| 赵俊燕.单位方体上沿曲面的振荡积分在Sobolev 空间上的有界性[J].数学年刊A辑,2017,38(4):405~418 |
| 单位方体上沿曲面的振荡积分在Sobolev 空间上的有界性 |
| The Boundedness of Certain Oscillatory Integrals on Unit Square Along Surfaces on Sobolev Spaces |
| Received:September 16, 2015 Revised:August 25, 2016 |
| DOI：10.16205/j.cnki.cama.2017.0033 |
| 中文关键词: Hyper singular oscillatory integral, Surface, Multiparameter, Unit square |
| 英文关键词:Hyper singular oscillatory integral, Surface, Multiparameter, Unit square |
| 基金项目:本文受到国家自然科学基金(No.11371316,No.11771388,No.11671363,No.11471288)的资助. |
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| 中文摘要: |
| 研究了欧氏空间$\mathbb{R}^{2}$中单位方体$Q^2=[0,1]^2$上沿曲面$(t,s,\gamma(t,s))$的振荡奇异积分算子
$$
\mathcal{T}_{\alpha,\beta}f(u,v,x)=\int_{Q^2}f(u-t,v-s,x-\gamma(t,s))\rme^{\rmi t^{-\beta_1}s^{-\beta_2}}t^{-1-\alpha_1}s^{-1-\alpha_2}\rmd t\rmd s
$$
从Sobolev空间$L_r^p(\mathbb{R}^{2+n})$到$L^p(\mathbb{R}^{2+n})$中的有界
性, 其中$x\in \mathbb{R}^{n}$, $(u,v)\in \mathbb{R}^{2}$,
$(t,s,\gamma(t,s))=(t,s,t^{p_1}s^{q_1},t^{p_2}s^{q_2},\cdots,t^{p_n}s^{q_n})$
~为$\mathbb{R}^{2+n}$上一个曲面, 且$\beta_1>\alpha_1\geq0$,
$\beta_2>\alpha_2\geq0$. 这些结果推广和改进了$\mathbb{R}^3$上的某些已知的结果. 作为应用, 得到了乘积空间上粗糙核奇异积分算子的Sobolev有界性.\\ |
| 英文摘要: |
| Let $Q^2=[0,1]^2$ be the unit square in two
dimensional Euclidean space $\mathbb{R}^{2}$. The author studies the
boundedness properties from Sobolev spaces
$L_r^p(\mathbb{R}^{2+n})$ to $L^p(\mathbb{R}^{2+n})$ of the
oscillatory singular integral operator
$\mathcal{T}_{\alpha,\beta}$ defined on the set
$\mathcal{S}(\mathbb{R}^{2+n})$ of Schwartz test funtions $f$
by
$$
\mathcal{T}_{\alpha,\beta}f(u,v,x)=\int_{Q^2}f(u-t,v-s,x-\gamma(t,s))\rme^{\rmi t^{-\beta_1}s^{-\beta_2}}t^{-1-\alpha_1}s^{-1-\alpha_2}\rmd t\rmd s,
$$
where $x\in \mathbb{R}^{n}$, $(u,v)\in \mathbb{R}^{2} $,
$(t,s,\gamma(t,s))=(t,s,t^{p_1}s^{q_1},t^{p_2}s^{q_2},\cdots,t^{p_n}s^{q_n})$
is a surface on $\mathbb{R}^{2+n}$, and
$\beta_1>\alpha_1\geq0$, $\beta_2>\alpha_2\geq 0$. The results
extend and improve some known results on $\mathbb{R}^3$. As
applications, the author obtains some Sobolev boundedness results of
rough singular integral operators on the product spaces. |
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