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

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.