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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.

On characterizations of Bloch spaces and Besov spaces of pluriharmonic mappings