A MEAN-VALUE THEOREM FOR SOME EIGENFUNCTIONS OF THE LAPLACE-BELTRAMI OPERATOR ON THE UPPER-HALF SPACE

In this paper we study a mean-value property for solutions of the eigenvalue equation of the Laplace-Beltrami operator Delta(lb)h = -(n - 1)h with respect to the volume and the surface integrals on the Poincare upper-half space R-+(n+1) = {(x(0), ... ,x(n)) is an element of Rn+1 : x(n) > 0} with the Riemannian metric ds(2) = dx(0)(2)+dx(1)(2)+ ... +dx(n)(2)/x(n)(2) .