I would like to compute explicitly the oscillatory integral
$$I(y):=\int_{\mathbb{R}^{n}}\frac{e^{\imath |x-y|^2}}{1+|x|^m}dx,$$ $m> n$. Notice that when $m>n$ the integral $I(y)$ converges absolutely.
In spherical coordinates
$$I(y)=\int_{\mathbb{S}^{n-1}}\int_{0}^{\infty}\frac{e^{\imath |r\omega-y|^2}}{1+r^m}r^{n-1}drd\omega=e^{\imath |y|^2}\int_{\mathbb{S}^{n-1}}e^{\imath (\omega\cdot y)^2}\int_{0}^{\infty}\frac{e^{\imath (r-\omega\cdot y)^2}}{1+r^m}r^{n-1}drd\omega$$
as we can complete the square $|r\omega-y|^2=r^{2}-2r\omega\cdot y+|y|^2=(r-\omega\cdot y)^2-(\omega\cdot y)^2+|y|^2$