  The method of using rearrangements to give sufficient conditions for Fourier
inequalities between weighted Lebesgue spaces is revisited. New results in the
case q < p are established and a comparison between two known sufficient
conditions is completed. In addition, examples are given to show that a simple
weight condition that is sufficient for the weighted Fourier inequality in the
cases 2 < q < p and 1 < q < p < 2 is no longer sufficient in the case 1 < q < 2
< p, contrary to statements in Theorems 1 and 4 of, "Weighted Fourier
inequalities: new proofs and generalizations", J. Fourier Anal. Appl. 9 (2003),
1-37. Several alternatives are given for strengthening the simple weight
condition to ensure sufficiency in that case.
