  We study inequalities of the form \begin{equation*}
  \rho ( \lvert \hat{f} \rvert) \leq C \sigma(f) < \infty, \end{equation*} with
$f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on
$\mathbb{R}^n$ and \begin{equation*}
  \hat{f}(\xi) := \int_{\mathbb{R}^n} f(x) \, e^{- 2 \pi i \xi \cdot x} dx, \ \
\ \xi \in \mathbb{R}^n. \end{equation*} The functionals $\rho$ and $\sigma$ are
so-called rearrangement-invariant (r.i.) norms on $M_{+}(\mathbb{R}^n)$, the
nonnegative measurable functions on $\mathbb{R}^n$.
  Results first proved in the general context of r.i. spaces are then both
specialized and expanded on in the special cases of Orlicz spaces and of
Lorentz Gamma spaces.
