沿超曲面的强奇异积分算子在调幅空间上的有界性

沿超曲面的强奇异积分算子在调幅空间上的有界性

Hypersingular Integrals along Hypersurface on Modulation Spaces

摘要/Abstract

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数学物理学报 ›› 2010, Vol. 30 ›› Issue (4): 959-967.

程美芳¹, 张震球²

1.安徽师范大学数学计算机科学学院安徽芜湖 241000||2.南开大学数学学院天津 300071

收稿日期:2009-03-06 修回日期:2010-05-11 出版日期:2010-07-25 发布日期:2010-07-25
基金资助:
国家自然科学基金(10671041, 10971039)、安徽省教育厅一般项目(KJ2008B244)和安徽师范大学青年科学基金(2007xqn50)资助

CHENG Mei-Fang¹, ZHANG Zhen-Qiu²

1.College of Mathematics and Computer Sciences, Anhui Normal University, Anhui Wuhu 241000;
2.School of Mathematical Sciences, Nankai University, Tianjin 300071

Received:2009-03-06 Revised:2010-05-11 Online:2010-07-25 Published:2010-07-25
Supported by:
国家自然科学基金(10671041, 10971039)、安徽省教育厅一般项目(KJ2008B244)和安徽师范大学青年科学基金(2007xqn50)资助

摘要：

设

T f (x, x_{n}) = p . v . \int \frac{f (x - y, x_{n} - γ (y)) e^{- 2 π i | y |^{- β}} Ω (y) h (y)}{| y |^{n + α - 1}} d y, x, y \in R^{n - 1}, x_{n} \in R .

$Tf(x,x_{n})={\rm p.v.}\int\frac{f(x-y,x_{n}-\gamma(y)){\rm e}^{-2\pi {\rm i}|y|^{-\beta}}\Omega(y)h(y)}{|y|^{n+\alpha-1}}{\rm d}y, ~~~x,y\in{{\Bbb R}}^{n-1}, x_{n}\in{{\Bbb R}}.$
该文主要讨论了上述奇异积分算子在广义的调幅空间上的有界性, 其中粗糙核

Ω \in L^{1} (S^{n - 2})

$\Omega\in L^1(S^{n-2})$ ,

h (y)

$h(y)$ 为有界的径向函数, 而

γ (y)

$\gamma(y)$ 是满足一定条件的超曲面.

关键词: 强奇异积分, 调幅空间, 超曲面

Abstract:

Let

T f (x, x_{n}) = p . v . \int \frac{f (x - y, x_{n} - γ (y)) e^{- 2 π i | y |^{- β}} Ω (y) h (y)}{| y |^{n + α - 1}} d y, x, y \in R^{n - 1}, x_{n} \in R .

$Tf(x,x_{n})={\rm p.v.}\int\frac{f(x-y,x_{n}-\gamma(y)){\rm e}^{-2\pi {\rm i}|y|^{-\beta}}\Omega(y)h(y)}{|y|^{n+\alpha-1}}{\rm d}y, ~~~x,y\in{{\Bbb R}}^{n-1}, x_{n}\in{\Bbb R}.$ The purpose of this paper is to investigate the boundedness of such integral operators on general modulation spaces. Here the rough kernel

Ω

$\Omega$ is in the

L^{1} (S^{n - 2})

$L^1(S^{n-2})$ ,

h (y)

$h(y)$ is a bounded radial fuction and

γ (y)

$\gamma(y)$ is an appropriate hypersurface.

Key words: Hyper-singular integral, Modulation space, Hypersurface

中图分类号:

42B20

程美芳, 张震球. 沿超曲面的强奇异积分算子在调幅空间上的有界性[J]. 数学物理学报, 2010, 30(4): 959-967.

CHENG Mei-Fang, ZHANG Zhen-Qiu. Hypersingular Integrals along Hypersurface on Modulation Spaces[J]. Acta mathematica scientia,Series A, 2010, 30(4): 959-967.

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