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

Abstract

Let $M^{(k)}$, $k=1,2,\dots ,n$, be the boundary of an unbounded polynomial domain ${\Omega }^{(k)}$ of finite type in ${\mathbb{C}}^2$, and let ${\square }_b^{(k)}$ be the Kohn Laplacian on $M^{(k)}$. In this paper, we study multivariable spectral multipliers $m({\square }_b^{(1)},\dots ,{\square }_b^{(n)})$ acting on the Shilov boundary $\tilde{M}=M^{(1)}\times \cdots \times M^{(n)}$ of the product domain ${\Omega }^{(1)}\times \cdots \times {\Omega }^{(n)}$. We show that if a function $F({\lambda }_1,\dots ,{\lambda }_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m({\square }_b^{(1)},\dots ,{\square }_b^{(n)})$ is a product Calderón--Zygmund operator of Journé type.