| 程美芳,孙 伟,束立生.一类振荡积分算子在Wiener 共合空间上的有界性[J].数学年刊A辑,2018,39(2):113~126 |
| 一类振荡积分算子在Wiener 共合空间上的有界性 |
| Boundedness Properties of Certain Oscillatory Integrals on Wiener Amalgam Space |
| Received:November 28, 2016 Revised:July 13, 2017 |
| DOI：10.16205/j.cnki.cama.2018.0011 |
| 中文关键词: Wiener共合空间, 强奇异积分算子, 调幅空间 |
| 英文关键词:Wiener amalgam space, Strongly singular integral operator, Modulation space |
| 基金项目:本文受到国家自然科学基金(No.11201003, No.11771223)和安徽省高校自然科学基金(No.KJ2017ZD27, No.KJ2015A117)的资助. |
| Author Name | Affiliation | E-mail | | CHENG Meifa | Corresponding author. School of Mathematics and Statistics,
Anhui Normal University, Wuhu 241002, Anhui, China. | cmf78529@mail.ahnu.edu.cn | | SUN W | School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, Anhui, China. | shulsh@mail.ahnu.edu.cn | | SHU Lishe | School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, Anhui, China. | shulsh@mail.ahnu.edu.cn |
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| 中文摘要: |
| 假设 $a, b>0$ 并且
$$
K_{a,b}(x)=\dfrac{\rme^{\rmi |x|^{-b}}}{|x|^{n+a}}.
$$
定义强奇异卷积算子$T$如下:
$$
Tf(x)=(K_{a,b}\ast f)(x),
$$
本文主要考虑了如上定义的算子$T$在Wiener共合空间$W(\mathcal{F}L^{p},L^{q})({\mathbb{R}}^{n})$上的有界性.
另一方面, 设$\alpha,\beta>0$ 并且 $\gamma(t)=|t|^{k}$ 或 $\gamma(t)={\rm sgn}(t)|t|^{k}$. 利用振荡积分估计, 本文还研究了算子
$$
T_{\alpha,\beta}f(x,y)=\text{p.v.}\int_{-1}^{1}f(x-t,y-\gamma(t))\frac{\rme^{-2\pi
\rmi|t|^{-\beta}}}{t|t|^{\alpha}}\rmd t
$$
及其推广形式
$$
\Lambda_{\alpha,\beta}f(x,y,z)=\int_{Q^{2}}f(x-t,y-s,z-t^{k}s^{j})\rme^{-2\pi\rmi t^{-\beta_1}s^{-\beta_2}}t^{-\alpha_1-1}s^{-\alpha_2-1}\rmd t\rmd s
$$
在Wiener共合空间$W(\mathcal{F}L^{p},L^{q})$上的映射性质. 本文的结论足以表明, Wiener共合空间是Lebesgue空间的一个很好的替代. |
| 英文摘要: |
| Suppose $a, b>0$ and
$$
K_{a,b}(x)=\dfrac{\rme^{\rmi |x|^{-b}}}{|x|^{n+a}}.
$$
The first task in this paper is to study the boundedness properties of the strongly singular convolution operator $Tf(x)=(K_{a,b}\ast f)(x)$ on Wiener amalgam spaces $W(\mathcal{F}L^{p},L^{q})({\mathbb{R}}^{n})$.
If $\alpha,\beta>0$ and $\gamma(t)=|t|^{k}$ or $\gamma(t)={\rm sgn}(t)|t|^{k}$, the second task of this paper is to investigate the mapping properties of the operator defined by
$$
T_{\alpha,\beta}f(x,y)=\text{p.v.}\int_{-1}^{1}f(x-t,y-\gamma(t))\frac{\rme^{-2\pi\rmi |t|^{-\beta}}}{t|t|^{\alpha}}\rmd t
$$
and its general form given by
$$
\Lambda_{\alpha,\beta}f(x,y,z)=\int_{Q^{2}}f(x-t,y-s,z-t^{k}s^{j})\rme^{-2\pi\rmi t^{-\beta_1}s^{-\beta_2}}t^{-\alpha_1-1}s^{-\alpha_2-1}\rmd t\rmd s
$$
on Wiener amalgam spaces $W(\mathcal{F}L^{p},L^{q})$. The essential tool of this paper is the oscillatory integral estimation.
The results of this paper show that Wiener amalgam spaces are good substitutions for Lebesgue spaces. |
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