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 $\mathsf{Z}\subset \mathbb{B}$ and satisfies the constraint $|f|\le \exp M$ 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 $\mathsf{Z}$ via the charge ${\nu }_M$ in terms of $(2n-2)$-Hausdorff measure of the set $\mathsf{Z}$, 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 $\mathbb{R}\subset \mathbb{C}$.

14 pages; in Russian