Q_p spaces and Dirichlet type spaces

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