On the Harmonic Zygmund Spaces

Abstract

In this paper we study a class ${\mathcal{Z}}_{H}$ of harmonic mappings on the open unit disk $\mathbb{D}$ in the complex plane that is an extension of the classical (analytic) Zygmund space. We extend to the elements of this class a characterisation that is valid in the analytic case. We also provide a similar result for a closed separable subspace of ${\mathcal{Z}}_{H}$ which we call the little harmonic Zygmund space.

Composition Operators on Some Banach Spaces of Harmonic Mappings

“…In [5], it is shown that the elements of the space Z H can be characterized in terms of the membership to the classical Zygmund class, and the corresponding norms are equivalent.…”

“…In [2], the first author pursued this study by extending several classes of Banach spaces, including the Bloch space B and its generalizations B α known as α-Bloch spaces introduced by Zhu in [3], the growth spaces A − α (where α > 0), the Zygmund space Z, and the analytic Besov spaces B p for p > 1. In particular, the linear structure and properties of the harmonic α-Bloch spaces B α H , the harmonic growth spaces A − α H (for α > 0), and the harmonic Zygmund space Z H were studied in [4,5]. e harmonic Besov spaces B p H for p > 1 were introduced in [2].…”

On Characterizations of Weighted Harmonic Bloch Mappings and Its Carleson Measure Criteria

“…In the last few decades, the Banach spaces of analytic functions on D have been gaining a great deal of attention, but for the harmonic extensions of analytic spaces, it is still limited. Besides [1] by F. Colonna, papers such as [2] for the study of the operator theory on some spaces of harmonic mappings, [3] for characterizations of Bloch-type spaces of harmonic mappings, [4] for composition operators on some Banach spaces of harmonic mappings, [5] for the study of harmonic Bloch and Besov spaces, [6] for the study harmonic Zygmund spaces, [7] for the study of harmonic ν-Bloch mappings and [8] for the study of harmonic Lipschitz-type spaces. For α > 0, α-Bloch space for harmonic mapping is defined such that…”

The Möbius Invariant QTH Spaces

“…for characterization of Bloch type spaces of harmonic mapping, see [6], for harmonic zygmund spaces.…”