Bounded $\lambda$-harmonic functions in domains of $\mathbb{H}^n$ with asymptotic boundary with fractional dimension

TL;DR

This paper investigates the existence of bounded $\lambda$-harmonic functions in unbounded hyperbolic domains, linking their existence to the fractional dimension of the domain's asymptotic boundary.

Contribution

It establishes conditions based on the Hausdorff measure of the asymptotic boundary that determine the existence or nonexistence of bounded $\lambda$-harmonic functions in hyperbolic space domains.

Findings

Nonexistence of bounded $\lambda$-harmonic functions when boundary measure is zero.

Existence of bounded $\lambda$-harmonic functions for boundaries with fractional dimension greater than $(n-1)/2$.

Comparison and maximum principles for these domains.

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.