I'm currently working on a proof about the following uncertainty principle for my end of degree project:
(Hardy's uncertainty principle): Suppose that $f$ is a measurable function such that $\forall x, \xi \in \mathbb{R}$ $$ |f(x)| \leq C e^{-\pi a x^2}, $$ and $$ |\hat{f}(\xi)| \leq C' e^{(-\pi \xi^2)/a}, $$ where $C, C', a > 0$ and $\hat{f}$ denotes the fourier transform of $f$. Then $f$ is a scalar multiple of the gaussian $e^{-\pi a x^2}$.
After appropiate normalizations, it can be shown that it suffices to show the case $C = C' = a = 1$. The strategy that I was told to follow revolves around considering the complex function $G(z) = e^{\pi z^2} \hat{f}(z)$ (wich turns out to be entire) and showing that it is bounded on the entire complex plane so that I can apply Liouville's theorem to conclude the proof. The boundedness of $G(z)$ on the first quadrant $S = \{z = x+iy : x >0, y>0\}$ can be shown using the following lemma:
(Hardy's lemma): Let $S = \{z = x+iy : x >0, y>0\}$ and $f$ holomorphic on $S$ and continuous on the border $\partial S$ such that $$ |f(z)| \leq A, \quad \forall z \in \partial S $$ and $$ |f(z)| \leq B e^{\beta x^2}, \quad \forall z \in S $$ where $A, B, \beta >0$. Then $$ |f(z)| \leq \max(A,B), \quad \forall z \in S $$
For the rest of the quadrants, I was told that I could extend the argument by symmetry considering $f(-z)$ and the conjugate function $\overline{f}$. I see that the symmetry makes sense intuitively, but I´m struggling to concretly apply it to prove the complete uncertainty principle. I could use some help, thanks in advance!