On Weighted Fourier InequalitieS –– Some New Scales of Equivalent Conditions

Abstract

For Lebesgue spaces on Rⁿ, we study two-weight p → q-inequalities for Fourier transform. Some sufficient conditions on weights for such inequalities are known for special ranges of parameters p and q. In the same ranges of parameters we show, that in every case each of those conditions can be replaced by infinitely many conditions, even by continuous scales of conditions. We also derive some new such characterizations concerning the Fourier transform in weighted Lorentz spaces.