Are Wigner functions of any Unitary operator in $B(L^{2}(\mathbb{R}))$ in $L^{\infty}(\mathbb{R}^{2})$?


Let $e^{iA} \in B(L^{2}(\mathbb{R}))$. Define the Wigner function (Wigner transform) as follows.

$W\{e^{iA} \}(x,p) = \int dy ~ \langle x - \frac{y}{2}|e^{iA} | x+ \frac{y}{2}\rangle ~ e^{ipy}$

I would like to know if

$$\sup_{x,p \in \mathbb{R}}|W\{e^{iA} \}(x,p)| < \infty$$

If I take the absolute value and use triangle inequality I get nowhere.

$$|W\{e^{iA} \}(x,p)| \leq\int dy ~ |\langle x - \frac{y}{2}|e^{iA} | x+ \frac{y}{2}\rangle |$$

where $\langle x - \frac{y}{2}|e^{iA} | x+ \frac{y}{2}\rangle |$ should just be one singe the term within the absolute value is just a complex exponential of magnitude 1. In such a case we end up with no bound.

I hypothesize that the term $\int dy ~ \langle x - \frac{y}{2}|e^{iA} | x+ \frac{y}{2}\rangle ~ e^{ipy} $ should be computed before estimating bounds. But if we do this it seems that we would end up with some Dirac delta functions in which case the supremum operation would not be well defined.



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