The goal of the argument is to estimate the following integral: $$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\varphi_k(|\xi|)^2\,d\xi$$ where $t(x):x\in B(0,1)\mapsto t(x)\in(0,1)$ is arbitrary and $\varphi_k(|\xi|)$ is a Paley-Littlewood dyadic frequency projection to frequencies $2^{k-1}\le|\xi|\le 2^{k+1}$. Bourgain then argues that by stationary phase in polar coordinates, we have $$K_1(x,y)=2^{k/2} \varphi_k\left(\frac{|x-y|}{2[t(x)-t(y)]}\right) \frac{e^{i\frac{|x-y|^2}{4(t(x)-t(y))}}}{|x-y|^{1/2}|t(x)-t(y)|^{1/2}}+ O\left(\frac{2^{k/2}}{|x-y|}\right).$$ I wrote the integral as $$\int_0^{2\pi}\int_0^\infty e^{i|x-y|r\cos\theta+i(t(x)-t(y))r^2}r\,\varphi^2_k(r)\,dr\,d\theta.$$ The phase is stationary at $r_0=\frac{|x-y|\cos\theta}{2[t(y)-t(x)]}$. Therefore, in the limit $|x-y|\to\infty$, I obtain the principal contribution to be $$\int_0^{2\pi}\exp\left({i\frac{\cos(\theta)^2|x-y|^2}{4(t(x)-t(y))}}\right)\int_0^\infty \exp\left({2i[t(x)-t(y)](r-r_0)^2}\right)r\varphi_k(r)^2\,dr\,d\theta.$$ I believe I somehow need to use the support restriction on $\varphi_k$. How do I get to Bourgain's result?
Edit: I have now posted the question on mathoverflow here.
