A note on composition operators acting on holomorphic Sobolev spaces

Boo Rim Choe, Hyungwoon Koo, Wayne Smith
2011 Proceedings of the American Mathematical Society  
A holomorphic self-map ϕ of the unit disk is constructed such that the composition operator induced by ϕ is bounded on the Hardy-Sobolev space H 1 2 of order 2 as well as on the ordinary holomorphic Lipschitz space Lip 1 but unbounded on the Zygmund class Λ 1 . Among these three function spaces we have embedding relations H 1 2 ⊂ Lip 1 ⊂ Λ 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
more » ... all the function spaces involved are standard ones.
doi:10.1090/s0002-9939-2011-10944-4 fatcat:zai543sslfe4vgup72v6ntwbfe