Maybe this problem is difficult to understand, because we lump integration and distribution together(usually we can distinguish them).
I have some trouble about the following proof concerning the estimate of kernel function of cutoff Schrodinger operators.
Suppose that $P_{N}$ is the classical Littlewood-Paley cutoff operators. $\phi(\xi)$ is the standard bump function. We have the estimates \begin{equation} \Big|P_{N}e^{it\Delta}(x,y)\Big|\lesssim_{m}\begin{cases}|t|^{-\frac{d}{2}}, {\rm when} |x-y|\sim N|t|\geq N^{-1},\\ \frac{N^{d}}{\langle N^{2}t\rangle^{m}\langle N|x-y|\rangle^{m}} \end{cases} \end{equation}
Proof: \begin{align}\Big(P_{N}e^{it\Delta}\Big)(x,y)&\simeq\int_{\mathbb{R}^{d}}e^{i(x-y)\cdot\xi}\phi(\frac{\xi}{N})e^{-i|\xi|^{2}t}d\xi\\&=N^{d}\int_{\mathbb{R}^{d}}e^{-iN^{2}|\xi|^{2}t+i(x-y)\cdot N\xi}\phi(\xi)d\xi\\&\simeq\Big[N^{d}\hat{\phi}(N\cdot)*(4\pi it)^{-\frac{d}{2}}e^{\frac{i|\cdot|^{2}}{4t}}\Big](x-y)\end{align} Then we always have the estimates $$\Big(P_{N}e^{it\Delta}\Big)(x,y)\lesssim \min\{|t|^{-\frac{d}{2}},N^{d}\},$$ Then we can get the final estimate in the domain $$\{(x,y):|x-y|\sim N|t|\geq N^{-1}\},$$$$\{(x,y):|x-y|\sim N|t|\leq N^{-1}\},$$ and $$\{(x,y):|x-y|\leq N^{-1}, N|t|\leq N^{-1}\}.$$
But for $|x-y|\not\sim N|t|$. Because the boundness of support of $\phi(\xi)$, we can easily see that $|x-y|\gg N|t||\xi|$ or $|x-y|\ll N|t||\xi|$ for $\xi\in{\rm supp}\phi$. By direct calculation, Let $$\Phi(\xi)=-N^{2}t|\xi|^{2}+N(x-y)\cdot\xi$$ we can get $$\nabla_{\xi}\Phi=-2N^{2}t\xi+N(x-y),$$ $$\nabla^{2}_{\xi}\Phi=-2dN^{2}tI_{d\times d}.$$ Construct $$L(D)f=\frac{[Ni(x-y)-2iN^{2}\xi t]\cdot\nabla_{\xi}}{|N(x-y)-2N^{2}\xi t|^{2}}f, \ |\xi|\sim1$$ thus we have $$L(D)^{m}e^{Ni(x-y)\cdot\xi-iN^{2}|\xi|^{2} t}=e^{Ni(x-y)\cdot\xi-iN^{2}|\xi|^{2} t}.$$ Note that \begin{align}\Big|\Big(P_{N}e^{it\Delta}\Big)(x,y)\Big|&\lesssim N^{d}\Big|\int_{\mathbb{R}^{d}}e^{Ni(x-y)\cdot\xi-iN^{2}|\xi|^{2} t}L^{*}(D)^{2m}\phi(\xi)d\xi \Big|\\&\lesssim\frac{N^{d}}{|N(x-y)-2N^{2}\xi t|^{2m}}.\end{align}
The first question is: Why $|L^{*}(D)^{2m}\phi(\xi)|\lesssim \frac{1}{|N(x-y)-2N^{2}\xi t|^{2m}}$? when doing calculus we find that $L^{*}(D)\phi(\xi)=\nabla_{\xi}\cdot\Big(\frac{Ni(x-y)-2iN^{2}\xi t}{|N(x-y)-2N^{2}\xi t|^{2}}\phi(\xi)\Big)$ which should $\lesssim \frac{N^{2}t}{|N(x-y)-2N^{2}\xi t|}$ so $|L^{*}(D)^{2m}\phi(\xi)|\lesssim \frac{N^{4m}t^{2m}}{|N(x-y)-2N^{2}\xi t|^{2m}}$, where does $N^{4m}t^{2m}$ go??
The second question is Why $\frac{N^{d}}{|N(x-y)-2N^{2}\xi t|^{2m}}\lesssim \frac{N^{d}}{\langle N^{2}t\rangle^{m}\langle N|x-y|\rangle^{m}} $? Does it come from that $|x-y|\gg N|t||\xi|$ or $|x-y|\ll N|t||\xi|$? What is the detail of the proof?
The third question is where can I find the material concerning this estimate, any paper?
Thank you!!! Any hint is welcome, God will bless us!