A NOTE ON COMPOSITION OPERATORS ACTING ON HOLOMORPHIC SOBOLEV SPACES

Abstract

A holomorphic self-map phi of the unit disk is constructed such that the composition operator induced by phi is bounded on the Hardy-Sobolev space H-2(1) of order 2 as well as on the ordinary holomorphic Lipschitz space Lip(1) but unbounded on the Zygmund class A(1). Among these three function spaces we have embedding relations H-2(1) subset of Lip(1) subset of A(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.