Heisenberg inequalities for Jacobi transforms

The classical Heisenberg uncertainty principle states that for f∈L2(R)f∈L2(R),∫Rx2|f(x)|2dx⋅∫Rξ2|fˆ(ξ)|2dξ⩾14‖f‖4.In this paper, we obtain this inequality and variants of it for the Jacobi transform. It implies an analogue for the Jacobi–Dunkl transform. The proof is based on ultracontractive properties of the semigroups generated by the Jacobi differential operator and on the estimate on the heat kernel.