Qp Spaces and Dirichlet Type Spaces. Issue 4 (1st December 2017) - Bao, Guanlong; Gökhan Gögüs, Nihat; Pouliasis, Stamatis
Abstract: 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 Borel measures μ on the open unit disk. A criterion for the equality between the space D_μ, p and the usual Dirichlet type space D_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.