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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?
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 -