The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality
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We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is connected to an old result of Mahler concerning the the measure of a convex body and its geometric polar being essentially reciprocal. An additional assumption, that the weights must belong to a reverse Hölder class, is used to formulate the sufficient condition.
KeywordsFourier transform Weights Hausdorff-Young Reverse Hölder Polar Mahler volume
Mathematics Subject ClassificationPrimary 42B10 Secondary 42A38
The first part of this work was completed while the author was taking a sabbatical at the University of Western Ontario, hosted by Professor Gord Sinnamon. The author is thankful for the hospitality and helpful insights. The author is also in debt to João Pedro Ramos who pointed out an error in a preprint.
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