The Babenko-Beckner inequality
$|| \mathcal F f ||_q \geq C(q,p)||f||_p$
is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its $p$-Norm, where $1/q+1/p=1$. For a constant
$C(q,p) = \sqrt{p^{1/p}/q^{1/q}}$
this bound is said to be sharp. In particular, it is said to be saturated by Gaussians. However, nobody seems to explicitly state that Gaussians are the only functions which allow saturation, including the original reference (I have to admit that I do not completely understand the original proof).
Therefore, my question is: Are Gaussians the only functions which saturate the Babenko-Beckner inequality or is equality in principle also possible for different kind of functions?