We consider the reducing subspaces of ${M}_{{z}^{N}}$ on ${A}_{\alpha}^{2}({\mathbb{D}}^{k})$, where $k\ge 3$, ${z}^{N}={z}_{1}^{{N}_{1}}\cdots {z}_{k}^{{N}_{k}}$, and ${N}_{i}\ne {N}_{j}$ for $i\ne j$. We prove that each reducing subspace of ${M}_{{z}^{N}}$ is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that $\alpha =0$ and $\alpha \in (-1,+\mathrm{\infty})\setminus \mathbb{Q}$, respectively. Finally, we give a complete description of minimal reducing subspaces of ${M}_{{z}^{N}}$ on ${A}_{\alpha}^{2}({\mathbb{D}}^{3})$ with $\alpha >-1$.

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