The author generalizes the theorems of G. H. Hardy [J. Lond. Math. Soc. 8, 227–231 (1933; Zbl 0007.30403)] and of A. Miyachi [A generalization of a theorem of Hardy. Harmonic analysis seminar (Japan, Izunagaoka) (1997)] for the Jacobi-Dunkl transform \(\mathcal F\). Let \(E_t\) denote the heat kernel associated with the Dunkl operator. If \(| f(x)| \leq ME_a(x)\) and \(| {\mathcal F}f(\lambda)| \leq Me^{-a\lambda^2}\), then \(f\) is a constant multiple of \(E_a\). Let \(L^p_{\alpha,\beta}\) denote the \(L^p\) space on \(\mathbb R\) with respect to \(\Delta_{\alpha,\beta}(x)dx\). Then, by replacing \(L^1+L^\infty\) and the Fourier transform in the classical Miyachi theorem by \(L_{\alpha,\beta}^1+L_{\alpha,\beta}^\infty\) and \(\mathcal F\) respectively, he obtains an analogous theorem for the Dunkl-Jacobi case.
In the estimate of \({\mathcal F}(\lambda)\) in 3.2, the case that \(E_a^{-1}f\) belongs to \(L^1_{\alpha,\beta}\) is missing.