A note on composition operators acting on holomorphic Sobolev spaces

Abstract:

A holomorphic self-map $\varphi$ of the unit disk is constructed such that the composition operator induced by $\varphi$ is bounded on the Hardy-Sobolev space $H^1_2$ of order $2$ as well as on the ordinary holomorphic Lipschitz space $\textrm {Lip}_1$ but unbounded on the Zygmund class $\Lambda _1$. Among these three function spaces we have embedding relations $H^1_2\subset \textrm {Lip}_1\subset \Lambda _1$. 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.