The global wave front set of tempered oscillatory integrals with inhomogeneous phase functions

We study certain families of oscillatory integrals I_φ(a), parametrised by phase functions φ and amplitude functions a globally defined on R^d, which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of I_φ(a) are described both from the point of view of the lack of smoothness as well as with respect to the decay at infinity. In particular, the latter will depend on a version of the set of stationary points of φ, including elements lying at the boundary of the radial compactification of R^d. As applications, we consider some properties of the two-point function of a free, massive, scalar relativistic field and of classes of global Fourier integral operators on R^d, with the latter defined in terms of kernels of the form I_φ(a).