Universal Teichmüller space and BMO

Holomorphic contractibility and other properties of the Weil-Petersson and VMOA Teichmüller spaces

“…To prove µ ∈ M v (D), we recall two results from [23] and [4]. Keep in mind that CM 0 (D) denotes the collection of vanishing Carleson measures on D. …”

“…In this paper, we use the following characterization of holomorphy ( [19], [23]): a continuous map f : U → Y is holomorphic if for each (u, x) ∈ (U, X) and each element y * from a total subset Y * 0 of the dual space Y * , y * (f (u + tx)) is a holomorphic function in a neighborhood of zero in the complex plane. Here a subset Y * 0 of Y * is total if y * (y) = 0 for all y * ∈ Y * 0 implies y = 0.…”

“…is also a group under the composition [23]. For more studies of T v (D), we refer to [1], [2] and [20].…”

“…In fact, the topology induced by · 2 (resp. [23]). Therefore, it is not obvious that the inclusion map from…”

“…These subspaces of T (D) are introduced for different purposes: T 0 (D) is used to define the asymptotic Teichmüller space of D; T 2 (D) plays an important role in the study of the Weil-Petersson geometry of the universal Teichmüller space [16,22,26]; T v (D) has applications in harmonic analysis (see [1], [23] and references therein). In this paper, we study complex properties of…”