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

Here we introduce the ð ‘› th weighted space on the upper half-plane Î + = { 𠑧 ∈ â„‚ ∶ I m 𠑧 > 0 } in the complex plane â„‚ . For the case ð ‘› = 2 , we call it the Zygmund-type space, and denote it by ð ’µ ( Î + ) . The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator ð ¶ ð œ‘ ð ‘“ ( 𠑧 ) = ð ‘“ ( 𠜑 ( 𠑧 ) ) from the Hardy space ð » ð ‘ ( Î + ) on the upper half-plane, to the Zygmund-type space, where 𠜑 is an analytic self-map of the upper half-plane.