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

Autor: Jinhua Cheng
Jazyk: angličtina
Rok vydání: 2024
Předmět:
Zdroj: Mathematics, Vol 12, Iss 11, p 1653 (2024)
Druh dokumentu: article
ISSN: 2227-7390
DOI: 10.3390/math12111653
Popis: 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.
Databáze: Directory of Open Access Journals
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