Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball

Abstract

For p > 0, let B p (B n ) and L p (B n ) respectively denote the p-Bloch and holomorphic p-Lipschitz spaces of the open unit ball B n in C n . It is known that B p (B n ) and L 1−p (B n ) are equal as sets when p ∈ (0, 1). We prove that these spaces are additionally norm-equivalent, thus extending known results for n = 1 and the polydisk. As an application, we generalize work by Madigan on the disk by investigating boundedness of the composition operator C φ from L p (B n ) to L q (B n ).

Weighted Composition Operators between Mixed Norm Spaces and Spaces in the Unit Ball

“…Recall that the β-Bloch space Ꮾ β (B) = Ꮾ β is the space of all f ∈ H(B) such that [6]), and the little β-Bloch space…”

Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces

“…J g C ϕ (or I g C ϕ ) : H(p, q, φ) → B w (or B w 0 ) Using the same methods as in Theorems 1-4, we can obtain the following theorems concerning the operators in (2). Since the proofs are similar to those of Theorems 1-4, they will be omitted.…”

“…It makes B w into a Banach space. When w(z) = (1 − |z| 2 ) α , α > 0, then B w = B α is the well-known α-Bloch space (e.g., see [1][2][3][4][5]). …”

On an Integral Operator From the Zygmund Space to the Bloch-Type Space on the Unit Ball

“…For the case of μ(z) = (1 − |z| 2 ), see, e.g., [4]. Also, it was proved that the little Bloch-type space is equivalent with the subspace of B μ consisting of all f ∈ H(‫)ނ‬ such that…”