An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

TL;DR

This paper investigates an extension problem for the Laplace-Beltrami operator on noncompact symmetric spaces, establishing Hardy inequalities for fractional powers, analyzing the extension operator, and proving Poincaré-Sobolev inequalities.

Contribution

It introduces a new extension problem framework and derives Hardy and Poincaré-Sobolev inequalities for fractional Laplace-Beltrami operators on these spaces.

Findings

Established Hardy inequalities for fractional Laplace-Beltrami operators.

Analyzed the mapping properties of the extension operator.

Proved Poincaré-Sobolev inequalities on symmetric spaces.

Abstract

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincar\'e-Sobolev inequalities on these spaces.