When it is possible to integrate an oscillatory integral?

Question

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral $$ I_\phi = \int\limits_{-\infty}^\infty e^{i\phi(x,t;\theta)} d\theta. $$ By [Hormander, ALPDO, I] $I_\phi$ is a distribution from $\mathscr D'\bigl(\Omega \times (0,+\infty) \bigr)$ of order $\leq 2$ and with $$ WF(I_\phi) \subset \bigl\{ (x,t,d_x\phi(t,x;\theta),d_t \phi(t,x;\theta)) \; \mid \; \theta \in \mathbb R \setminus 0, \; d_\theta \phi(x,t;\theta)=0 \bigr\}. $$ Is it possible to say in terms of $\phi$ and its derivatives when $I = \int_0^{+\infty} I_\phi \, dt$ will be a distribution from $\mathscr D'(\mathbb R^k)$? For example, if $I_\phi \in \mathscr E'(\Omega \times (0,+\infty))$ it is the case and by [Hormander, ALPDO, I] we have $$ WF(I) \subset \bigl\{ (x,d_x \phi(t,x;\theta)) \; \mid \; t \in (0,+\infty), \; \theta \in \mathbb R \setminus 0, \; d_{\theta,t} \phi(x,t;\theta)=0 \bigr\}. $$