Lemma 4.23 . [045X]

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Lemma 4.23.

(Estimating integrands II) For $|y_{1}|+|y_{2}|\gtrsim 1$ and $|x_{1}|,|x_{2}|\leq\frac{1}{2}$ , we have the estimate

|\gamma_{p3}+\gamma_{p4}-\frac{\sqrt{-1}a_{p\bar{q}}y_{q}\sqrt{\mathbb{A}}}{2\pi A^{3/2}|y|_{a}^{2}}|\leq\frac{C}{A^{1/2}|y|_{a}^{2}}.

Morever,

|\gamma_{p3}-I_{p3}|+|\gamma_{p4}-I_{p4}|\leq\frac{C|\mu|}{|y|_{a}^{2}\varrho}.