Zeros of Holomorphic Functions in the Unit Ball and Subspherical Functions

Abstract

We continue our previous results from the functions of one complex variable in the unit disk to the functions of several variables in the unit ball. Let M be a δ-subharmonic function with Riesz charge µ_M on the unit ball \(\mathbb{B}\) in ℂⁿ. Let f be a nonzero holomorphic function on \(\mathbb{B}\) such that f vanishes on Z ⊂ \(\mathbb{B}\), and satisfies the inequality ∣f∣ ≤ exp M on \(\mathbb{B}\). Then restrictions on the growth of µ_M near the boundary of B imply certain restrictions on the distribution of Z. We give a quantitative study of this phenomenon in terms of (2n − 2)-Hausdorff measure of zero subset Z, and special non-radial test subharmonic functions constructed using ρ-subspherical functions.