A generalization of Miyachi's theorem

Abstract

The classical Hardy theorem on $\mathit{R}$, which asserts $f$ and the Fourier transform of $f$ cannot both be very small, was generalized by Miyachi in terms of $L^1+L^{\infty }$ and ${log}^{+}$-functions. In this paper we generalize Miyachi's theorem for ${\mathit{R}}^d$ and then for other generalized Fourier transforms such as the Chébli-Trimèche and the Dunkl transforms.