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 = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(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?
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-1-8,
author = {Richard Penney and Roman Urban},
title = {Unbounded harmonic functions on homogeneous manifolds of negative curvature},
journal = {Colloquium Mathematicae},
volume = {91},
year = {2002},
pages = {99-121},
zbl = {0983.22008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-1-8}
}
Richard Penney; Roman Urban. Unbounded harmonic functions on homogeneous manifolds of negative curvature. Colloquium Mathematicae, Tome 91 (2002) pp. 99-121. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-1-8/