  We calculate the norm of the Fourier operator from $L^p(X)$ to $L^q(\hat{X})$
when $X$ is an infinite locally compact abelian group that is, furthermore,
compact or discrete. This subsumes the sharp Hausdorff-Young inequality on such
groups. In particular, we identify the region in $(p,q)$-space where the norm
is infinite, generalizing a result of Fournier, and setting up a contrast with
the case of finite abelian groups, where the norm was determined by Gilbert and
Rzeszotnik. As an application, uncertainty principles on such groups expressed
in terms of R\'enyi entropies are discussed.
