesz charge $\mu_M$. We give a scale of integral uniform constraints from above on the distribution of the set ${\sf Z}$ via the charge $\nu_M$ in terms of $(2n-2)$-Hausdorff measure of the set $\sf Z$, as well as test convex radial functions and $\rho$-subspherical functions on the unit sphere $\mathbb S \subset \mathbb C^n$, which at $n=1$ can be interpreted as $2\pi$-periodic $\rho$-trigonometrically convex functions on the real axis $\mathbb R \subset \mathbb C$.