How to compute the asymptotics of this oscillatory integral?

Question

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Consider the following integral, for $\epsilon\ne 0:$

$$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[-ay+bx-yb]}\,dx\,da\,dy\,db\,,$$

where $\Omega$ is some compact neighborhood of the origin (you can assume the integral is over $\mathbb{R}^4$ and that there is a smooth cutoff function). The exponent $-ay+bx-yb$ has a nondegenerate and unique critical point at $(x,a,y,b)=(0,0,0,0)\,,$ and therefore the stationary phase approximation says that

$$\displaystyle\lim\limits_{\epsilon\to 0}\frac{1}{(2\pi)^2\epsilon^2}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[-ay+bx-yb]}\,dx\,da\,dy\,db=0\,.$$

My guess is that it's also true that $$\lim\limits_{\epsilon\to 0}\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[-ay+bx-yb]}\,dx\,da\,dy\,db=0\,,$$

but I'm not sure how to show it. Does anyone have any suggestions?

Answer 1

As Fedor Petrov suggested, you can diagonalize the quadratic form.

Writing $A = a+y$, $B = (a-y)$, $C = (b+x-y)$ and $D = (b-x +y)$, up to some negligible constants (and restoring the smooth cut-off) your integral is

$$ \int \varphi \cdot (C+D)(B-A) e^{\frac{i}{\epsilon} (- A^2 + B^2 + C^2 - D^2)} $$

This you can write as (up to negligible constants) $$ \epsilon^2 \int \varphi \cdot (\partial_C - \partial_D)(\partial_A + \partial_B) e^{\frac{i}{\epsilon} (- A^2 + B^2 + C^2 - D^2)}$$

After integration by parts, the standard argument shows that the leading order term is, up to a proportionality constant, given by $$ \epsilon^4 (\partial_C - \partial_D)(\partial_A + \partial_B)\varphi(0) $$

If your cut-off function is really just a cut-off function (so it is constant in a neighborhood of the origin), then in fact your integral is effectively supported away from the stationary point of the phase, and so should actually decay faster than any polynomial.