Fourier inequalities in Morrey and Campanato spaces

Abstract

We study norm inequalities for the Fourier transform, namely, ||(f) over cap||(lambda)(Xp,q) less than or similar to ||f||Y, (0,1) where Xis either a Morrey or Campanato space and Y is an appropriate function space. In the case of the Morrey space we sharpen the estimate ||(f) over cap ||M-p,q(lambda) less than or similar to ||f||L-s',L-q, s >= 2, 1/s= 1/p-lambda/n. We also show that (0.1) does not hold when both X and Y are Morrey spaces.