Fourier inequalities in Morrey and Campanato spaces

TL;DR

This paper investigates Fourier transform inequalities within Morrey and Campanato spaces, providing sharper estimates for Morrey spaces and establishing conditions under which these inequalities hold or fail.

Contribution

It refines Fourier inequalities for Morrey spaces and identifies when such inequalities are valid or invalid for these and Campanato spaces.

Findings

Sharpened the Fourier inequality estimate for Morrey spaces.

Proved the non-existence of the inequality when both spaces are Morrey.

Established the validity of the inequality for Campanato spaces with truncated Lebesgue spaces.

Abstract

We study norm inequalities for the Fourier transform, namely, \begin{equation}\label{introduction} \|\widehat f\|_{X_{p,q}^\lambda} \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 $ \|\widehat f\|_{M_{p,q}^\lambda} \lesssim \|f\|_{L_{s',q}},$ $ s\geq 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.