Classification of Nonnegativeg-Harmonic Functions in Half-Spaces

In this paper we present a short proof of the following classification Theorem forg-harmonic functions in half-spaces. Assume thatuis a nonnegative solution to Delta(g)u= 0 in {x(n)> 0} that continuously vanishes on the flat boundary {x(n)= 0}. Then, modulo normalization,u(x) =x(n)in {x(n)>= 0}. Our proof depends on a recent quantitative version of the Hopf-Olei & x306;nik Lemma proven by the authors in Braga and Moreira (Adv. Math.334, 184-242,2018). Moreover, in this paper, we show how to adapt the proofs in the literature to extend Carleson Estimate, Boundary Harnack Inequality and Schwartz Reflection Principle to the context of nonnegativeg-harmonic functions. These results are also ingredients for the proof of the main result.