Bounded -Harmonic Functions in Domains of with Asymptotic Boundary with Fractional Dimension

The existence and nonexistence of.-harmonic functions in unbounded domains of Hn are investigated. We prove that if the ( n -1)/ 2 Hausdorff measure of the asymptotic boundary of a domain is zero, then there is no bounded.-harmonic function of for.. [0,.1( Hn)], where.1( Hn) = ( n -1) 2/ 4. For these domains, we have comparison principle and some maximum principle. Conversely, for any s > ( n-1)/ 2, we prove the existence of domains with asymptotic boundary of dimension s for which there are bounded.1-harmonic functions that decay exponentially at infinity.