The Fourier transform in weighted rearrangement invariant spaces

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