Fourier inequalities in Morrey and Campanato spaces

We study norm inequalities for the Fourier transform, namely,(0.1)‖fˆ‖Xp,qλ≲‖f‖Y, where X is either a Morrey or Campanato space and Y is an appropriate function space. In the case of the Morrey space we sharpen the estimate ‖fˆ‖Mp,qλ≲‖f‖Ls′,q, s≥2, 1s=1p−λn. We also show that (0.1) does not hold when both X and Y are Morrey spaces. If X is a Campanato space, we prove that (0.1) holds for Y being the truncated Lebesgue space.