Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball (Q2471851)

For \(p>0\), let \({\mathcal B}^p(\mathbb B_n)\) and \({\mathcal L}_p(\mathbb B_n)\) denote, respectively, the \(p\)-Bloch and holomorphic \(p\)-Lipschitz spaces of the open unit ball \(\mathbb B_n\) in \(\mathbb C^n\). It is known that \({\mathcal B}^p(\mathbb B_n)\) and \({\mathcal L}_{1-p}(\mathbb B_n)\) are equal as sets when \(p\in(0,1)\). We prove that these spaces are additionally norm-equivalent, thus extending known results for \(n=1\) and the polydisk. As an application, we generalize work by \textit{K.\,M.\thinspace Madigan} [Proc.\ Am.\ Math.\ Soc.\ 119, No.\,2, 465--473 (1993; Zbl 0793.47037)] on the disk by investigating boundedness of the composition operator \({\mathfrak C}_\varphi\) from \({\mathcal L}_p(\mathbb B_n)\) to \({\mathcal L}_q(\mathbb B_n)\).