Characterizations of Bloch-Type Spaces of Harmonic Mappings

Munirah Aljuaid,Flavia Colonna

doi:10.1155/2019/5687343

Munirah Aljuaid, Flavia Colonna

Open Access

https://doi.org/10.1155/2019/5687343

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Abstract

We study the Banach space BHα (α>0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)<∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α>0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α>1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0<α<1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

Highlights

Given a region Ω in the complex plane C, a harmonic mapping with domain Ω is a complex-valued function h defined on Ω satisfying the Laplace equationΔh fl 4hzz ≡ 0 on Ω, (1)having denoted by hzz the mixed complex second partial derivatives of h.It is well known that a harmonic mapping h admits a representation of the form f + g, where f and g are analytic functions
We study the Banach space BαH (α > 0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D(1 −
Since analytic functions are clearly harmonic, a natural question is whether such spaces X are subspaces of some Banach space XH of harmonic mappings on D in such a way that the norm on the larger space agrees with the norm of X when restricting to the elements of X

Summary

Introduction

Given a region Ω in the complex plane C, a harmonic mapping with domain Ω is a complex-valued function h defined on Ω satisfying the Laplace equation. In the last several decades, much research has been carried out on the study of Banach spaces of analytic functions on the open unit disk D in the complex plane. A space that has been thoroughly studied in complex function theory is the classical Bloch space B defined as the set of analytic functions f on D such that βf fl sup z∈D This topic is treated in [14]

Preliminaries and Background

Harmonic Growth Spaces and α-Bloch Spaces

18. For α norms are