Consider the oscillatory integral of the form $$I(\lambda) = \int_D a(\mathbf x) e^{i\lambda \mathbf x^T A \mathbf x} d\mathbf x,$$ where $D\subset \mathbb R^n$ is a box (or more generally a polygon), $a(\mathbf x)$ is a smooth function, $A$ is a constant matrix. If this helps, one may assume that $n$ is even and $A$ is block-off-diagonal as $$A = \begin{pmatrix} 0_{n/2\times n/2} & * \\ * & 0_{n/2\times n/2} \end{pmatrix}.$$ I am interested in the asymptotic behavior of $I(\lambda)$ as $\lambda \to \infty$.
A standard tool is to use the stationary phase method, but in the multi-dimensional settings, as far as I found, there is not so much results on the boundary contribution. Even $D$ is a simple shape of a box, there would be corner contribution, etc.
Is there any general theory on this topic?