Uniqueness theorems for weighted harmonic functions in the upper half-plane

Abstract We consider a class of weighted harmonic functions in the open upper half-plane known as α -harmonic functions. Of particular interest is uniqueness problem for such subject to vanishing Dirichlet boundary value on real line and an appropriate condition at infinity. find that non-classical case ( ≠ 0) allows considerably more relaxed infinity compared classical = usual half-plane. The reason behind this dichotomy different geometry zero sets certain polynomials naturally derived from binomial series. These findings shed new light theory functions, which we provide sharp results under conditions along geodesics or rays emanating origin.