THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

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THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

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doi:10.1007/s10473-021-0207-0

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (2): 426-436.doi: 10.1007/s10473-021-0207-0

王光庆¹, 杨杰², 陈文艺¹

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. College of Mathematics and System Science, Xinjiang University, Xinjiang 830046, China

收稿日期:2019-11-29 修回日期:2020-04-29 出版日期:2021-04-25 发布日期:2021-04-29
通讯作者: Jie YANG E-mail:yangjie1106@xju.edu.cn
作者简介:Guangqing WANG,E-mail:2017102010002@whu.edu.cn;Wenyi CHEN,E-mail:wychencn@whu.edu.cn

Guangqing WANG¹, Jie YANG², Wenyi CHEN¹

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. College of Mathematics and System Science, Xinjiang University, Xinjiang 830046, China

Received:2019-11-29 Revised:2020-04-29 Online:2021-04-25 Published:2021-04-29
Contact: Jie YANG E-mail:yangjie1106@xju.edu.cn
About author:Guangqing WANG,E-mail:2017102010002@whu.edu.cn;Wenyi CHEN,E-mail:wychencn@whu.edu.cn

摘要： Let $T_{a,\varphi}$ be a Fourier integral operator defined by the oscillatory integral

\begin{array}{rcl} T_{a, φ} u (x) & = & \frac{1}{(2 π)^{n}} \int_{R^{n}} e^{i φ (x, ξ)} a (x, ξ) \hat{u} (ξ) d ξ, \end{array}

$\begin{eqnarray*} T_{a,\varphi}u(x) &=&\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{ {\rm i} \varphi(x,\xi)}a(x,\xi) \hat{u}(\xi){\rm d}\xi, \end{eqnarray*}$ where

a \in S_{ϱ, δ}^{m}

$a\in S^{m}_{\varrho,\delta}$ and

φ \in Φ^{2}

$\varphi\in \Phi^{2}$ , satisfying the strong non-degenerate condition. It is shown that if

0 < ϱ \leq 1

$0<\varrho\leq1$ ,

0 \leq δ < 1

$0\leq\delta<1$ and

m \leq \frac{ϱ^{2} - n}{2}

$m\leq \frac{\varrho^{2}-n}{2}$ , then

T_{a, φ}

$T_{a,\varphi}$ is a bounded operator from

L^{\infty} (R^{n})

$L^{\infty}(\mathbb{R}^n)$ to

B M O (R^{n}) .

${\rm BMO}(\mathbb{R}^n).$

关键词: Fourier integral operators, phase function, BMO spaces

Abstract: Let $T_{a,\varphi}$ be a Fourier integral operator defined by the oscillatory integral

\begin{array}{rcl} T_{a, φ} u (x) & = & \frac{1}{(2 π)^{n}} \int_{R^{n}} e^{i φ (x, ξ)} a (x, ξ) \hat{u} (ξ) d ξ, \end{array}

$\begin{eqnarray*} T_{a,\varphi}u(x) &=&\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{ {\rm i} \varphi(x,\xi)}a(x,\xi) \hat{u}(\xi){\rm d}\xi, \end{eqnarray*}$ where

a \in S_{ϱ, δ}^{m}

$a\in S^{m}_{\varrho,\delta}$ and

φ \in Φ^{2}

$\varphi\in \Phi^{2}$ , satisfying the strong non-degenerate condition. It is shown that if

0 < ϱ \leq 1

$0<\varrho\leq1$ ,

0 \leq δ < 1

$0\leq\delta<1$ and

m \leq \frac{ϱ^{2} - n}{2}

$m\leq \frac{\varrho^{2}-n}{2}$ , then

T_{a, φ}

$T_{a,\varphi}$ is a bounded operator from

L^{\infty} (R^{n})

$L^{\infty}(\mathbb{R}^n)$ to

B M O (R^{n}) .

${\rm BMO}(\mathbb{R}^n).$

Key words: Fourier integral operators, phase function, BMO spaces

中图分类号:

42B20

王光庆, 杨杰, 陈文艺. THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS[J]. 数学物理学报(英文版), 2021, 41(2): 426-436.

Guangqing WANG, Jie YANG, Wenyi CHEN. THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS[J]. Acta mathematica scientia,Series B, 2021, 41(2): 426-436.

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