Abstract: This talk is on the mapping properties of the Fourier transform between Banach function spaces. These are generalizations of Hausdorff-Young and Pitt's inequalities.

We provide several relations between weight functions, that guarantee the boundedness of the Fourier series coefficients, viewed as a map between weighted Lorentz spaces. As a useful machinery, we briefly introduce the quasi concave functions and generalize a number of known inequalities. Finally, we apply our results to Fourier inequalities in weighted Lebesgue spaces and Lorentz-Zygmund spaces.