Marcinkiewicz multipliers associated with the Kohn Laplacian on the Shilov boundary of the product domain in $\mathbb {C}^{2n}$

Abstract

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.