Stein-Weiss inequality on non-compact symmetric spaces

TL;DR

This paper characterizes conditions for the Stein-Weiss inequality on non-compact symmetric spaces, extending classical inequalities and their admissible parameters beyond Euclidean limits.

Contribution

It provides a comprehensive characterization of Stein-Weiss inequality conditions on symmetric spaces, expanding the range of admissible parameters for related inequalities.

Findings

Conditions for Stein-Weiss inequality validity are fully characterized.

Weighted inequalities like Heisenberg and Gagliardo-Nirenberg are established in this setting.

Admissible index sets are larger than in Euclidean space.

Abstract

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.