Unbounded harmonic functions on homogeneous manifolds of negative curvature

Richard Penney; Roman Urban

Colloquium Mathematicae (2002)

  • Volume: 91, Issue: 1, page 99-121
  • ISSN: 0010-1354

Abstract

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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?

How to cite

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Richard Penney, and Roman Urban. "Unbounded harmonic functions on homogeneous manifolds of negative curvature." Colloquium Mathematicae 91.1 (2002): 99-121. <http://eudml.org/doc/283895>.

@article{RichardPenney2002,
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 = 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? },
author = {Richard Penney, Roman Urban},
journal = {Colloquium Mathematicae},
keywords = {unbounded harmonic function; asymptotic expansion; Poisson integral},
language = {eng},
number = {1},
pages = {99-121},
title = {Unbounded harmonic functions on homogeneous manifolds of negative curvature},
url = {http://eudml.org/doc/283895},
volume = {91},
year = {2002},
}

TY - JOUR
AU - Richard Penney
AU - Roman Urban
TI - Unbounded harmonic functions on homogeneous manifolds of negative curvature
JO - Colloquium Mathematicae
PY - 2002
VL - 91
IS - 1
SP - 99
EP - 121
AB - 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?
LA - eng
KW - unbounded harmonic function; asymptotic expansion; Poisson integral
UR - http://eudml.org/doc/283895
ER -

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