关于S^p空间上加权复合算子的有界性及嵌入映射的紧性

On the Boundedness of Weighted Composition Operators on S^p Spaces and the Compactness of Inclusion Maps

S^p(1≤p≤∞)空间为导数属于Hardy空间H^p的复平面单位圆盘D上所有解析函数组成的空间.令函数φ和φ是D上的解析函数且φ(D)■D,则将算子W_(φ,φ):f→φf■φ称为加权复合算子.文章给出了当1≤q≤p≤∞,φ∈S~∞时,加权复合算子W_(φ,φ)从空间S^p到S^q上的有界性的充要条件.然后通过推广经典的Fejer-Riesz不等式证明了当1<p≤∞时,S^p到圆盘代数A上的嵌入映射是紧的.

S^p（1≤p≤∞） is the space of all analytic functions on the unit disk D whose derivatives belong to the Hardy space H^p. Let φ and φ be two analytic functions defined on D such that φ（D） D, then the operator defined by W（φ,φ）： f→φfoφis called the weighted composition operator. In this paper, we give the necessary and sufficient conditions for the boundedness of W（φ,φ） from S^p into Sq when 1≤q≤p≤∞,φ∈S^∞. Then, by generalizing the classical Fejer-Riesz inequality, we prove that the inclusion map of S^p into the disk algebra A is compact when 1 p≤∞.