Asymptotic expansions of Fourier integrals with light cone singularities

Let the integral transforms φ̃(y) and φ̃(y1,y2) be defined through φ̃(y)=∫d4xeixys(x)φ(x) and φ̃(y1,y2)=∫d4x1eix1y1∫d4x2eix2y2s(x1,x2)φ(x1,x2), respectively. Here all variables x and y are 4-vectors in Minkowski space. The functions φ are elements of S, and the factors s contain certain types of light cone singularities. The integral transforms φ̃ are investigated with respect to their characteristic properties implied by these light cone singularities using the method of van der Corput's neutralizers. It turns out that the behavior of φ̃ is determined by the light cone singularities if one goes to infinity in the space of the y variables along an arbitrary straight line. All characteristically different cases are classified and for each case a complete asymptotic expansion is derived.