Unbounded harmonic functions on homogeneous manifolds of negative curvature
Volume 91 / 2002
Colloquium Mathematicum 91 (2002), 99-121
MSC: 22E25, 43A85, 53C30, 31B25.
DOI: 10.4064/cm91-1-8
Abstract
We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group $N$ and $A={\mathbb R}^+.$ We prove that if $F$ is harmonic and satisfies some growth condition then $F$ has an asymptotic expansion as $a\to 0$ with coefficients from ${\cal D}^\prime (N).$ Then we single out a set of at most two of these coefficients which determine $F.$
Then using asymptotic expansions we are able to prove some theorems answering partially the following question.
Is a given harmonic function the Poisson integral of “something" from the boundary $N$?
