Division of holomorphic functions and growth conditions

Abstract

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.