An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type (Q2114122)

The authors study an extension problem for the Laplace-Beltrami operator on symmetric spaces of the noncompact type. The solution of such an extension problem, which can be written by means of a Poisson kernel, gives an expression for the fractional Laplace-Beltrami operator. The authors prove asymptotic estimates for the Poisson kernel which are then used to deduce Hardy-type inequalities for the fractional powers of the Laplace-Beltrami operator. The authors also deduce mapping properties for the Poisson operator that go beyond the Euclidean ones, as a consequence of the Kunze-Stein phenomenon. Finally, analogs of the Poincaré-Sobolev inequality for the fractional Laplace-Beltrami operator are proved.