A model of degenerate and singular oscillatory integral operators

Abstract

The purpose of this paper is to determine the optimal order of decrease of the L² norm for the following degenerate and singular oscillatory integral operators defined by

formula here

for f∈C^∞₀(ℝ), where ψ(x, y) is a smooth compactly supported function on ℝ², S(x, y) is a homogeneous polynomial of degree n

formula here

and K(x, y) is a distribution kernel which satisfies the following hypotheses:

K is a C² function away from the diagonal and the estimates

formula here

hold for 0<μ<1 and i=1, or 2.

This paper basically uses the methods of [6], which are very general and can be widely used. The main idea of these methods is that the sizes of operator norms after the decomposition are estimated in different ways and the decomposition is then summed back by balancing the estimates of two types. The first type estimate takes into account the support sizes of the integrand after the decomposition. The second type estimate is delicate, depending on how to keep track of the distance to the singular varieties of phase function. In our case, we have to include the singular variety arising from distribution kernel as well. The analytic tool of the second type estimate is the operator version of the Van der Corput lemma [7]. We notice that the crux of matters is to locate the singular varieties of phase function and kernel. For higher dimensional situations, the problem is very difficult. One expects that the powerful theory of algebraic geometry may play a significant role. Here we would like to remark that Mather's results [2] may be useful although the geometrical shapes of singular varieties are basically undecidable.