Distance of a Bloch Function to the Little Bloch Space

Abstract

Motivated by a formula of P. Jones that gives the distance of a Bloch function to BMOA, the space of bounded mean oscillations, we obtain several formulas for the distance of a Bloch function to the little Bloch space, B o . Immediate consequences are equivalent expressions for functions in Bo-We also give several examples of distances of specific functions to Bo. We comment on connections between distance to Bo and the essential norm of some composition operators on the Bloch space, B. Finally we show that the distance formulas in B have Bloch type spaces analogues.

Duality and distance formulas in spaces defined by means of oscillation

“…We have thus verified the condition of Theorem 2.4 for Z = M 0 (X, L) * , proving the corollary. This improves a result previously obtained in [4] and [27]. Corollary 1.5 says furthermore that the the Bloch space has a unique predual, reproducing a result found in [23].…”

“…For the Bloch space B and little Bloch space B 0 the situation is similar. B * * 0 = B and the distance from f ∈ B to B 0 has been characterized by Attele [4] and Tjani [27]. Numerous people have explored the validity of the biduality Hv 0 (Ω) * * = Hv(Ω), where Hv(Ω) is a weighted space, consisting of analytic functions bounded under the weighted supremum norm given by a weight function v on Ω ⊂ C, and Hv 0 (Ω) denotes the corresponding little space (see Example 4.4 for details).…”

Embedding Theorems for Dirichlet Type Spaces

“…The following lemma, which characterized the distance from the Bloch function to the little Bloch space was proved by Attele [3] and Tjani [24]. Here and afterward, we denote g r (z) = g(rz) with 0 < r < 1.…”

Distances from Bloch Functions to Q K -Type Spaces

“…(2) If p 2 > p 1 > 1, K satisfies conditions (a) and (b), then ( For the distances from Bloch functions to the space Q K,0 (p, p − 2), we have the following theorem which can be compared with a similar result in [5]. Theorem 2.…”