Uniqueness for the n-dimensional half space Dirichlet problem

In R(n), we prove uniqueness for the Dirichlet problem in the half space x(n) > 0, with continuous data, under the growth condition u = o(\x\sec(gamma) theta) as \x\ --> infinity (x(n) = \x\ cos theta, gamma is an element of R. Under the natural integral condition for convergence of the Poisson integral with Dirichlet data, the Poisson integral will satisfy this growth condition with gamma = n - 1. A Phragmen-Lindelof principle is established under this same growth condition. We also consider the Dirichlet problem with data of higher order growth, including polynomial growth. In this case, if u = o(\x\(N+1) secY(gamma) theta) (gamma is an element of R, N greater than or equal to 1), we prove solutions are unique up to the addition of a harmonic polynomial of degree N that vanishes when x(n) = 0.