Composition property of holomorphic functions on the ball (Q584461)

Let \(\phi\) be a homogeneous polynomial in \({\mathbb{C}}^ n\) such that \(\phi (B_ n)=B_ 1\) (here \(B_ k=\{z\in {\mathbb{C}}^ k:\) \(\| z\| <1\})\) and let g be a Bloch function in \(B_ 1.\) The author proves that \(g\circ \phi\) is in \(H^ p(B_ n)\) for every \(p\in (0,\infty)\). Moreover, if \(\phi\) is a homogeneous polynomial, then, as it is shown, one may find such an exponent \(\alpha =\alpha (\phi)\leq n-2\) that \(h\circ \phi \in H^ p(B_ n)\) for every \(p\in (0,\infty)\) and every \(h\in A(B_ 1)\) with the property \(\int_{B_ 1}| h|^ p(1-| z|^ 2)^{\alpha}dm<\infty.\)