For the two dimensional vectors $x=(x_1,x_2)$ and $\xi=(\xi_1,\xi_2)$, define the following operator $$e^{it\Delta}f(x):=\int_{\mathbb{R}^2}e^{i x\cdot\xi+it|\xi|^2}\hat{f}(\xi)\mathrm{d}\xi, \quad x\cdot\xi:=x_1\xi_1+x_2\xi_2.$$ For the functions ${\psi}_1$ and ${\psi}_2$, assume that $$\mathrm{supp}(\widehat{\psi}_1) \subset\{\xi\in\mathbb{R}^2:|\xi|\sim M\}, \quad \mathrm{supp}(\widehat{\psi}_2) \subset\{\xi\in\mathbb{R}^2:|\xi|\sim N\}.$$ Here $\widehat{\psi}_1$ denotes the Fourier transform of $\psi_1$ and the notation $|\xi|\sim M$ means that $$c_1 M\leq |\xi|\leq c_2 M$$ for some universal constants $c_1$ and $c_2$. Furthermore, let $\delta_0$ be the Dirac delta function. Then my question is how to use Cauchy-Schwarz to conclude the following estimate: \begin{align} \left\|e^{it\Delta}\psi_1 e^{it\Delta}\psi_2\right\|_{L_{t,x}^2(\mathbb{R}^3)}&=\int_{\mathbb{R}_{\tau,\xi}^3} \left|\int_{\mathbb{R}_{\xi'}^2}\widehat{\psi}_1(\xi')\widehat{\psi}_2(\xi-\xi') \delta_0(\tau-|\xi'|^2-|\xi-\xi'|^2)\mathrm{d}\xi'\right|^2\mathrm{d}\tau\mathrm{d}\xi \\ &\leq \left\|\psi_1\right\|_{L^2}^2 \left\|\psi_2\right\|_{L^2}^2 \left[\sup_{\tau,|\xi|\sim N} \left|\left\{\xi'\in\mathbb{R}^2: |\xi'|\sim M, |\xi'|^2+|\xi-\xi'|^2=\tau\right\}\right|\right] \\ &\leq C \frac{M}{N}. \end{align} This estimate appears in Bourgian's paper Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity in the proof of Lemma 111. Further details can be seen there.
As far as I know, up to a constant, there holds $$\widehat{e^{it\Delta}\psi}(\tau,\xi)=\delta_0(\tau-|\xi|^2)\widehat{\psi}(\xi),$$ where $\widehat{e^{it\Delta}\psi}(\tau,\xi)$ is the space-time Fourier transform of $e^{it\Delta}\psi(t,x)$. Therefore, Plancherel theorem can give the identity $$\left\|e^{it\Delta}\psi_1 e^{it\Delta}\psi_2\right\|_{L_{t,x}^2(\mathbb{R}^3)}=\int_{\mathbb{R}_{\tau,\xi}^3} \left|\int_{\mathbb{R}_{\xi'}^2}\widehat{\psi}_1(\xi')\widehat{\psi}_2(\xi-\xi') \delta_0(\tau-|\xi'|^2-|\xi-\xi'|^2)\mathrm{d}\xi'\right|^2\mathrm{d}\tau\mathrm{d}\xi.$$ However, I am not very familar with the computations for delta functions and confused about using Cauchy-Schwarz to compute further. In fact, the last inequality $$\left\|\psi_1\right\|_{L^2}^2 \left\|\psi_2\right\|_{L^2}^2 \left[\sup_{\tau,|\xi|\sim N} \left|\left\{\xi'\in\mathbb{R}^2: |\xi'|\sim M, |\xi'|^2+|\xi-\xi'|^2=\tau\right\}\right|\right] \leq C \frac{M}{N}$$ also confuse me very much.
Any comments would be helpful. Thanks in advance for your comments and answers!