Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball
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For p > 0, let Ꮾ p (Bn ) and ᏸ p (Bn ) denote, respectively, the p-Bloch and holomorphic p-Lipschitz spaces of the open unit ball Bn in Cn . It is known that Ꮾ p (Bn ) and ᏸ1− p (Bn ) are equal as sets when p ∈ (0,1). We prove that these spaces are additionally normequivalent, thus extending known results for n = 1 and the polydisk. As an application, we generalize work by Madigan on the disk by investigating boundedness of the composition operator Cφ from ᏸ p (Bn ) to ᏸq (Bn ). Copyright © 2006 D. D. Clahane and S. Stevi´c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Background and terminology Let n ∈ N, and suppose that D is a domain in Cn . Denote the linear space of complexvalued, holomorphic functions on D by Ᏼ(D). If ᐄ is a linear subspace of Ᏼ(D) and φ : D → D is holomorphic, then one can define the linear operator Cφ : ᐄ → Ᏼ(D) by Cφ ( f ) = f ◦ φ for all f ∈ ᐄ. Cφ is called the composition operator induced by φ. The problem of relating properties of symbols φ and operators such as Cφ that are induced by these symbols is of fundamental importance in concrete operator theory. However, efforts to obtain characterizations of self-maps that induce bounded composition operators on many function spaces have not yielded completely satisfactory results in the several-variable case, leaving a wealth of basic open problems. In this paper, we try to make progress toward the goal of characterizing the holomorphic self-maps of the open unit ball Bn in Cn that induce bounded composition operators between holomorphic p-Lipschitz spaces ᏸ p (Bn ) for 0 < p < 1 by translating the problem to (1 − p)-Bloch spaces Ꮾ1− p (Bn ) via an auxiliary Hardy/Littlewood-type norm-equivalence result of potential independent interest. This method was also used in [7] for B1 and in [3] for the unit polydisk Δn . The function-theoretic characterization of analytic self-maps of B1 that induce bounded composition operators on ᏸ p (B1 ) for 0 < p < 1 is due to Madigan [7], and the case of Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 61018, Pages 1–11 DOI 10.1155/JIA/2006/61018
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Norm equivalence and composition operators
Δn was handled in a joint paper by the present authors with Zhou [3], in which a full characterization of the holomorphic self-maps φ of Δn that induce bounded composition operators between ᏸ p (Δn ) and ᏸq (Δn ), and, more generally, between Bloch spaces Ꮾ p (Δn ) and Ꮾq (Δn ), is obtained for p, q ∈ (0,1), along with analogous characterizations of compact composition operators between these spaces. Although our main results concerning composition operators, Theorem 2.1 and Corollary 2.2, are not full characterizations, they do generalize Madigan’s result for the disk to Bn ; on the other hand, we obtain a complete Hardy-Littlewood norm-equivalence result for p-Bloch and (1 − p)-Lipschitz spaces of Bn fo
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