Oscillatory integrals of subanalytic functions

In several papers, R. Cluckers and D. Miller have built and investigated a class of real functions which contains the subanalytic functions and which is closed under parameterized integration. This class does not allow any oscillatory behavior, nor stability under Fourier transform. On the other hand, the behavior of oscillatory integrals, in connection with singularity theory, has been heavily investigated for decades. In this talk, we explain how to build a class of complex functions, which contains the subanalytic functions and their complex exponentials, and which is closed under parameterized integration and under Fourier transform.Our techniques involve appropriate preparation theorems for subanalytic functions, and some elements of the theory of uniformly distributed families of maps.(joint work with R. Clucker, G. Comte, D. Miller and T. Servi).