The classical Hardy theorem on $\rr$ was generalized by Miyachi and Bonami, Demange, and Jaming. In this paper we show that Miyachi's theorem and Bonami-Demange-Jaming's one can be reformulated for the Jacobi transform in terms of the heat kernel.
"Generalized Hardy's theorem for the Jacobi transform." Hiroshima Math. J. 36 (3) 331 - 337, November 2006. https://doi.org/10.32917/hmj/1171377076