Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates? \begin{equation} \left|\int e^{2\pi i v\cdot \xi}e^{-iw|\xi|^2}\eta\Big({\frac{\xi}{N}}\Big)\frac{1}{\big(1+|\xi|^2\big)^s} d\xi\right|\lesssim \frac{1}{|v|^m} \end{equation} where we want the implicit constant independent of $w, N$.
My attempts:
We first tackle the case $w=0$. Denote $\big((1+|\xi|^2)^{\frac{s}{2}}\big)^{\vee}$ by $G_s$. $G_s(x)$ behaves like $\frac{1}{|x|^{d-s}}$ when $|x|\leq 2$ and has exponential decay when $|x|>2$. We proceed with
\begin{equation*}
\begin{split}
LHS=\left|\int N^d\check{\eta}(Nx)G_{2s}(v-x) dx\right|&\lesssim \left|\int \frac{N^d}{(1+N|x|)^L}G_{2s}(v-x) dx\right| \\
&=\int_{|v-x|\leq \frac{|v|}{2}}+\int_{|v-x|>\frac{|v|}{2}} \\
&\lesssim \frac{N^d}{(1+N|v|)^L}\int_{|v-x|\leq \frac{|v|}{2}}\frac{ dx}{|v-x|^{d-2s}} +\frac{1}{|v|^{d-2s}}
\end{split}
\end{equation*}
Let $L=d$, we get a bound $\frac{1}{|v|^{2d-2s-1}}$. (Note that we have $\varphi(x)\varphi(z)$ in $K$, so $|v|$ has an upper bound.) This is not bad in low dimensions.
For general $w\neq 0$, we want to prove
\begin{equation*}
\left|\bigg(e^{-iw|\xi|^2}\eta\Big({\frac{\xi}{N}}\Big)\bigg)^{\vee}(x)\right|\lesssim \frac{N^d}{(1+N|x|)^L}
\end{equation*}
still holds, so that one can proceed as above.
This leads
\begin{equation*}
\left|\int e^{i\lambda\cdot \xi}e^{-i\alpha|\xi|^2}\eta(\xi) d\xi\right|\lesssim \frac{1}{|\lambda|^L}
\end{equation*}
But this seems not true for large $L$.
This is an important step in P. Sjolin's article "Regularity of solutions to the Schrodinger equation” (P11 (15)).
