Zygmund type spaces of harmonic functions on the real unit ball

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 $\mathcal{HZ}^\alpha$ ($0<\alpha\leq2$) 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 $\mathcal{HZ}$. Finally, an integral characterization of $\mathcal{HZ}^\alpha$ is also discussed. All obtained results can be viewed as counterparts of the known results for Bloch spaces.