Hyperbolic mean growth of bounded holomorphic functions in the ball

Abstract:

We consider the hyperbolic Hardy class $\varrho H^{p}(B)$, $0<p<\infty$. It consists of $\phi$ holomorphic in the unit complex ball $B$ for which $\vert \phi \vert < 1$ and \begin{equation*}\sup _{0<r<1} \int _{\partial B} \left \{ \varrho (\phi (r\zeta ), 0)\right \}^{p} d\sigma (\zeta ) ~<~ \infty ,\end{equation*} where $\varrho$ denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type $g$-function and the area function are defined in terms of the invariant gradient of $B$, and membership of $\varrho H^{p}(B)$ is expressed by the $L^{p}$ property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator $\mathcal {C}_{\phi }$, defined by $\mathcal {C}_{\phi }f = f\circ \phi$, from the Bloch space into the Hardy space $H^{p}(B)$.