Hardy famously showed that a function $f \in L^2(\mathbb{R})$ satisfying the time-frequency bounds
\[ |f(x)| \lesssim (1 + |x|)^N e^{-\frac12 x^2} \quad \text{and} \quad |\widehat{f}(\xi)| \lesssim (1 + |\xi|)^N e^{-\frac12 \xi^2} \]
must necessarily take the form $f(x) = p(x) e^{-\frac12 x^2}$ for some polynomial $p$ of degree at most $N$. Multidimensional analogs were later established; in particular, Bonami, Demange, and Jaming showed that coordinate-wise estimates suffice to reach the same conclusion. In this talk, we will show that time-frequency decay of the form
\[ |f(x)| \lesssim e^{-\frac12 |x|^2 + \omega(|x|)} \quad \text{and} \quad |\widehat{f}(\xi)| \lesssim e^{-\frac12 |\xi|^2 + \omega(|\xi|)} \]
for a weight function $\omega$ dominated by $t^2$, imposes a specific structure on $f$, expressed through bounds on the coefficients of its Hermite expansion. We will put particular emphasis on the proof of the multidimensional case. Among other things, we will determine exactly when coordinate-wise time-frequency estimates suffice in terms of the weight function $\omega$.