LP BOUNDEDNESS OF FOURIER OPERATORS WITH SP-m AMPLITUDE FUNCTION

In this paper we discuss the problem of preserving Lboundedness for the Fourier integral operator under more general conditions. In other words, we assume that the rank of the Hessian matrix of the non-degenerate phase function Φ（χ,ξ） is at least r (0≤r≤（n-1）) at the critical points and the amplitude function a（x, ξ） ∈S（1/2<ρ≤1）. The main result of this paper says that for m≥[n（3-ρ）-2-r] |（1/2）-（1/p）|, （1<p<∞）, the Fourier integral operator （1） can be extended to be a bounded operator: （L）→（L）. In a sense, this result is obviously best possible.