In the second chapter we study the properties of the weighted dirichlet - type spaces and composition operators on them . we not only characterize these spaces by taylor series , but also give sufficient and necessary conditions in terms of carleson measure for the boundedness and compactness of composition operator . moreover , we apply the comparability propositions which induced from above sufficient and necessary conditions to discuss the relationships between the compactness of composition operator cv and angular derivative or innerness of the inducing function , and so on 本文討論了一類函數空間相互的包含關系,而對其中的加權dirichlet型空間,不但給出了空間的相互包含關系,并且對它進行了級數刻畫;利用carleson測度,刻畫了加權dirichlet型空間上復合算子的有界性及緊性,并利用由此得到的比較性命題討論了復合算子的緊性與角導數,內函數等的關系。