Fourier inequalities in Morrey and Campanato spaces

Abstract

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 $\Vert \hat{f}{\Vert }_{M_{p,q}^{\lambda }}\lesssim \Vert f{\Vert }_{L_{s^{\prime },q}},$ $s\ge 2,$ $\frac{1}{s}=\frac{1}{p}-\frac{\lambda }{n}.$ 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.