Lp-Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators

Abstract

In this paper, I explore a specific class of bi-parameter pseudo-differential operators characterized by symbols σ(x1,x2,ξ1,ξ2) falling within the product-type Hörmander class Sρ,δm. This classification imposes constraints on the behavior of partial derivatives of σ with respect to both spatial and frequency variables. Specifically, I demonstrate that for each multi-index α,β, the inequality |∂ξα∂xβσ(x1,x2,ξ1,ξ2)|≤Cα,β(1+|ξ|)m∏i=12(1+|ξi|)−ρ|αi|+δ|βi| is satisfied. My investigation culminates in a rigorous analysis of the Lp-boundedness of such pseudo-differential operators, thereby extending the seminal findings of C. Fefferman from 1973 concerning pseudo-differential operators within the Hörmander class.