L p -Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators

In this paper, I explore a specific class of bi-parameter pseudo-differential operators characterized by symbols σ ( x 1 , x 2 , ξ 1 , ξ 2 ) falling within the product-type Hörmander class S ρ , δ m . 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 | ∂ ξ α ∂ x β σ ( x 1 , x 2 , ξ 1 , ξ 2 ) | ≤ C α , β ( 1 + | ξ | ) m ∏ i = 1 2 ( 1 + | ξ i | ) − ρ | α i | + δ | β i | is satisfied. My investigation culminates in a rigorous analysis of the L p -boundedness of such pseudo-differential operators, thereby extending the seminal findings of C. Fefferman from 1973 concerning pseudo-differential operators within the Hörmander class.