ON HARMONIC BLOCH SPACES IN THE UNIT BALL OF ℂⁿ

Abstract

In this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball B n . First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from B n to C in terms of weighted Lipschitz functions. Then we prove the existence of a Landau-Bloch constant for a class of vector-valued harmonic Bloch mappings from B n to C n .2010 Mathematics subject classification: primary 30C65; secondary 30C45, 30C20.

Lipschitz-Type and Bloch-Type Spaces of Pluriharmonic Mappings in a Hilbert Space

“…For the related results of harmonic functions, we refer to [4,[12][13][14] and the references therein.…”

On the Harmonic Zygmund Spaces

“…While harmonic mappings have been studied extensively in a wide variety of settings, particularly analysing the function-theoretic aspects, the study of the harmonic extensions of classical Banach spaces of analytic functions has begun fairly recently and is still in a phase of development. Besides the aforementioned papers [1,6], references focusing on spaces of harmonic mappings in one and several complex variables include [3,4,12]. The study of composition operators on several harmonic weighted-type spaces, including the harmonic Zygmund space, is carried out in [2].…”

Characterizations of Bloch-Type Spaces of Harmonic Mappings

“…Besides [3], references in the unit disk setting include [5] for the study of harmonic Bloch and Besov spaces, [6] for the study of harmonic ]-Bloch mappings, [7] on planar harmonic Lipschitz and Hardy classes, and [8] for the study of harmonic Lipschitz-type spaces. In the setting of the unit ball in C , see [9] for the study of the harmonic Bloch spaces, [10,11] for the study of the harmonic Bergman spaces and [12] for extensions of the main results in [5]. For a general reference on harmonic mappings in the plane we refer the interested reader to [13].…”