A generalization of Miyachi’s theorem

Abstract

The classical Hardy theorem on R, which asserts f and the Fourier transform of f cannot both be very small, was generalized by Miyachi in terms of L¹+L^∞ and log⁺-functions. In this paper we generalize Miyachi’s theorem for R^d and then for other generalized Fourier transforms such as the Chébli-Trimèche and the Dunkl transforms.