The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality

Abstract

1 where E is any bounded, measurable set containing the origin, and F is any translation of its geometric polar. In fact, the set E may be translated as well, and the sets E and F may even be interchanged. Further, we discuss that this class of weights is essentially the same if E and F are replaced with a rectanguloid and its polar, or an ellipsoid and its polar. Previously in [5], we examined this inequality where E and F were cubes of reciprocal measure. The current condition is a generalization because having reciprocal measure and being polar are essentially the same for cubes.Sufficiency. We assume a weaker version of the necessary condition plus a reverse Hölder condition to prove weighted restricted weak-type estimates for the Fourier transform. This is accomplished by comparing either the measure u or v to Lebesgue measure and then applying the Hausdorff-Young inequality. The result actually implies restricted weak-type for a range of p and q; we are then able to interpolate and obtain the (strong-type) weighted Fourier inequality.