# On an optimal estimate of a certain pseudo-differential operator

Concerning a pseudo-differential operator of which the symbol is $e^{-i\xi^{m}}/\hat{f}(\xi)$, where $m$ is a positive integer. Can we choose a ceratain $f$ such that there exists an estimate of $L^p$-$L^q$ or $L^p$-$L^p$ type with some optimal weight? The optimal weight refers to a certain form with some optimal parameter, for example, $e^{|x|^m/\alpha^2}$, and $\alpha$ is made to be minimum.

Actually this is related to a problem that the user 'shanlin' raised, which is about how to apply the Hardy's uncertainty principle to dispersive equations.

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This question ought to be improved: what is $f$? What do you mean by an optimal weight? What is the precise reference to a question by shanlin? You cannot ignore that your question is simply impossible to answer in its present form. Thanks in advance for clarifications. –  Bazin Aug 10 at 14:28
$f$ is a function such that $\hat{f}$ doesn's vanish. Now the map from $L^2(\phi dx)$ to $L^2(\psi d\xi)$: $u\to \mathcal{F}^{-1}(e^{-i\xi ^m}\hat{u}/\hat{f})$ is waiting to be made bounded, as long as we choose an appropriate $f$. But at the same time I also want $\phi$ and $\psi$ to be parameterized and rather clear, for example, can $\phi$ and $\psi$ be $e^{|x|^m/\alpha^2}$ and $e^{|x|^m/\beta^2}$? Further for optimality, can $\alpha\beta$ has a sharp upper bound over which the map cannot be bounded (I guess 'the upper bound' for $\alpha\beta$ may be a proper manner here)? –  Tianxiao Huang Aug 10 at 16:17
Summing up, I want some specific $\phi$, $\psi$ and $f$ to be determined. Just drop any reference, I wish anyone who is familar with Fourier mutipler will give me some hints. Thank Bazin, and the question that 'shanlin' asks is from the original view but mine has been transformed, btw, now he has just asked only 1 question that I refered to –  Tianxiao Huang Aug 10 at 16:45