Illinois Journal of Mathematics

Division of holomorphic functions and growth conditions

William Alexandre and Emmanuel Mazzilli

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Let $D$ be a strictly convex domain of $\mathbb{C}^{n}$, $f_{1}$ and $f_{2}$ be two holomorphic functions defined on a neighbourhood of $\overline{D}$ and set $X_{l}=\{z,f_{l}(z)=0\}$, $l=1,2$. Suppose that $X_{l}\cap bD$ is transverse for $l=1$ and $l=2$, and that $X_{1}\cap X_{2}$ is a complete intersection. We give necessary conditions when $n\geq2$ and sufficient conditions when $n=2$ under which a function $g$ can be written as $g=g_{1}f_{1}+g_{2}f_{2}$ with $g_{1}$ and $g_{2}$ in $L^{q}(D)$, $q\in[1,+\infty)$, or $g_{1}$ and $g_{2}$ in $\operatorname{BMO}(D)$. In order to prove the sufficient condition, we explicitly write down the functions $g_{1}$ and $g_{2}$ using integral representation formulae and new residue currents.

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Illinois J. Math., Volume 57, Number 3 (2013), 629-664.

First available in Project Euclid: 3 November 2014

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Primary: 32A22: Nevanlinna theory (local); growth estimates; other inequalities {For geometric theory, see 32H25, 32H30} 32A26: Integral representations, constructed kernels (e.g. Cauchy, Fantappiè- type kernels) 32A27: Local theory of residues [See also 32C30] 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] 32A40: Boundary behavior of holomorphic functions 32A55: Singular integrals


Alexandre, William; Mazzilli, Emmanuel. Division of holomorphic functions and growth conditions. Illinois J. Math. 57 (2013), no. 3, 629--664. doi:10.1215/ijm/1415023504.

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