| Abstract: | The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited, a comparison between two known sufficient conditions is completed, and the method is extended to provide sufficient conditions for a new range of indices. When (1, simple conditions on weights ensure that the Fourier transform will map a weighted (L^p) space into a weighted (L^q) space. These are established in Theorems 1 and 4 of Benedetto and Heinig (J Fourier Anal Appl 9(1):1–37, 2003). The proofs apply when (2 and (1 but not in the remaining case, (1. Here, counterexamples are given to show that these simple conditions are no longer sufficient when (1. Also, various additional conditions are presented, any of which will restore sufficiency in that case. |
