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We prove the $L^p$ boundedness of the Marcinkiewicz integral operators ${\mu }_{\Omega }$ on ${ℝ}^{n₁}\times \cdots \times {ℝ}^{n_k}$ under the condition that $\Omega \in L{\left(logL\right)}^{k/2}{(}^{n₁-1}\times \cdots {\times }^{n_k-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.