Let $\Delta$ be the Laplace-Beltrami operator on a non-compact symmetric space of any rank, and denote the bottom of its $L^2$-spectrum as $-|\rho|^{ 2}$. In this paper, we provide a comprehensive characterization of both the sufficient and necessary conditions ensuring the validity of the Stein-Weiss inequality for the entire family of operators $\lbrace{ (-\Delta+b)^{ -\frac{ \sigma}{ 2}}}\rbrace_{ \sigma\ge0,\,b\ge-|\rho|^{ 2}}$. As an application, some weighted functional inequalities, such as Heisenberg's uncertainty principle, Gagliardo-Nirenberg's interpolation inequality, Pitt's inequality, etc., become available in this context. In particular, their sets of admissible indices are larger than those in the Euclidean setting.
Categories: math.AP