Weights induced by homogeneous polynomials (Q1101852)

Let \(B\) be the unit ball and \(S\) the unit sphere in \({\mathbb{C}}^ n\) \((n\geq 2)\). Let \(\sigma\) be the unique normalized rotation-invariant Borel measure on \(S\) and \(m\) the normalized area measure on \({\mathbb{C}}.\) We first prove that if \(\Lambda\) is a holomorphic homogeneous polynomial on \({\mathbb{C}}^ n \)normalized so that \(\Lambda\) maps \(B\) onto the unit disk \(U\) in \({\mathbb{C}}\) and if \(\mu =\sigma [(\Lambda |_ S)^{-1}]\), then \(\mu \ll m\) and the Radon-Nikodym derivative \(d\mu\) /dm is radial and positive on \(U\). Then we obtain the asymptotic behavior of \(d\mu\) /dm for a certain, but not small, class of functions \(\Lambda\). These results generalize two recent special cases of P. Ahern and P. Russo. As an immediate consequence we enlarge the class of functions for which Ahern-Rudin's Paley-type gap theorems hold.