Models of Degenerate Fourier Integral Operators and Radon Transforms

Abstract

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