Zeros of Holomorphic Functions in the Unit Ball and Subspherical Functions

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 B in ℂ n . Let f be a nonzero holomorphic function on B such that f vanishes...

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Bibliographic Details
Published in:Lobachevskii journal of mathematics Vol. 40; no. 5; pp. 648 - 659
Main Authors: Khabibullin, B. N., Khabibullin, F. B.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01.05.2019
Springer Nature B.V
Subjects:
Algebra
Analysis
Analytic functions
Complex variables
Geometry
Harmonic functions
Mathematical analysis
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
uniqueness theorem
Hausdorff measure
holomorphic function
zero set
subspherical function
subharmonic function
Riesz measure
ISSN:1995-0802, 1818-9962
Online Access:Get full text
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Summary: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 B in ℂ n . Let f be a nonzero holomorphic function on B such that f vanishes on Z ⊂ B , and satisfies the inequality ∣ f ∣ ≤ exp M on 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 (2 n − 2)-Hausdorff measure of zero subset Z, and special non-radial test subharmonic functions constructed using ρ -subspherical functions.
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ISSN:1995-0802
1818-9962
DOI:10.1134/S199508021905010X