加权复合算子在Dirichlet空间上的有界性

Boundedness of Weighted Composition Operators on Dirichlet Spaces

Dirichlet空间D^p(0<p<∞)是经典Dirichlet空间D^2的一种自然推广.设函数φ和Ф是Δ上的解析函数且φ(Δ)Δ,则将加权复合算子定义为Wφ,Ф:f■Ф(fφ).当2<q≤p<∞时,本文给出了D^p中元素的基于有限零点的分解并得到了当φ是Δ上的共形满射时该加权复合算子W_(φ,Ф)的有界性的充要条件.

D^p（0〈p〈∞）is the generalization of the classical Dirichlet space D^2. Let φ and Ф be two analytic functions defined on △ such that φ（Δ）Ф Δ, then the weighted composition operator is the operator defined by Wφ,Ф：f→Ф（f^°φ）. When2〈q≤p〈∞, the faetorizafions are given for the elements of D^p based on the zeros and the necessary and sufficient conditions for the boundedness of Wφ,Ф are obtained when φ is a conformal mapping of △ onto itself.