A note on Hausdorff-Young inequalities in function spaces

The classical Hausdorff-Young inequalities for the Fourier transform acting between appropriate $L_p$ spaces are cornerstones of Fourier analysis. Here we extend it to weighted spaces of Besov or Sobolev type where the weight has the form $w(x)=(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{\alpha \mathrm{/}2}$. This note is not a paper or draft but a sketchy complement to some earlier results where we dealt with mapping properties of the Fourier transform.