Bounded composition operators on Hardy space and Bergman space

Let Ω1 and Ω2 be bounded domains in the complex plane C and ϕ:Ω1→Ω2 be a holomorphic map. When Ω1=Ω2=D, the unit disk, the composition operator Cϕ is always bounded on Hardy space Hp(D) and weighted Bergman space Aαp(D) over D for α>−1. In this paper, we initially generalize these results by investigating the boundedness of Cϕ:Hp(Ω2)→Hp(Ω1) and Cϕ:Aαp(Ω2)→Aαp(Ω1) for Ωj(j=1,2), which have strong Lipschitz boundaries. Such a collection of these domains encompasses those with C1,α boundaries. Furthermore, we present a counterexample involving a special bounded domain Ω where the composition operator Cϕ:Ap(Ω)→Ap(D) fails to be bounded. Although this Ω possesses Cα boundary for any α