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 $$1<q<p<\infty $$ , 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 $$2<q<p$$ and $$1<q<p<2$$ but not in the remaining case, $$1<q<2<p$$ . Here, counterexamples are given to show that these simple conditions are no longer sufficient when $$1<q<2<p$$ . Also, various additional conditions are presented, any of which will restore sufficiency in that case.
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