Division of holomorphic functions and growth conditions (Q470843)

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 \(D\).