Let $\chi$ be a $C_c^\infty$ function which equals $1$ on $|x|<1.$ Define $$ I_{\Phi, \epsilon} = \int e^{i\Phi(x,\theta)}a(x, \theta) \chi(\epsilon \theta) d \theta $$ for a phase function $\Phi$ with suitable assumptions so that $\lim_{\epsilon} I_{\Phi, \epsilon}$ converges to a distribution in $\mathcal D'(\mathbb R^n).$ The distributional limit is defined to be the oscillatory integral.
(In more details, $\Phi$ is smooth, and homogeneous of degree 1 in $\theta.$ $d\Phi$ is non vanishing.)
My question is, what sort of distributions can be written as an oscillatory integral? Surely all tempered distributions do (via Fourier Transform), but I could not see what happens in the general case.