Models of degenerate Fourier integral operators and Radon transforms
Models of degenerate Fourier integral operators and Radon transforms
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DOI：
10.2307/2118622
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发表时间：
1994-11
影响因子：
4.9
通讯作者：
D. H. Phong;E. Stein
中科院分区：
文献类型：
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作者：
D. H. Phong;E. Stein
傅里叶积分算子的拉格朗日投影与奇点经常出现在许多领域的分析和几何[3]，[5]，[9]，[10]，[18]，[20]。然而，至今很少有分析工具可用于他们的研究，甚至他们最简单的正则性关于拉格朗日奇点仍然是模糊的[4]，[5]，[13]，[16]。本文研究了一类振荡积分TA和Fourier积分算子R，它们可以用来模拟拉格朗日量的高阶奇异性。它们在两个n阶变量中具有齐次多项式相位，并且实际上n = 3的情况是拉格朗日函数的模型，其投影为惠特尼折叠[3]，[10]，[11]。将高阶退化情况n > 4与低阶退化情况n = 2，3区分开来的主要困难在于，在那里出现的临界簇通常不是光滑流形。对这些振荡积分的系统研究始于[14]。这项工作的主要思想是将临界簇视为远离低维子簇的光滑流形，并跟踪到这个低维子簇的距离。为了实现这一点，我们介绍了一种固定相的方法，它清楚地表现出对临界点之间的距离的依赖性。该方法给出了达特 * 的核K（x，y）的大小的精确界限，但没有关于其相位和所产生的抵消的信息。似乎不可能将该方法细化到相位水平，并且这建议转而寻找可以间接地合并所需取消的算子TA的分解。本文的主要目标是介绍这样的分解。这些分解，我们将试图描述的时刻，反映了奇异性的临界点，并强大到足以产生上述振荡积分和Radon变换的尖锐的界限。尽管考虑中的模型非常
Fourier integral operators whose Lagrangians project with singularities arise frequently in many areas of analysis and geometry [3], [5], [9], [10], [18], [20]. However, there are as yet few analytic tools available for their study, and even their simplest regularity properties with respect to the singularities of the Lagrangian are still obscure [4], [5], [13], [16]. In this paper we study a class of oscillatory integrals TA and Fourier integral operators R which can be expected to model the higher order singularities of the Lagrangian. They have homogeneous polynomial phases in two variables of order n, and indeed the case n = 3 is the model for Lagrangians which project with Whitney folds [3], [10], [11]. The main difficulty which sets the higher order degeneracy cases n > 4 apart from the the lower ones n = 2, 3 is that the critical varieties which arise there are usually not smooth manifolds. A systematic study of these oscillatory integrals was begun in [14]. The main idea in that work was to treat the critical varieties as smooth manifolds away from a lower-dimensional subvariety, and to keep track of the distance to this lower-dimensional subvariety. To achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points. The method gave sharp bounds on the size of the kernel K(x, y) of TAT*, but no information on its phase and the resulting cancellations. It does not seem possible to refine this approach to the phase level, and this suggests looking instead for decompositions of the operators TA which can incorporate indirectly the required cancellations. The main goal of this paper is to introduce such decompositions. These decompositions, which we will try to describe momentarily, reflect the singular nature of the critical points, and are powerful enough to yield sharp bounds for the above oscillatory integrals and Radon transforms. Although the models under consideration are very