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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: NORM EQUIVALENCE AND COMPOSITION OPERATORS BETWEEN BLOCH/LIPSCHITZ SPACES OF THE BALL | NORM EQUIVALENCE AND COMPOSITION OPERATORS BETWEEN BLOCH LIPSCHITZ SPACES OF THE BALL DANA D. CLAHANE AND STEVO STEVIC Received 11 October 2005 Revised 30 January 2006 Accepted 12 February 2006 For p 0 let p Bn and ẩp Bn denote respectively the p-Bloch and holomorphic p-Lipschitz spaces of the open unit ball B in C . It is known that p Bn and ẩ1 -p B are equal as sets when p e 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 Cộ from ẩp Bn to ẩ Bn . Copyright 2006 D. D. Clahane and S. Stevic. 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 e N and suppose that D is a domain in C . Denote the linear space of complexvalued holomorphic functions on D by D . If is a linear subspace of D and ộ D - D is holomorphic then one can define the linear operator Cộ D by Cộ f f ộ for all f e . Cộ is called the composition operator induced by ộ. The problem of relating properties of symbols ộ and operators such as Cộ 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 B in C that induce bounded composition operators between holomorphic p-Lipschitz spaces ẩp Bn for 0 p 1 by translating the problem to 1 - p -Bloch spaces 1-p Bn via an auxiliary Hardy Littlewood-type norm-equivalence result of potential .
