BOUNDEDNESS OF COMPOSITION OPERATORS ON Dt SPACES FOR τ > 1

Abstract

In this note, some conditions of composition operators on Dτ spaces to be bounded are given by means of Carleson measures and pointwise multipliers, for some ranges of τ. The authors prove that (i) Let 1<τ≤n+2 and 2k<τ≤2k+1 (or 2k-1<τ≤2k) for some positive integer k. Suppose φ=(φ1,⋯,φn) be a univalent mapping from B into itself, denote dμjl(z)=|R(l)φj(z)|2(k-1+2)(1-|z|2)2k-τ+1dv(z) for l=1,2,⋯,k+1. If μj(l)φ-1 are (τ-2k+2l-4)-Carleson measures for all then the composition operator Cφ on Dτ is bounded; (ii) Let 1<τ≤n+2, φ=(φ1⋯φn) be univalent and the Fréchet derivative of φ-1 be bounded on φ(B). If Rφj∈M(Dτ-2) for all j, then the composition operator Cφ on Dτ is bounded; (iii) Let τ >n + 2 and φ as in (ii). If φj∈Dτ for all j, then the composition operator Cφ on Dτ is bounded.