  The Fourier coefficient map is considered as an operator from a weighted
Lorentz space on the circle to a weighted Lorentz sequence space. For a large
range of Lorentz indices, necessary and sufficient conditions on the weights
are given for the map to be bounded. In addition, new direct analogues are
given for known weighted Lorentz space inequalities for the Fourier transform.
Applications are given that involve Fourier coefficients of functions in LlogL
and more general Lorentz-Zygmund spaces.
