R^3和PDE应用中曲面测度的傅里叶变换的估计
Estimates for Fourier transforms of surface measures in R^3 and PDE applications
摘要：
利用局部二维奇点的奇点的分解定理和基于Van der Corput引理的论证，给出了R^3中局部定义的光滑超曲面测度的傅里叶变换衰减率的新估计，并对一些已知估计给出了新的证明.然后，这些被用来给出某些PDE问题的解的L^q界，这些界是根据它们的初始数据的L^p范数，对于不同的p和q值。不像许多早期的工作，在这个问题上，没有使用已被经常用于研究二维振荡积分的适应坐标系，所有需要的信息是由奇点的奇点的分解定理提供。
英文摘要：
A local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in R^3, as well as to provide new proofs of some known estimates. These are then used to give L^q bounds on solutions to certain PDE problems in terms of the L^p norms of their initial data for various values of p and q. Unlike much of the earlier work in this subject, no use is made of the adapted coordinate systems that have been often been used to study two-dimensional oscillatory integrals; all of the needed information is furnished by the resolution of singularities theorem.
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