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

TL;DR

This paper investigates spectral multipliers associated with the Kohn Laplacian on the Shilov boundary of product domains in complex space, establishing Marcinkiewicz-type conditions for boundedness as Calderón–Zygmund operators.

Contribution

It introduces multivariable spectral multiplier results for the Kohn Laplacian on product boundaries, extending Calderón–Zygmund theory to this complex setting.

Findings

Spectral multipliers satisfying Marcinkiewicz conditions are bounded as Calderón–Zygmund operators.

The operators are of Journé type, suitable for product space analysis.

The results generalize classical multiplier theorems to complex boundary domains.

Abstract

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.