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Abstract and Applied Analysis
Volume 2009 (2009), Article ID 161528, 8 pages
Research Article

Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane

Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11001 Beograd, Serbia

Received 14 December 2008; Accepted 23 February 2009

Academic Editor: Simeon Reich

Copyright © 2009 Stevo Stević. 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.


Here we introduce the 𝑛 th weighted space on the upper half-plane Ξ  + = { 𝑧 ∈ β„‚ ∢ I m 𝑧 > 0 } in the complex plane β„‚ . For the case 𝑛 = 2 , we call it the Zygmund-type space, and denote it by 𝒡 ( Ξ  + ) . The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator 𝐢 πœ‘ 𝑓 ( 𝑧 ) = 𝑓 ( πœ‘ ( 𝑧 ) ) from the Hardy space 𝐻 𝑝 ( Ξ  + ) on the upper half-plane, to the Zygmund-type space, where πœ‘ is an analytic self-map of the upper half-plane.