Dirichlet and VMOA domains via Schwarzian derivative

Abstract

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