In this paper, we show that the Möbius invariant function space Qp can be generated by variant Dirichlet type spaces Dμ, 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 Dμ, p is given. We obtain a sufficient condition to construct different Dμ, p spaces and provide examples. We establish decomposition theorems for Dμ, p spaces and prove that the non-Hilbert space Qp is equal to the intersection of Hilbert spaces Dμ, p. As an application of the relation between Qp and Dμ, p spaces, we also obtain that there exist different Dμ, p spaces; this is a trick to prove the existence without constructing examples.
- Dirichlet type space
- Möbius invariant function space
- Qp space