单位方体上沿曲面的振荡积分在Sobolev空间上的有界性

研究了欧氏空间R~2中单位方体Q~2=0,1]~2上沿曲面(t,s,γ(t,s))的振荡奇异积分算子T_(α,β)f(u,v,x)=∫_(Q~2)f(u-t,v-s,x-γ(t,s))e~(it~(-β_1)s~(-β_2))t~(-1-α_1)s~(-1-α_2)dtds从Sobolev空间L_τ~p(R~(2+n))到L~p(R~(2+n))中的有界性,其中x∈R~n,(u,v)∈R~2,(t,s,γ(t,s))=(t,s,t~(P_1)s~(q_1),t~(p_2)s~(q_2),…,t~(p_n)s~(q_n))为R~(2+n)上一个曲面,且β_1α_1≥0,β_2α_20.这些结果推广和改进了R~3上的某些已知的结果.作为应用,得到了乘积空间上粗糖核奇异积分算子的Sobolev有界性.

The Boundedness of Certain Oscillatory Integrals on Unit Square Along Surfaces on Sobolev Spaces

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.