Qp Spaces and Dirichlet Type Spaces

In this paper, we show that the Mobius invariant function space Q(p) can be generated by variant Dirichlet type spaces D-mu,D-p induced by finite positive Borel measures mu on the open unit disk. A criterion for the equality between the space D-mu,D-p and the usual Dirichlet type space D-p is given. We obtain a sufficient condition to construct different D-mu,D-p spaces and provide examples. We establish decomposition theorems for 9) D-mu,D-p spaces and prove that the non-Hilbert space Q(p) is equal to the intersection of Hilbert spaces D-mu,D-p. As an application of the relation between Q(p) and D-mu,D-p spaces, we also obtain that there exist different D-mu,D-p spaces; this is a trick to prove the existence without constructing examples.