Bounded $\lambda$-Harmonic Functions in Domains of ${\mathbb {H}}^n$ with Asymptotic Boundary with Fractional Dimension

Abstract

The existence and nonexistence of $\lambda$-harmonic functions in unbounded domains of ${{\mathbb {H}}}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $\Omega$ is zero, then there is no bounded $\lambda$-harmonic function of $\Omega$ for $\lambda \in [0,\lambda _1({\mathbb {H}}^n)]$, where $\lambda _1({\mathbb {H}}^n)=(n-1)^2/4$. 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 $\lambda _1$-harmonic functions that decay exponentially at infinity.