Superposition operators, Hardy spaces, and Dirichlet type spaces

For 0−1 the space of Dirichlet type Dαp consists of those functions f which are analytic in the unit disc D and satisfy ∫D(1−|z|)α|f′(z)|pdA(z)<∞. The space Dp−1p is the closest one to the Hardy space Hp among all the Dαp. Our main object in this paper is studying similarities and differences between the spaces Hp and Dp−1p (0