Abstract
A holomorphic self-map φ \varphi of the unit disk is constructed such that the composition operator induced by φ \varphi is bounded on the Hardy-Sobolev space H 2 1 H^1_2 of order 2 2 as well as on the ordinary holomorphic Lipschitz space Lip 1 \textrm {Lip}_1 but unbounded on the Zygmund class Λ 1 \Lambda _1 . Among these three function spaces we have embedding relations H 2 1 ⊂ Lip 1 ⊂ Λ 1 H^1_2\subset \textrm {Lip}_1\subset \Lambda _1 . So, the main points here are that our construction provides a composition operator which is bounded on smaller spaces, but not on a larger space and that all the function spaces involved are standard ones.
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