Let $\Phi(x,y,\theta)$ be a phase function defined on $X \times X \times (\mathbb R^n-0)$ where $X$ is some domain in $\mathbb R^n$, let $A(x,y,\theta)$ be an amplitude function. As usually an integral Fourier operator is defined by the formula $$ (Fu)(y)=\frac{1}{(2\pi)^n} \int\limits_{\mathbb R^n}\int\limits_{X} e^{i\Phi(x,y,\theta)} A(x,y,\theta)u(x) \; dxd\theta $$ I'm currenly looking for the generalization of the integral Fourier operator on the case when $\theta$ belongs to some open convex cone $\Gamma$ in $\mathbb R^n$, for example $\Gamma = \mathop{\mathsf{int}} \mathbb R^n_+$: $$ (F_{\Gamma}u)(y) = \frac{1}{(2\pi)^n} \int\limits_{\Gamma}\int\limits_{X} e^{i\Phi(x,y,\theta)} A(x,y,\theta)u(x) \; dxd\theta $$ At first I thought that this "little" modification of definition will not sufficiently affect ideas of proofs of some results in theory of integral Fourier operators and I can easily modify them. But it was a mistake. So my question is if there are some results on this generalization yet?
An example of arising problem is further. I have an operator $$ (Au)(y)= \frac{1}{(2\pi)^n} \int\limits_{\mathbb R^n_+} \int\limits_{\mathbb R^n_+} e^{i \theta(x-y)+iH(x,y,\theta)} u(x) \; dx d\theta $$ where $H(x,y,\theta)$ is a function with some specific properties. I have to show that $A = I +K$, where $I$ is the identity operator, $K$ is a compact linear operator from $L^2_{0}(\mathbb R^n_+)$ to $L^2_{loc}(\mathbb R^n_+)$. If $\Gamma$ is equal to $\mathbb R^n$ then it can be done using representation $$ (Au)(y) = I + \frac{i}{(2 \pi)^n}\int\limits_{0}^{1} \int\limits_{\mathbb R^n} \int\limits_{\mathbb R^n_+} e^{i \theta(x-y) + itH(x,y,\theta)} H(x,y,\theta) u(x) \; dxd\theta $$ where $$ I = \frac{1}{(2\pi)^n} \int\limits_{\mathbb R^n} \int\limits_{X} e^{i \theta (x-y)} u(x) \; dx d\theta. $$ If we try to do tha same in the case of our "generalized" Fourier integral operator we will obtain instead $I$ in this formula an operator $$ \frac{1}{(2 \pi)^n} \int\limits_{\Gamma} \int\limits_{X} e^{i \theta(x-y)}u(x) \, dxd\theta. $$
