Consider the operator $$(A_\Delta\psi)(p) = \int_\Delta \int_{\mathbb{R}^n} e^{ix\cdot(p-k) + i\phi(x,p,k)} a(x,p,k) \psi(k) dk\: dx$$ where $\Delta$ is a Borel or Lebegue set in $\mathbb{R}^n$, in general different from $\mathbb{R}^n$ itself and $\psi$ ranges in $C_c^\infty(\mathbb{R}^n)$. Above $a$ is complex valued and $\phi$ is real valued, the measures $dk$, $dx$ are the Lebesgue one on $\mathbb{R}^n$.
Are there conditions on $\phi$ and $a$ such that $A_\Delta$ defines an operator $C_c^\infty(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ which is also bounded? (I know that the function $a$ I'm considering does not tend to $0$ sufficiently fast as $|p| \to +\infty$ to assure through rough estimates of the integral that the function $A_\Delta\psi$ is $L^2$, for instance when $|\Delta| <+\infty$.)
I am finding literature about this issue but $\Delta$ it is always the whole $\mathbb{R}^m$ (or sometimes an open set therein).
I am interested in the case where $a: \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^n \to \mathbb{C}$ is not compactly supported (I can assume that it is smooth and maybe with all of derivatives bounded), and the set $\Delta$ may change. I can also assume that the measure of $\Delta$ is finite or that the set is bounded.