Zygmund type spaces of harmonic functions on the real unit ball

Abstract

In this paper, we obtain several characterizations of harmonic Zygmund type spaces on the real unit ball $\mathbb{B}$ of ${\mathbb{R}}^n$. First, we characterize the harmonic Zygmund space ${\mathcal{H}\mathcal{Z}}^{\alpha }$ ($0<\alpha \le 2$) in terms of Zygmund type conditions. Our result extends Theorem 3.4 of [M. Aljuaid and F. Colonna, On the harmonic Zygmund spaces, Bull. Aust. Math. Soc. 101 (2020), 466--476.] to the setting of the real unit ball $\mathbb{B}$. Second, we establish an analogue of the Holland-Walsh characterization of the harmonic Zygmund space $\mathcal{H}\mathcal{Z}$. Finally, an integral characterization of ${\mathcal{H}\mathcal{Z}}^{\alpha }$ is also discussed. All obtained results can be viewed as counterparts of the known results for Bloch spaces.

17 pages