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

2024/09/27 by Cheng, Jinhua
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.18413

Abstract

In this paper, we 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 S^m_{ρ, δ}. This classification imposes constraints on the behavior of partial derivatives of σ with respect to both spatial and frequency variables. Specifically, we demonstrate that for each multi-index α, β, the inequality | ∂_ξ^α∂_x^βσ(x₁,x₂,ξ₁,ξ₂)| ≤ C_{α, β}(1+|ξ|)^m∏_i=1² (1+|ξ_i|)^{-ρ|α_i|+δ|β_i|} is satisfied. Our 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.

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