Shifted Pitt and uncertainty inequalities on Riemannian symmetric spaces of noncompact type

Our primary objective is to study Pitt-type inequalities on Riemannian symmetric spaces $\mathbb{X}$ of noncompact type, as well as within the framework of Jacobi analysis. Inspired by the spectral gap of the Laplacian on $\mathbb{X}$, we introduce the notion of a \textit{shifted} Pitt's inequality as a natural and intrinsic analogue tailored to symmetric spaces, capturing key aspects of the underlying non-Euclidean geometry. In the rank one case (in particular, for hyperbolic spaces), we show that the sufficient condition for the \textit{shifted} Pitt's inequality matches the necessary condition in the range $p \leq q \leq p'$, yielding a sharp characterization of admissible polynomial weights with non-negative exponents. In the Jacobi setting, we modify the transform so that the associated measure exhibits polynomial volume growth. This modification enables us to fully characterize the class of polynomial weights with non-negative exponents for which Pitt-type inequalities hold for the modified Jacobi transforms. As applications of the \textit{shifted} Pitt's inequalities, we derive $L^2$-type Heisenberg-Pauli-Weyl uncertainty inequalities and further establish generalized $L^p$ versions. Moreover, the geometric structure of symmetric spaces allows us to formulate a broader version of the uncertainty inequalities previously obtained by Ciatti-Cowling-Ricci in the setting of stratified Lie groups.