Divison problem in generalized growth spaces on the unit ball in \({\mathbb C}^n\) (Q2799669)

The authors solve the ``baby'' corona problem for certain growth spaces of holomorphic functions on the unit ball \(\mathbb{B}\) of \(\mathbb{C}^n\).NEWLINENEWLINEA positive function \(\omega\) on \((0,1]\) is almost increasing (resp., decreasing) if there exists a constant \(C>0\) such that \(\omega(s)\leq C\omega(t)\) (resp., \(\omega(s)\geq C\omega(t)\)) for all \(0<s<t\leq 1\). It is of order \(\alpha\) if NEWLINE\[NEWLINE \alpha=\sup\left\{\gamma:\,\frac{\omega(t)}{t^{\gamma}\,} \,\,\text{is almost increasing}\right\} =\inf\left\{\delta:\,\frac{\omega(t)}{t^{\delta}} \,\,\text{is almost decreasing}\right\}. NEWLINE\]NEWLINENEWLINENEWLINEFor a positive weight \(\omega\) on \((0,1]\), let NEWLINE\[NEWLINE A^{\omega}(\mathbb{B})=\left\{f\in \mathrm{Hol}(\mathbb{B}):\,\sup_{z\in \mathbb{B}} |f(z)|/\omega(1-|z|^2)<\infty\right\}. NEWLINE\]NEWLINENEWLINENEWLINEThe main result of this paper is the following division theorem:NEWLINENEWLINELet \(g_1,\ldots, g_m\in H^\infty(\mathbb{B})\) satisfy the corona condition \(\sum_{j=1}^m |g_j|^2\geq \delta^2>0\). Let \(\omega\) be a weight of order \(\alpha\in(-1, 0]\) and let \(f\in A^\omega(\mathbb{B})\).NEWLINENEWLINEThen there exist \(u_1,\ldots,u_m\in A^\eta(\mathbb{B})\) such that \( \sum_{j=1}^m g_j\,u_j=f, \) where \(\eta(t)=\omega(t) \) if \(-1<\alpha<0\), and \(\eta(t)=\omega(t)\log(1/t)\) if \(\alpha=0\) and \(\omega\) is almost decreasing.