Marcinkiewicz multipliers associated with the Kohn Laplacian on the Shilov boundary of the product domain in C²ⁿ

Let M⁽ⁱ⁾, i= 1 , 2 , … , n, be the boundaries of unbounded domains Ω ⁽ⁱ⁾ of finite type in C², and let □b(i) be the Kohn Laplacian on M⁽ⁱ⁾. In this paper, we study multivariable spectral multipliers m(□b(1),…,□b(n)) acting on the Shilov boundary M~ = M⁽¹⁾× ⋯ × M⁽ⁿ⁾ of the product domain Ω ⁽¹⁾× ⋯ × Ω ⁽ⁿ⁾. We show that if a function m(λ₁, … , λ_n) satisfies a Marcinkiewicz-type smoothness condition defined using Sobolev norms, then the spectral multiplier operator m(□b(1),…,□b(n)) is a product Calderón–Zygmund operator of Journé type.