When I reading a book concerning the proof of Kato local smoothing effect, I'm confused by the following inequality:
$$\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\hat{a}(\eta-\xi){\color{red}{\delta(|\eta|^{2}-|\xi|^{2})|\xi|^{1/2}|\eta|^{1/2}}}\hat{f}(\xi)\overline{\hat{f}(\eta)}d\xi d\eta=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\hat{a}(\eta-\xi)\color{red}{\delta(|\eta|-|\xi|)\frac{|\xi|^{1/2}|\eta|^{1/2}}{|\xi|+|\eta|}}\hat{f}(\xi)\overline{\hat{f}(\eta)}d\xi d\eta$$ I want to know the red part of the above identity. The $\delta$ in above formula means Dirac generalization function, i.e. a distribution!
On the support of the measure $\delta(\lvert\xi\rvert^2-\lvert\eta\rvert^2)$, you have $\lvert \xi\rvert=\lvert\eta\rvert$. Moreover, $\delta$ is $-1$ homogeneous, meaning that $\delta(\lambda t)=\lambda^{-1}\delta(t)$ for all $\lambda>0$. The above formula follows from these two properties.
More precisely, the $d\eta$ part of your double integral reads $$\begin{split} \int_{\mathbb R^d} \delta(\lvert\xi\rvert^2-\lvert\eta\rvert^2)F(\eta)\, d\eta&= \int_{\mathbb R^d}\delta ((\lvert\xi\rvert -\lvert \eta\rvert)(\lvert\xi\rvert +\lvert \eta\rvert)) F(\eta)\, d\eta \\&= \int_{\mathbb R^d}\frac{\delta (\lvert\xi\rvert -\lvert \eta\rvert)}{2\lvert \xi\rvert} F(\eta)\, d\eta=\int_{\mathbb R^d}\frac{\delta (\lvert\xi\rvert -\lvert \eta\rvert)}{\lvert \xi\rvert+\lvert\eta\rvert} F(\eta)\, d\eta, \end{split}$$ for a certain function $F(\eta)$. Multiplying by the remaining factors, which are functions of $\xi$, and integrating against $d\xi$ you get the desired formula.