It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on
R
n
\mathbb R^n
, modified by a weight, then the weight is bounded above and below and the space is equivalent to
L
2
L^2
. Also, if it is bounded from
L
p
L^p
to
L
q
L^q
, each modified by the same weight, then the weight is bounded above and below and
1
≤
p
=
q
′
≤
2
1\le p=q’\le 2
. Applications prove the non-boundedness on these spaces of an operator related to the Schrödinger equation.