For a non-trivial function $f \in L^{2}(\mathbb{R})$ and its Fourier transform $\widehat{f}(\xi) = (2 \pi)^{-1/2} \int_{-\infty}^{\infty} f(t) e^{- i t \xi} dt$, the classical Hardy uncertainty principle states that if
$$ |f(t)| \lesssim e^{- \eta x^{2}} \quad \text{and} \quad |\widehat{f}(\xi)| \lesssim e^{- \eta \xi^{2}} , $$
then necessarily $0 \leq \eta \leq 1/2$. Moreover, in the extreme case that $\eta = 1/2$, the function $f$ must be a constant multiple of the Gaussian $e^{-\frac{1}{2} x^{2}}$. In this talk, we will be interested in those functions $f$ for which the above time-frequency estimate holds for any $\eta < 1/2$. In particular, we will show that it is true if and only if
$$ |H(f, n)| \lesssim e^{- r n} , \qquad \forall r > 0 , $$
where $H(f, n)$ denotes the $n$th Hermite coefficient of $f$. Moreover, we will see that more refined bounds correspond to a sharper time-frequency decay of $f$, and vice-versa. Our main tools are the Bargmann transform and some optimal weighted forms of the Phragmén-Lindelöf principle on sectors.
This talk is based on collaborative work with Joachim Toft and Jasson Vindas.