We consider the function u whose Fourier transform is the positive part of the Fourier transform of a function / on R". If « < 2 and / satisfies simple regularity conditions (in particular if / is in the Schwartz space Sf(R"), then u lies in L'(R n ) . If n > 3, then simple counterexamples exist; for example, if/(x) = |x| 2 exp(-|x| 2 ), then u does not lie in L l (H").