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

Jinhua Cheng ()
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Jinhua Cheng: School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China

Mathematics, 2024, vol. 12, issue 11, 1-10

Abstract: 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.

Keywords: bi-parameter pseudo-differential operators; L p -boundedness; cone decomposition; BMO space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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