Division of holomorphic functions and growth conditions
type: Item
instance of: scholarly article
title: Division of holomorphic functions and growth conditions
zbMATH Open document ID: 1307.32007
author: William Alexandre
author: Emmanuel Mazzilli
published in: Illinois Journal of Mathematics
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.1565
full work available at URL: https://projecteuclid.org/euclid.ijm/1415023504
reviewed by: Joan Fàbrega
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 32A40
zbMATH DE Number: 6369184
zbMATH Keywords: strictly convex domains in \(\mathbb C^n\)
zbMATH Keywords: bounded holomorphic functions
zbMATH Keywords: division of holomorphic functions
zbMATH Keywords: BMOA-space
zbMATH Keywords: Bergman space
MaRDI profile type: Publication
arXiv ID: 1207.1565
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cites work: Currents of residual type associated with a complete intersection
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cites work: Estimates on the Bergman kernels of convex domains
cites work: BMO functions and the \(\overline\partial\)-equation
review text: Let \(D\) be a strictly convex domain in \(\mathbb{C}^n\), \(n\geq 2\), with smooth boundary. Let \(f_1\) and \(f_2\) be holomorphic functions on a neighbourhood of \(\overline{D}\) such that the sets \(X_j=\{z: f_j(z)=0\}\), \(j=1,2\), satisfy \(X_j\cap bD\) is transverse and \(X_1\cap X_2\) is a complete intersection.NEWLINENEWLINEThe authors prove that if \(g_1, g_2\in H^\infty(D)\), then the function \(g=f_1g_1+f_2g_2\) satisfies the obvious condition \(|g(z)|/\max\{|f_1(z)|,|f_2(z)|\}\in L^\infty(D)\) and the divided differences of the functions \(g/f_1\) on \(X_2\setminus X_1\) and \(g/f_2\) on \(X_1\setminus X_2\) satisfy certain growth conditions. Conversely, if \(n=2\) and \(g\) is a function in the ideal of \(\mathcal{O}(D)\) generated by \(f_1\) and \(f_2\) satisfying the above necessary conditions, then there exist \(g_1, g_2\in BMOA(D)\) such that \(g=f_1g_1+f_2g_2\).NEWLINENEWLINEThe paper also contains a similar version of these results for functions \(g_1\) and \(g_2\) in the Bergman space on \(D\).
description: scientific article; zbMATH DE number 6369184