Fourier inequalities in Morrey and Campanato spaces

We study norm inequalities for the Fourier transform, namely, \begin{equation}\label{introduction} |\widehat f|{X{p,q}^λ} \lesssim |f|_{Y}, \end{equation} 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 $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{M_{p,q}^{\lambda }}\lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{L_{s^{\mathrm{\prime }},q}},$ $s\ge 2,$ $\frac{1}{s}=\frac{1}{p}-\frac{\lambda }{n}\mathrm{.}$ We also show that \eqref{introduction} does not hold when both $X$ and $Y$ are Morrey spaces. If $X$ is a Campanato space, we prove that \eqref{introduction} holds for $Y$ being the truncated Lebesgue space.