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

 Author Name Affiliation E-mail ZHAO Junyan Department of Mathematics, Zhejiang University, Hangzhou 310027, China; Department of Mathematics, Zhejiang Normal University, Jinhua 321004,linebreak Zhejiang, China. zhaojunyan929@163.com
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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.\\
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.