$\underline{\textbf{Background:}}$
I am currently trying to estimate the following $L^p$ norm:
\begin{align} & \Big{|}\Big{|} A \Big{|}\Big{|}_p \\ & = \int_{\mathbb{R}^d} \Big{|} \int_{|\xi|<2t^{1/2}} \exp\Big{(} ix\cdot\xi \Big{)} \Big{(}1 - i \sqrt{\frac{4t}{|\xi|^2} -1 } \Big{)} \exp\Big{(}-\frac{|\xi|^2}{2} \big{(}1+i\sqrt{ \frac{4t}{|\xi|^2} -1 }\big{)}\Big{)} \text{d}\xi \Big{|}^p \text{d}x \Big{\} }^{1/p} , \end{align}
where $t > 0$, and $d \geq 2$.
The above complicated function was obtained from part of an inverse Fourier transform (which needs to be split into small and large $|\xi|$, as the $\sqrt{\frac{4t}{|\xi|^2} - 1}$ part blows up at $|\xi| = 2t^{1/2}$).
As such, I would like to treat it as a Fourier transform still (formally), by rewriting as
\begin{align} \Big{|}\Big{|} A \Big{|}\Big{|}_p = \Big{|}\Big{|} \mathcal{F}^{-1}\Big{[} \exp\Big{(}i t^{1/2} |\xi|^2 \sqrt{ \frac{4}{|\xi|^2} -1 }\Big{)} \hat{f} \Big{]} \Big{|}\Big{|}_p , \end{align}
where
\begin{align} \hat{f}(t,\xi) := \Big{(}1 - i \sqrt{\frac{4t}{|\xi|^2} -1 } \Big{)} \exp\Big{(}-\frac{|\xi|^2}{2} \Big{)}\Bigg{|}_{|\xi|<2t^{1/2}} \end{align}
(I hope to get around the issue of smoothness by taking the limit of $\hat{f}$ times some test function.)
$\underline{\textbf{Problem:}}$
I would like to simplify the problem further. Note that the multiplier in $\hat{f}$ is pointwise bounded above and below:
\begin{align} C^{-1} \frac{t^{1/2}}{|\xi|} \leq \Big{|}1 - i \sqrt{\frac{4t}{|\xi|^2} -1 } \Big{|} \leq C \frac{t^{1/2}}{|\xi|}, \text{ for all } C \geq 4. \end{align}
I am wondering if this permits the following estimate:
\begin{align} & \Big{|}\Big{|} \mathcal{F}^{-1}\Big{[} \exp\Big{(}i t^{1/2} |\xi|^2 \sqrt{ \frac{4}{|\xi|^2} -1 }\Big{)} \hat{f} \Big{]} \Big{|}\Big{|}_p \\ & \leq C \Big{|}\Big{|} \mathcal{F}^{-1}\Big{[} \exp\Big{(}i t^{1/2} |\xi|^2 \sqrt{ \frac{4}{|\xi|^2} -1 }\Big{)} \frac{t^{1/2}}{|\xi|} \exp\Big{(}-\frac{|\xi|^2}{2} \Big{)} \Big{]} \Big{|}\Big{|}_p. \end{align}
In more general terms, if we have pointwise boundedness above and below for a Fourier multiplier
\begin{align} C^{-1}\psi(\xi) \leq \phi(\xi) \leq C\psi(\xi), \text{ for all } \xi \in \mathbb{R}^d, \end{align}
can we then say
\begin{align} \Big{|}\Big{|} \mathcal{F}^{-1} \Big{[} \phi \hat{g} \Big{]} \Big{|}\Big{|}_p \leq C \Big{|}\Big{|} \mathcal{F}^{-1} \Big{[} \psi \hat{g} \Big{]} \Big{|}\Big{|}_p, \end{align}
for sufficiently well behaved $\hat{g}, \phi, \psi$?
\begin{align} \end{align}