Stein-Weiss inequality on non-compact symmetric spaces

Let $\mathrm{\Delta }$ be the Laplace-Beltrami operator on a non-compact symmetric space of any rank, and denote the bottom of its $L^2$-spectrum as $-\mathrm{\mid }\rho {\mathrm{\mid }}^2$. 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 $\{(-\mathrm{\Delta }+b{)}^{-\frac{\sigma }{2}}{\}}_{\sigma \ge 0,b\ge -\mathrm{\mid }\rho {\mathrm{\mid }}^2}$. 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.