Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball. (English) Zbl 1131.47018

Clahane, Dana D.; Stević, Stevo

J. Inequal. Appl. 2006, Article ID 61018, 11 p. (2006).

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 K. M. 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 \({\mathcal L}_q(\mathbb B_n)\).

MSC:

47B33	Linear composition operators
32A37	Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

\(p\)-Bloch space; \(p\)-Lipschitz space; boundedness; composition operator

Citations:

Zbl 0793.47037

References:

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