On the failure of the factorization condition for non-degenerate Fourier integral operators

In this paper we give examples of polynomial phase functions for which the factorization condition of Seeger, Sogge and Stein (Ann. Math. 134 (1991)) fails. The corresponding Fourier integral operators turn out to be still continuous in L p L^p . We also give examples of the failure of the factorization condition for translation invariant operators. In this setting the frequency space must be at least 5-dimensional, which shows that the examples are optimal. We briefly discuss the stationary phase method for the corresponding operators.