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Question on estimate in one of Jean Bourgain's 1992 papers

The paper in question is A Remark on Schrodinger Operators. 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}\...
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