On characterizations of Bloch spaces and Besov spaces of pluriharmonic mappings

Abstract

We characterize the Bloch spaces and Besov spaces of pluriharmonic mappings on the unit ball of ${\mathbb{C}}^{n}$ by using the following quantity: $\sup_{\rho(z,w)< r,z\neq w}\frac{(1-|z|^{2})^{\alpha}(1-|w|^{2})^{\beta}|\hat{D}^{(m)}f(z)-\hat {D}^{(m)}f(w)|}{|z-w|}$ , where $\alpha+\beta=n+1$ , $\hat{D}^{(m)}=\frac{\partial ^{m}}{\partial z^{m}}+\frac{\partial^{m}}{\partial\bar{z}^{m}}$ , $|m|=n$ . This generalizes the main results of (Yoneda in Proc. Edinb. Math. Soc. 45:229-239, 2002) in the higher dimensional case.