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Weighted Fourier Inequalities via Rearrangements

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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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Acknowledgements

Supported by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Department of Mathematics, University of Western Ontario, London, ON, Canada

    Javad Rastegari & Gord Sinnamon

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  1. Javad Rastegari
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  2. Gord Sinnamon
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Correspondence to Gord Sinnamon.

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Communicated by Hans G. Feichtinger.

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Rastegari, J., Sinnamon, G. Weighted Fourier Inequalities via Rearrangements. J Fourier Anal Appl 24, 1225–1248 (2018). https://doi.org/10.1007/s00041-017-9565-3

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  • DOI: https://doi.org/10.1007/s00041-017-9565-3

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