Let $M^{ (k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial domain $\Omega^{ (k)}$ of finite type in $\mathbb C ^2$, and let $\Box_b^{ (k)}$ be the Kohn Laplacian on $M^{ (k)}$. In this paper, we study multivariable spectral multipliers $m(\Box_b^{ (1)},\ldots, \Box_b^{ (n)})$ acting on the Shilov boundary $\widetilde{ M}=M^{ (1)} \times\cdots\times M^{ (n)}$ of the product domain $\Omega^{ (1)}\times\cdots\times \Omega^{ (n)}$. We show that if a function $F(\lambda_1, \ldots ,\lambda_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m(\Box_b^{ (1)}, \ldots, \Box_b^{ (n)})$ is a product Calder\'on--Zygmund operator of Journ\'e type.
Categories: math.CV