  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.
