Analytic Fourier integral operators, Monge-Ampère equation and holomorphic factorization

Abstract

We will show that the factorization condition for the Fourier integral operators I X Y leads to a parametrized parabolic MongeAmpere equation In case of an analytic operator the bration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases by considering more general continuation problem for the level sets of a holomorphic mapping The results are applied to obtain Lpcontinuity for translation invariant operators in Rn with n and for arbitrary Rn with dXY j n