Hardy's uncertainty principle states that a real function $f$ and its Fourier transform $\widehat{f}$ may not both decay faster at infinity than the standard Gaussian $e^{-\pi t^2}$, unless $f = 0$. In a sense, $e^{- \pi t^2}$ is the closest a non-zero function can come to having both $f$ and $\widehat{f}$ almost compactly supported.

Restrict now to Schwartz functions normalized by $f(0) = \widehat{f}(0) = 1$. The standard Gaussian $e^{-\pi t^2}$ is the unique Gaussian function respecting this normalization, and it is self-dual with the optimal asymptotic decay, which is asymptotically much faster than $t^{-A}$ for any $A$. However, for $A > 10$ say, this Gaussian is not everywhere dominated by $t^{-A}$.

**Question.** *Is there an upper bound on the values $A > 0$ for which there exists a Schwartz function $f$ having $f(0) = \widehat{f}(0) = 1$ and $|f(t)|,|\widehat{f}(t)| < t^{-A}$ for all $t \in \mathbb{R}$*?