Let \(M^{(i)}\), \(i=1,2,\ldots , n\), be the boundaries of unbounded domains \(\Omega ^{(i)}\) of finite type in \(\mathbb {C}^2\), and let \(\Box _b^{(i)}\) be the Kohn Laplacian on \(M^{(i)}\). 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 \(m(\lambda _1, \ldots ,\lambda _n)\) satisfies a Marcinkiewicz-type smoothness condition defined using Sobolev norms, then the spectral multiplier operator \(m(\Box _b^{(1)}, \ldots , \Box _b^{(n)})\) is a product Calderón–Zygmund operator of Journé type.

设\(M^{(i)}\) , \(i=1,2,\ldots , n \)是有限类型的无界域\(\Omega ^{(i)}\)的边界\(\mathbb {C}^2\)，让\(\Box _b^{(i)}\)成为\(M^{(i)}\)上的 Kohn Laplacian 。在本文中，我们研究了作用在 Shilov 边界上的多变量谱乘数\(m(\Box _b^{(1)},\ldots , \Box _b^{(n)})\)作用于 Shilov 边界\({\widetilde{M }}=M^{(1)} \times \cdots \times M^{(n)}\)的乘积域\(\Omega ^{(1)}\times \cdots \times \Omega ^{( n)}\)。我们证明如果一个函数\(m(\lambda _1, \ldots ,\lambda _n)\)满足使用 Sobolev 范数定义的 Marcinkiewicz 型平滑条件，则谱乘子算子\(m(\Box _b^{(1)}, \ldots , \Box _b^{(n)})\)是 Calderón 的乘积–Journé 类型的 Zygmund 算子。