Singularities of affine fibrations in the regularity theory of Fourier integral operators

We consider regularity properties of Fourier integral operators in various function spaces. The most interesting case is the $L^p$ spaces, for which survey of recent results is given. For example, sharp orders are known for operators satisfying the so-called smooth factorization condition. Here this condition is analyzed in both real and complex settings. In the letter case, conditions for the continuity of Fourier integral operators are related to singularities of affine fibrations in $\mathbb C^n$ (or subsets of $\mathbb C^n$) specified by the kernels of Jacobi matrices of holomorphic maps. Singularities of such fibrations are analyzed in this paper in the general case. In particular, it is shown that if the dimension $n$ or the rank of the Jacobi matrix is small, then all singularities of an affine fibration are removable. The fibration associated with a Fourier integral operator is given by the kernels of the Hessian of the phase function of the operator. On the basis of an analysis of singularities for operators commuting with translations we show in a number of cases that the factorization condition is satisfied, which leads to $L^p$ estimates for operators. In other cases, examples are given in which the factorization condition fails. The results are applied to deriving $L^p$ estimates for solutions of the Cauchy problem for hyperbolic partial differential operators.