$L^p$-Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929517322305536 |
|---|---|
| author | Cheng, Jinhua |
| author_facet | Cheng, Jinhua |
| contents | In this paper, we 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}
$\mathbf{S}^m_{ρ, δ}$. 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
$| \partial_ξ^α\partial_x^βσ(x_1,x_2,ξ_1,ξ_2)| \le C_{α, β}(1+|ξ|)^m\prod_{i=1}^2 (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$-Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators Cheng, Jinhua Classical Analysis and ODEs In this paper, we 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} $\mathbf{S}^m_{ρ, δ}$. 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 $| \partial_ξ^α\partial_x^βσ(x_1,x_2,ξ_1,ξ_2)| \le C_{α, β}(1+|ξ|)^m\prod_{i=1}^2 (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. |
| title | $L^p$-Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2409.18413 |