Dirichlet and VMOA domains via Schwarzian derivative

Short proofs of the following results concerning a bounded conformal map g of the unit disc D are presented: (1) logg′ belongs to the Dirichlet space if and only if the Schwarzian derivative Sg of g satisfies Sg(z)(1−2|z|)∈L2(D); (2) logg′∈VMOA if and only if 2|Sg(z)|3(1−2|z|) is a vanishing Carleson measure on D. Analogous results for Besov and Qp,0 spaces are also given.