  In this paper we address the problem of estimating the operator norm of the
embeddings between multidimensional weighted Paley-Wiener spaces. These can be
equivalently thought as Fourier uncertainty principles for bandlimited
functions. By means of radial symmetrization mechanisms, we show that such
problems can all be shifted to dimension one. We provide precise asymptotics in
the general case and, in some particular situations, we are able to identify
the sharp constants and characterize the extremizers. The sharp constant study
is actually a consequence of a more general result we prove in the setup of de
Branges spaces of entire functions, addressing the operator given by
multiplication by $z^k$, $k \in \mathbb{N}$. Applications to sharp higher order
Poincar\'{e} inequalities and other related extremal problems are discussed.
